Right Triangle Calculator
Calculate missing right triangle sides, angles, area, perimeter, and hypotenuse using common side and angle combinations.
Enter what you know
Choose a calculation type, enter the known measurements, and the calculator will solve the missing parts of the right triangle.
a² + b² = c²
Area = (a × b) ÷ 2
Perimeter = a + b + c
sin(A) = opposite ÷ hypotenuse, cos(A) = adjacent ÷ hypotenuse, tan(A) = opposite ÷ adjacent
Right triangles are everywhere — in the slope of a staircase, the pitch of a roof, the angle of a ramp, the diagonal of a screen, and the path of a navigation bearing. Being able to quickly solve for any unknown side or angle is an essential skill for students, engineers, architects, carpenters, surveyors, and anyone who works with physical measurements in the real world. This free right triangle calculator gives you instant answers for any combination of known values: two sides, one side and one angle, or any other valid input. Enter what you know and the tool resolves the rest.
Below this calculator, you will find an in-depth guide covering everything from the fundamental trigonometric ratios (SOH-CAH-TOA) to special triangle rules, worked examples, angle-finding techniques, real-world use cases, and the most common mistakes students and professionals make when solving right triangles. This page is part of the broader math calculator collection available here, where you can find dozens of tools for geometry, algebra, statistics, and more.
Table of Contents
Jump directly to any section of this guide.
What Is a Right Triangle?
A right triangle is a triangle that contains exactly one interior angle measuring 90 degrees — called the right angle. Because the sum of all interior angles in any triangle always equals 180 degrees, the remaining two angles in a right triangle must be acute (each less than 90 degrees), and they must add up to exactly 90 degrees. This constraint makes right triangles uniquely predictable: knowing just one acute angle immediately tells you the other, and knowing any two sides immediately defines the shape completely.
Right triangles sit at the very foundation of trigonometry. The entire field of trigonometric functions — sine, cosine, tangent, and their inverses — was originally defined in terms of the ratios of sides within right triangles. From this starting point, mathematics extended those definitions to circles, waves, complex numbers, and eventually the underpinnings of modern physics and engineering. Every time you see a sine or cosine in a formula, the ghost of a right triangle is somewhere in its ancestry.
In everyday geometry, right triangles appear any time two lines meet perpendicularly — horizontal floors meeting vertical walls, ladder rungs meeting supports, or horizontal ground meeting a sloped ramp. They also appear as half of rectangles (any diagonal cuts a rectangle into two congruent right triangles) and half of equilateral triangles (any altitude of an equilateral triangle bisects it into two 30-60-90 right triangles). This ubiquity is exactly why the ability to solve a right triangle efficiently is so practically valuable.
One Right Angle
Exactly one 90° interior angle — always opposite the hypotenuse.
Angles Sum to 180°
The two acute angles always add up to 90° between them.
Hypotenuse Is Longest
The side opposite the right angle is always the longest of the three.
Parts of a Right Triangle: Sides and Angles
To solve a right triangle accurately, you need to correctly identify each of its components. Using the wrong label for a side — especially confusing a leg with the hypotenuse — is one of the most common sources of calculation errors. Here is a precise breakdown of every element.
The Three Sides
The Hypotenuse (c)
The hypotenuse is the longest side of any right triangle. It is always the side directly opposite the 90-degree angle. In formulas, it is almost universally labeled c. You can visually identify it because it never touches the corner where the right angle lives — it spans between the two acute angle vertices instead. The hypotenuse is always longer than either individual leg but shorter than the sum of both legs.
The Two Legs (a and b)
The legs are the two shorter sides that form the right angle. They are labeled a and b in the Pythagorean Theorem and in most textbooks. In trigonometry, these same sides are referred to as “opposite” and “adjacent” relative to a specific angle. Which leg counts as opposite or adjacent depends on which acute angle you are currently considering — these labels shift with perspective, which is why understanding the angle reference point is critical.
The Three Angles
A right triangle has three angles: the right angle (always exactly 90°), and two acute angles conventionally labeled A (or α) and B (or β). Because they must sum to 90°, knowing one acute angle immediately gives you the other: if angle A is 35°, then angle B must be 55°. In this calculator, angle A is typically measured at the corner where side a (the opposite leg) meets the hypotenuse, and angle B is measured at the corner where side b meets the hypotenuse.
Opposite, Adjacent, and Hypotenuse: Perspective Matters
When using trigonometric ratios, “opposite” and “adjacent” are defined relative to the angle you are working with — not relative to the triangle as a fixed object. If you are working from angle A: the opposite side is the leg that does not touch angle A (side a), the adjacent side is the leg that does touch angle A (side b), and the hypotenuse (c) is always the hypotenuse regardless of perspective. Shift your focus to angle B and the labels for sides a and b swap: what was opposite becomes adjacent and vice versa. The hypotenuse never changes.
Quick Memory Tip: Draw your right triangle and mark the right angle with a small square. Pick the angle you want to work with and label it. The side directly across from it (not touching it) is “opposite.” The side right next to it (touching it, but not the hypotenuse) is “adjacent.” The longest side that bridges the two acute angles is always the hypotenuse.
SOH-CAH-TOA: The Core of Right Triangle Trigonometry
SOH-CAH-TOA is the most widely taught mnemonic in trigonometry, and for good reason — it encodes all three primary trigonometric ratios in a single memorable phrase. Each syllable corresponds to a ratio of two sides relative to a given angle. Once you have this memorized, you can set up any right triangle equation from scratch.
SOH — sin(θ) = Opposite / Hypotenuse
CAH — cos(θ) = Adjacent / Hypotenuse
TOA — tan(θ) = Opposite / Adjacent
θ represents either of the two acute angles; the labels “opposite” and “adjacent” shift depending on which angle you choose.
Understanding Each Ratio
Sine (sin)
Sine compares the side across from the angle to the hypotenuse. It tells you what fraction of the hypotenuse’s length the opposite leg represents. Sine values always range between −1 and 1 for any angle. For acute angles in a right triangle, sine is always a positive value between 0 and 1.
Cosine (cos)
Cosine compares the side next to the angle (the adjacent leg) to the hypotenuse. It is essentially sine’s complement — cos(θ) = sin(90° − θ). Cosine is widely used in physics to find horizontal components of forces and velocities, and in navigation for bearing calculations.
Tangent (tan)
Tangent compares the two legs directly without involving the hypotenuse. It is extremely practical: the tangent of a slope angle equals the rise divided by the run, making it the go-to ratio for ramp design, roof pitch, and surveying grade calculations. Tangent can be any positive or negative number — it is unbounded.
The Reciprocal Ratios: Cosecant, Secant, and Cotangent
Beyond SOH-CAH-TOA, right triangle trigonometry also uses three reciprocal functions. The cosecant (csc) is the reciprocal of sine: csc(θ) = hyp / opp. The secant (sec) is the reciprocal of cosine: sec(θ) = hyp / adj. The cotangent (cot) is the reciprocal of tangent: cot(θ) = adj / opp. These appear less often in basic triangle-solving but become important in calculus, physics, and advanced trigonometric identities. If your calculator or course requires any of these, simply take the reciprocal of the corresponding primary function.
Working with coordinate geometry alongside trigonometry? The midpoint calculator on WalDev makes it easy to find center points between coordinate pairs, a common companion task when solving triangles on a coordinate plane.
All the Formulas You Need to Solve a Right Triangle
Solving a right triangle means finding every unknown side and angle from the information given. Depending on which values you already have, a different formula is the most direct path to the solution. The table below is a complete reference covering every scenario.
| What You Know | What You Want | Formula to Use |
|---|---|---|
| Both legs (a and b) | Hypotenuse (c) | c = √(a² + b²) |
| Hypotenuse (c) and one leg (a) | Other leg (b) | b = √(c² − a²) |
| Hypotenuse (c) and angle θ | Opposite side | opposite = c × sin(θ) |
| Hypotenuse (c) and angle θ | Adjacent side | adjacent = c × cos(θ) |
| Adjacent side and angle θ | Opposite side | opposite = adjacent × tan(θ) |
| Adjacent side and angle θ | Hypotenuse | hypotenuse = adjacent / cos(θ) |
| Opposite side and angle θ | Adjacent side | adjacent = opposite / tan(θ) |
| Opposite side and angle θ | Hypotenuse | hypotenuse = opposite / sin(θ) |
| Opposite and hypotenuse | Angle θ | θ = arcsin(opposite / hypotenuse) |
| Adjacent and hypotenuse | Angle θ | θ = arccos(adjacent / hypotenuse) |
| Opposite and adjacent | Angle θ | θ = arctan(opposite / adjacent) |
| One acute angle (A) | Other acute angle (B) | B = 90° − A |
The Pythagorean Theorem: The Foundational Equation
At the core of every right triangle calculation is the Pythagorean Theorem: a² + b² = c². This equation states that the sum of the squares of both legs always equals the square of the hypotenuse. It works in all three rearrangements: solve for the hypotenuse by adding the squared legs and taking the square root; solve for a missing leg by subtracting the squared known leg from the squared hypotenuse and taking the square root. If you need a deep dive into this specific formula with worked examples and history, see the dedicated Pythagorean Theorem calculator guide.
a² + b² = c²
c = √(a² + b²) · a = √(c² − b²) · b = √(c² − a²)
Where c is the hypotenuse and a, b are the two legs. All values must be positive real numbers.
How to Use the Right Triangle Calculator
The right triangle calculator accepts any valid combination of inputs — two sides, one side and one angle, or other valid pairs — and instantly computes all remaining unknowns. Here is the standard workflow for getting accurate results.
Before entering anything, determine which values you have. Do you know two sides? One side and one angle? Label your triangle clearly: mark the right angle (90°), identify which side is the hypotenuse (opposite the right angle), and label the two legs relative to your working angle as “opposite” and “adjacent.” Misidentifying your sides is the most common source of errors.
Input your known side lengths and/or angles into the appropriate fields. Sides should be entered in the same unit (all centimeters, all inches, all meters — never mix). Angles should be entered in degrees unless the tool specifies radians. The calculator needs a minimum of two pieces of information to produce a full solution.
Confirm you are working in degrees for most practical problems. Radians are used primarily in calculus, physics, and advanced mathematics. If you enter 1.047 radians when you meant 60 degrees, the calculator will produce a completely different (and correct-for-radians but wrong-for-your-intent) answer. When in doubt, use degrees.
The calculator will display all remaining sides and angles. Review them for reasonableness: the hypotenuse should be longer than both legs, both acute angles should be positive and less than 90°, and all sides should be positive. A quick sanity check — squaring the two legs and confirming they sum to the hypotenuse squared — costs only seconds and confirms your answer.
Note the units you used for input — the calculator output will be in those same units. Round to an appropriate number of decimal places for your application. For a woodworking cut, rounding to the nearest millimeter is fine. For a structural engineering calculation, more decimal places may be warranted depending on the tolerance requirements of the project.
Finding a Missing Side of a Right Triangle
The method you use to find a missing side depends on exactly which values you already have. There are two primary pathways: the Pythagorean Theorem (when you have two sides and want the third) or a trigonometric ratio (when you have one side and one angle). Both are equally rigorous — the Pythagorean approach is often faster when two sides are known, while the trigonometric approach is the only option when only one side and one angle are given.
Path 1: Two Known Sides → Use the Pythagorean Theorem
When you know both legs, the hypotenuse is c = √(a² + b²). When you know the hypotenuse and one leg, the missing leg is √(c² − a²) or √(c² − b²) respectively. This approach produces exact results whenever the inputs are integers or simple fractions, and precise decimal results for all other inputs.
📘 Worked Example A — Finding the Hypotenuse from Two Legs
Given: leg a = 9 m, leg b = 12 m. Find: hypotenuse c.
Step 1: Square both legs — a² = 81, b² = 144.
Step 2: Add the squares — 81 + 144 = 225.
Step 3: Take the square root — c = √225 = 15 m.
Answer: The hypotenuse is exactly 15 meters. (This is a 3-4-5 triple scaled by 3.)
📘 Worked Example B — Finding a Leg from the Hypotenuse and One Leg
Given: hypotenuse c = 25 ft, leg a = 7 ft. Find: leg b.
Step 1: Square both known values — c² = 625, a² = 49.
Step 2: Subtract — c² − a² = 625 − 49 = 576.
Step 3: Take the square root — b = √576 = 24 ft.
Answer: The missing leg is 24 feet. (7-24-25 is a Pythagorean triple.)
Path 2: One Known Side + One Known Angle → Use Trigonometry
When only one side length is available along with one acute angle, the trigonometric ratios are your tools. Identify the relationship between the known side, the unknown side, and the given angle, then match that relationship to sine, cosine, or tangent.
📘 Worked Example C — Finding a Side Using Sine
Given: hypotenuse c = 20 cm, angle A = 37°. Find: the side opposite angle A.
Setup: sin(A) = opposite / hypotenuse, so opposite = hypotenuse × sin(A).
Calculate: opposite = 20 × sin(37°) = 20 × 0.6018 ≈ 12.04 cm.
Answer: The opposite side is approximately 12.04 cm.
Bonus: Adjacent side = 20 × cos(37°) = 20 × 0.7986 ≈ 15.97 cm.
📘 Worked Example D — Finding a Side Using Tangent
Given: adjacent leg = 10 m, angle A = 52°. Find: the opposite leg.
Setup: tan(A) = opposite / adjacent, so opposite = adjacent × tan(A).
Calculate: opposite = 10 × tan(52°) = 10 × 1.2799 ≈ 12.80 m.
Answer: The opposite leg is approximately 12.80 meters.
Prisms and pyramids with triangular bases require both triangle-solving and volume skills. The volume calculator handles three-dimensional shapes and pairs naturally with right triangle calculations for real-world geometry tasks.
Finding a Missing Angle in a Right Triangle
Finding an unknown angle in a right triangle requires either the simple complement rule (if you already know one acute angle) or an inverse trigonometric function (if you only know side lengths). Understanding both approaches gives you the flexibility to handle any given combination of inputs.
Method 1: Use the Complement Rule
Since the three angles of any triangle sum to 180°, and one angle is already fixed at 90°, the remaining two acute angles must sum to 90°. This means: if you know angle A, then angle B = 90° − A. No calculator or formula is needed beyond simple subtraction. This is the fastest possible angle-finding method and should always be your first instinct when one acute angle is already known.
Method 2: Inverse Trigonometric Functions
When only side lengths are available, inverse trig functions work backward from a ratio to produce the angle. Your calculator’s arcsin (or sin⁻¹), arccos (or cos⁻¹), and arctan (or tan⁻¹) buttons perform this reversal. Choose the function based on which two sides you know:
Use arcsin when you know the opposite side and the hypotenuse: θ = arcsin(opposite / hypotenuse). For example, if the opposite side is 6 and the hypotenuse is 10, θ = arcsin(0.6) ≈ 36.87°.
Use arccos when you know the adjacent side and the hypotenuse: θ = arccos(adjacent / hypotenuse). For example, if the adjacent side is 8 and the hypotenuse is 10, θ = arccos(0.8) ≈ 36.87°. (Note: this gives the angle at the other end from the arcsin example.)
Use arctan when you know both legs: θ = arctan(opposite / adjacent). This is the most common scenario in surveying and construction. For example, if the opposite leg is 3 and the adjacent leg is 4, θ = arctan(0.75) ≈ 36.87°.
📘 Worked Example — Finding Both Angles from Two Sides
Given: leg a = 5 (opposite to angle A), leg b = 12 (adjacent to angle A). Find: angles A and B.
Find angle A: A = arctan(5/12) = arctan(0.4167) ≈ 22.62°.
Find angle B: B = 90° − 22.62° = 67.38°. (Or verify: arctan(12/5) = arctan(2.4) ≈ 67.38°. ✓)
Find hypotenuse: c = √(5² + 12²) = √(25 + 144) = √169 = 13.
Complete solution: a = 5, b = 12, c = 13, A ≈ 22.62°, B ≈ 67.38°, right angle = 90°.
⚠ Degree vs. Radian Mode: Always verify your calculator is in degree mode before using inverse trig functions for practical problems. If your calculator is in radian mode, arctan(1) will return 0.785 (which is π/4 radians) rather than 45°. This is a silent error — the calculator will not warn you, and the resulting angle will be completely wrong for your application.
Special Right Triangles: 30-60-90 and 45-45-90
Two specific right triangle types appear so frequently in mathematics, architecture, and engineering that their side ratios are worth memorizing. These “special right triangles” let you solve for all sides without a calculator whenever you encounter their specific angle combinations.
The 45-45-90 Triangle (Isosceles Right Triangle)
A 45-45-90 triangle has two equal acute angles of 45° each, which forces the two legs to be equal in length. If each leg has length n, then the hypotenuse is n√2 (approximately 1.414n). Conversely, if the hypotenuse is h, each leg is h/√2 = h√2/2 (approximately 0.707h).
Side Ratio
The legs are always equal; the hypotenuse is the leg multiplied by √2 ≈ 1.41421.
Where It Appears
This triangle appears whenever you draw a diagonal across a square — the diagonal creates two congruent 45-45-90 triangles. It also appears in roof trusses designed at 45°, in drafting and CAD work, and in the geometry of certain crystal structures. Any square tile cut diagonally produces this triangle.
The 30-60-90 Triangle
A 30-60-90 triangle has angles of 30°, 60°, and 90°. Its side ratio is 1 : √3 : 2 — the shortest side (opposite 30°) has length n, the longer leg (opposite 60°) has length n√3 (≈ 1.732n), and the hypotenuse (opposite 90°) has length 2n. This is the triangle you get when you cut an equilateral triangle exactly in half with an altitude.
Side Ratio
If the shortest side is n, the other leg is n × 1.732, and the hypotenuse is n × 2.
Where It Appears
This triangle appears in equilateral triangle geometry, hexagonal grids, certain roof pitches, electrical engineering (three-phase power), and in the unit circle at 30° and 60° reference angles. The ratio of 1:2 between the shortest leg and the hypotenuse is its signature feature — the hypotenuse is always exactly twice the shortest leg.
Common Values: Sine, Cosine, and Tangent at Key Angles
| Angle | sin(θ) | cos(θ) | tan(θ) |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 = 0.500 | √3/2 ≈ 0.866 | 1/√3 ≈ 0.577 |
| 45° | √2/2 ≈ 0.707 | √2/2 ≈ 0.707 | 1.000 |
| 60° | √3/2 ≈ 0.866 | 1/2 = 0.500 | √3 ≈ 1.732 |
| 90° | 1 | 0 | undefined |
These exact values are worth memorizing if you work with trigonometry regularly. Notice the symmetry: sin(30°) = cos(60°) and cos(30°) = sin(60°), reflecting the co-function identity. Tan(45°) = 1 because the legs of the isosceles right triangle are equal. Tan(90°) is undefined because the adjacent side would be zero, creating a division by zero.
Worked Examples: Solving Right Triangles Step by Step
The most reliable way to build confidence with right triangle calculations is to work through realistic problems from start to finish. The following examples cover every common input scenario and demonstrate the full solution process.
Example 1: Roof Pitch Design (Two Legs Known)
📐 Construction Scenario
Scenario: A roofer needs to cut a rafter for a house. The horizontal run (adjacent leg) is 14 feet and the vertical rise (opposite leg) is 6 feet. Find the rafter length (hypotenuse) and the pitch angle.
Step 1 — Hypotenuse: c = √(14² + 6²) = √(196 + 36) = √232 ≈ 15.23 ft.
Step 2 — Pitch angle (A at the eave): A = arctan(rise/run) = arctan(6/14) = arctan(0.4286) ≈ 23.20°.
Step 3 — Ridge angle (B at the ridge): B = 90° − 23.20° = 66.80°.
Complete solution: Rafter length ≈ 15.23 ft, eave angle ≈ 23.20°, ridge angle ≈ 66.80°.
Example 2: Navigation Bearing (Angle and One Side Known)
🧭 Navigation Scenario
Scenario: A ship travels 50 nautical miles on a bearing of 40° east of north. How far east and how far north has it traveled?
Setup: The hypotenuse is the total distance (50 nm), the northward distance is adjacent to the 40° angle, and the eastward distance is opposite.
Northward (adjacent): = 50 × cos(40°) = 50 × 0.7660 ≈ 38.30 nm north.
Eastward (opposite): = 50 × sin(40°) = 50 × 0.6428 ≈ 32.14 nm east.
Answer: Approximately 38.30 nautical miles north and 32.14 nautical miles east.
Example 3: Ladder Safety (Hypotenuse and One Angle Known)
🪜 Safety Scenario
Scenario: Safety guidelines recommend that a ladder be placed at a 75° angle from the ground. If the ladder is 6 meters long, how high up the wall does it reach, and how far from the wall should the base be placed?
Height (opposite): = 6 × sin(75°) = 6 × 0.9659 ≈ 5.80 m up the wall.
Base distance (adjacent): = 6 × cos(75°) = 6 × 0.2588 ≈ 1.55 m from the wall.
Answer: The ladder reaches approximately 5.80 m up the wall with its base placed 1.55 m out from the wall.
Example 4: Screen Diagonal (Hypotenuse from Two Sides)
📺 Consumer Electronics Scenario
Scenario: A monitor measures 27 inches wide and 15.19 inches tall. What is its diagonal measurement (the advertised screen size)?
Diagonal: c = √(27² + 15.19²) = √(729 + 230.74) = √959.74 ≈ 30.98 inches.
Answer: This is a 31-inch monitor (rounded to the nearest inch as is conventional). TV and monitor sizes are always measured as the diagonal of the screen, a direct application of the right triangle formula.
Many engineering and construction problems involve both geometric calculations and percentage-based ratios. The percentage decrease calculator and the percent error calculator are both available at WalDev for those companion calculations.
Real-World Applications of Right Triangle Calculations
Right triangle mathematics is not confined to textbooks. It underpins the design and operation of virtually every built environment and many natural systems. Understanding where these calculations arise in practice deepens your intuition for which formula to reach for and why the results matter.
Construction and Architecture
Every roof pitch involves a right triangle: the horizontal run, the vertical rise, and the sloping rafter form the three sides. Carpenters use the tangent ratio to calculate pitch angles and the Pythagorean Theorem to determine rafter lengths. Staircase design similarly requires right triangles: the tread depth is the adjacent leg, the riser height is the opposite leg, and the stringer (the angled board supporting the stairs) is the hypotenuse. According to Khan Academy’s trigonometry curriculum, solving right triangles is one of the most immediately applicable geometric skills taught at the secondary school level, precisely because of its pervasive presence in physical construction.
Ensuring walls are truly square — perpendicular to each other and to the floor — also uses right triangle geometry. The classic “3-4-5 check” used by builders worldwide involves measuring 3 feet along one wall, 4 feet along the adjacent wall, and confirming the diagonal between those two endpoints is exactly 5 feet. If it is, the corner is a perfect right angle. This is the Pythagorean Theorem applied without a calculator directly on a job site.
Surveying and Civil Engineering
Surveyors measure land boundaries, elevation changes, and site layouts using instruments that report horizontal angles and distances. Converting a sloped distance to a true horizontal distance requires the cosine of the slope angle. Computing the elevation change between two points from a known slope distance and angle uses the sine ratio. Building a road up a hill with a specific grade percentage — say, a 5% grade — means the rise-to-run ratio is 0.05, directly expressed as the tangent of the incline angle (approximately 2.86° for a 5% grade).
Navigation and GPS
Pilots and sailors decompose their velocity vectors into horizontal and vertical (or north-south and east-west) components using exactly the same cosine and sine decomposition shown in Example 2 above. A plane flying at a heading of 30° east of north at 500 knots is moving 500 × cos(30°) ≈ 433 knots northward and 500 × sin(30°) = 250 knots eastward simultaneously. GPS systems constantly perform these decompositions to translate distance and bearing data into coordinate changes.
Physics and Engineering
Force vectors, electric field components, and velocity vectors in physics are routinely decomposed using the same right triangle formulas. A force of 100 newtons applied at 30° above horizontal has a horizontal component of 100 × cos(30°) ≈ 86.6 N and a vertical component of 100 × sin(30°) = 50 N. Structural engineers analyzing diagonal bracing members in a frame, electrical engineers resolving AC circuit phasors, and mechanical engineers computing load components on inclined planes all rely on this same trigonometric decomposition daily.
Computer Graphics and Game Development
Every time two objects in a game move toward each other, the engine calculates a distance using the Pythagorean formula applied to coordinate differences. Collision detection, projectile trajectory calculation, camera angle adjustment, and shadow ray casting all use right triangle geometry continuously at runtime. The distance formula d = √((x₂ − x₁)² + (y₂ − y₁)²) is simply the Pythagorean Theorem expressed in Cartesian coordinates, calculated millions of times per second in modern 3D engines.
📐 Roof Pitch Calculation
A 6/12 roof pitch has a rise of 6 inches and a run of 12 inches. The angle = arctan(6/12) ≈ 26.57° and the rafter factor (hypotenuse per foot of run) = √(6² + 12²)/12 = √180/12 = √(180)/12 ≈ 1.118. Multiply any horizontal run in feet by 1.118 to get the exact rafter length.
⚡ Electrical Phase Calculations
In three-phase AC power, the phase voltage equals the line voltage divided by √3 (≈ 1.732). This ratio comes directly from the 30-60-90 triangle geometry of the phasor diagram. A 208 V three-phase system delivers 208 / √3 ≈ 120 V per phase — the standard US household voltage, derived from right triangle trigonometry.
Area and Perimeter of a Right Triangle
Beyond side and angle solving, two other measurements frequently needed for right triangles are area and perimeter. Both are straightforward to compute once all three sides are known.
Area
The area of any triangle is (1/2) × base × height, where base and height must be perpendicular. In a right triangle, the two legs are already perpendicular to each other, making the formula particularly clean:
Area = (1/2) × a × b
Where a and b are the two legs (not the hypotenuse).
For the 3-4-5 triangle: Area = (1/2) × 3 × 4 = 6 square units. For the 5-12-13 triangle: Area = (1/2) × 5 × 12 = 30 square units.
Perimeter
The perimeter is simply the sum of all three side lengths. Find any unknown sides first using the Pythagorean Theorem or trigonometric ratios, then add all three:
Perimeter = a + b + c
Where c is the hypotenuse.
For the 3-4-5 triangle: Perimeter = 3 + 4 + 5 = 12 units. For a 45-45-90 triangle with legs of 10: Perimeter = 10 + 10 + 10√2 ≈ 10 + 10 + 14.14 ≈ 34.14 units.
Altitude to the Hypotenuse
A right triangle has a special property regarding the altitude drawn from the right angle vertex to the hypotenuse. If the altitude has length h and divides the hypotenuse into segments of length p and q, then: h² = p × q (the altitude is the geometric mean of the two segments), a² = p × c (each leg squared equals the product of the full hypotenuse and the adjacent segment), and the area can also be computed as (1/2) × h × c. This geometric mean relationship is tested in standardized exams and appears in problems about similar triangles formed by the altitude.
📘 Example — Altitude to the Hypotenuse
Given: A right triangle with legs a = 6 and b = 8. Find the altitude to the hypotenuse.
Step 1: Find hypotenuse: c = √(36 + 64) = √100 = 10.
Step 2: Find altitude using area: Area = (1/2) × 6 × 8 = 24. Also: Area = (1/2) × h × c = (1/2) × h × 10.
Step 3: Solve: 24 = 5h → h = 24/5 = 4.8.
Answer: The altitude from the right angle to the hypotenuse is 4.8 units.
Common Mistakes to Avoid When Solving Right Triangles
Even students and professionals who understand the formulas conceptually make procedural errors that produce wrong answers. Recognizing the most frequent mistakes in advance helps you develop habits that keep your calculations reliable.
Misidentifying the hypotenuse. The hypotenuse is always the side opposite the right angle — not simply the longest-looking side from a poorly drawn diagram. Always identify the right angle first, then locate its opposite side. The right angle is conventionally marked with a small square in geometric diagrams, and the hypotenuse is the side that does not touch that square.
Confusing opposite and adjacent. The labels “opposite” and “adjacent” shift depending on which angle you are working with. If you label angle A but accidentally use the sides that are opposite and adjacent to angle B, your sine, cosine, and tangent ratios will produce values for the wrong angle. Always re-label “opposite” and “adjacent” at the start of each new calculation for each new reference angle.
Forgetting calculator degree mode. A calculator in radian mode will silently produce wrong answers for degree-based problems. sin(30) in degree mode = 0.5; sin(30) in radian mode ≈ −0.988. These results are vastly different and there is no automatic warning. Before every trigonometric calculation, check your mode display (usually shows DEG or RAD in a corner of the screen).
Applying the Pythagorean Theorem to non-right triangles. a² + b² = c² applies only to right triangles. If your triangle has no 90° angle, using this formula will produce an incorrect result. For non-right triangles, use the Law of Cosines (c² = a² + b² − 2ab·cos(C)) instead. Always confirm you have a right triangle before applying the Pythagorean formula.
Entering a leg value larger than the hypotenuse. In a valid right triangle, the hypotenuse is strictly the longest side. If you attempt to compute a leg using b = √(c² − a²) and the value of a is larger than c, the expression under the square root becomes negative — there is no real solution. This always means one of your input values is wrong.
Mixing units. Entering one side in centimeters and another in inches (or meters and feet) will produce a nonsensical result. All side lengths must be in the same unit before entering them. If your measurements come from different sources using different units, convert them to a common unit first.
Rounding intermediate values prematurely. If you round a calculated intermediate value (like a calculated leg) before using it in a second calculation, rounding error compounds. Carry full precision through all intermediate steps and round only the final answer to the precision required by your application.
Skipping the reasonableness check. After computing a result, spend five seconds verifying it makes sense. The hypotenuse should be longer than both legs. Both acute angles should be positive and less than 90°. If you computed a 30-60-90 triangle, the hypotenuse should be exactly twice the shortest leg. A quick check can catch errors that would otherwise propagate into a larger project.
For data-driven work that involves geometric and statistical analysis together, the standard deviation calculator on WalDev offers fast variance and distribution calculations without requiring any spreadsheet software.
Frequently Asked Questions
Click any question to expand the full answer.
What is a right triangle?
A right triangle is a triangle that contains exactly one interior angle measuring 90 degrees — called the right angle. The side directly opposite the right angle is the hypotenuse, which is always the longest side. The other two sides are called legs, and they form the right angle at one of the triangle’s corners. Because the sum of all interior angles in any triangle equals 180°, the two remaining angles in a right triangle must both be acute (less than 90°) and must together sum to exactly 90°. Right triangles are the foundational shape for the entire field of trigonometry.
What does SOH-CAH-TOA mean?
SOH-CAH-TOA is a mnemonic device that encodes the three primary trigonometric ratios. SOH stands for Sine = Opposite over Hypotenuse. CAH stands for Cosine = Adjacent over Hypotenuse. TOA stands for Tangent = Opposite over Adjacent. The words “opposite,” “adjacent,” and “hypotenuse” describe the sides of a right triangle relative to a specific reference angle. Once you identify which angle you are working with, label its opposite side (the one across from it), its adjacent side (the one next to it that is not the hypotenuse), and the hypotenuse (always the side opposite the right angle), and you can set up any trig equation directly from the mnemonic.
How do I find a missing side when I know two sides?
Use the Pythagorean Theorem: a² + b² = c², where c is the hypotenuse and a and b are the legs. If you know both legs, the hypotenuse is √(a² + b²). If you know the hypotenuse and one leg, the missing leg is √(c² − a²) or √(c² − b²). For example, if the legs are 8 and 15, the hypotenuse is √(64 + 225) = √289 = 17. If the hypotenuse is 13 and one leg is 5, the other leg is √(169 − 25) = √144 = 12.
How do I find a missing side when I know one side and one angle?
Use the appropriate trigonometric ratio from SOH-CAH-TOA. First, identify the relationship between your known side, your unknown side, and the given angle. If the known side is the hypotenuse and you want the opposite side, use: opposite = hypotenuse × sin(θ). If the known side is the hypotenuse and you want the adjacent side, use: adjacent = hypotenuse × cos(θ). If you know the adjacent side and want the opposite, use: opposite = adjacent × tan(θ). If the known side is the opposite and you want the hypotenuse, use: hypotenuse = opposite / sin(θ). Rearrange any SOH-CAH-TOA equation algebraically to isolate the unknown side.
How do I find a missing angle when I only know two sides?
Use an inverse trigonometric function. If you know the opposite side and the hypotenuse, use arcsin(opposite / hypotenuse). If you know the adjacent side and the hypotenuse, use arccos(adjacent / hypotenuse). If you know both legs, use arctan(opposite / adjacent) — this is the most common scenario. For example, if the opposite leg is 7 and the adjacent leg is 24, the angle is arctan(7/24) = arctan(0.2917) ≈ 16.26°. Then the other acute angle is simply 90° − 16.26° = 73.74°. Always ensure your calculator is in degree mode when performing these calculations for practical applications.
What are the rules for a 30-60-90 triangle?
In a 30-60-90 triangle, the three sides are always in the fixed ratio 1 : √3 : 2. The shortest side (opposite the 30° angle) has some length n. The medium side (opposite the 60° angle) has length n√3 ≈ 1.732n. The hypotenuse (opposite the 90° angle) has length 2n — always exactly double the shortest side. To use these rules: if you know the shortest side, multiply by √3 to get the medium side and multiply by 2 to get the hypotenuse. If you know the hypotenuse, divide by 2 to get the shortest side. If you know the medium side, divide by √3 to get the shortest side. This triangle appears whenever an equilateral triangle is bisected by an altitude.
What are the rules for a 45-45-90 triangle?
A 45-45-90 triangle has two equal 45° acute angles, which forces both legs to be equal in length. If each leg is n, the hypotenuse is n√2 (approximately 1.41421n). If the hypotenuse is h, each leg is h/√2 = h√2/2 (approximately 0.70711h). This triangle is sometimes called an isosceles right triangle. It appears every time you draw a diagonal across a square — the diagonal divides the square into two congruent 45-45-90 triangles. At 45°, sine and cosine have the same value (√2/2 ≈ 0.707), and tangent equals exactly 1.
Can I solve a right triangle with only one known value?
Not completely. A right triangle has six elements (three sides and three angles), and one is already fixed at 90°. To find the remaining five unknowns, you need at least two additional pieces of information. Knowing only one side without any angle gives you infinitely many similar triangles — they all have the same angles but different scales. Knowing only one angle (beyond the 90°) gives you the shape but not the size. You always need at least two independent pieces of information: either two sides, or one side and one non-right angle, to fully determine a right triangle.
What is the area of a right triangle?
The area of a right triangle is (1/2) × a × b, where a and b are the two legs (not the hypotenuse). Because the legs are perpendicular, one naturally serves as the base and the other as the perpendicular height, making the formula identical to the standard (1/2) × base × height. For example, a right triangle with legs of 5 cm and 12 cm has an area of (1/2) × 5 × 12 = 30 square centimeters. If you know the hypotenuse and both legs, you can also compute the area using (1/2) × hypotenuse × altitude, where the altitude is the perpendicular drawn from the right angle vertex to the hypotenuse.
What is the difference between sine, cosine, and tangent in a right triangle?
All three are trigonometric ratios that relate a specific angle to two sides of a right triangle. Sine (sin θ) = opposite / hypotenuse — it compares the opposite leg to the longest side. Cosine (cos θ) = adjacent / hypotenuse — it compares the adjacent leg to the longest side. Tangent (tan θ) = opposite / adjacent — it compares the two legs directly. While sine and cosine always produce values between 0 and 1 for acute angles, tangent can produce any positive value and grows toward infinity as the angle approaches 90°. The choice of which ratio to use depends entirely on which two sides you know or need.
How does a right triangle differ from obtuse or acute triangles?
A right triangle has exactly one 90-degree angle. An acute triangle has all three angles less than 90°. An obtuse triangle has exactly one angle greater than 90°. The key mathematical consequence of these differences is formula applicability: the Pythagorean Theorem and the standard sine/cosine/tangent ratios as expressed in SOH-CAH-TOA apply only to right triangles. For non-right triangles, the Law of Sines (a/sin A = b/sin B = c/sin C) and the Law of Cosines (c² = a² + b² − 2ab·cos C) are the appropriate tools. A right triangle can also be tested from its sides: it satisfies a² + b² = c² exactly. If a² + b² > c², the triangle is acute; if a² + b² < c², it is obtuse.
What are inverse trigonometric functions and when should I use them?
Inverse trigonometric functions work backward: instead of taking an angle and returning a ratio, they take a ratio and return the corresponding angle. Arcsin (sin⁻¹) reverses sine, arccos (cos⁻¹) reverses cosine, and arctan (tan⁻¹) reverses tangent. Use arcsin when you know the opposite side and hypotenuse and need the angle: θ = arcsin(opp/hyp). Use arccos when you know the adjacent side and hypotenuse: θ = arccos(adj/hyp). Use arctan when you know both legs: θ = arctan(opp/adj). These functions always return the acute angle (between 0° and 90°) for positive side length ratios, which is the appropriate range for right triangle work. On scientific calculators, they are accessed via the “2nd” or “Shift” key followed by the regular sin, cos, or tan key.
Why is the tangent of 90° undefined?
The tangent of an angle is defined as the ratio of the opposite side to the adjacent side (tan θ = opposite / adjacent). As an angle in a right triangle approaches 90°, the adjacent side shrinks toward zero while the opposite side approaches the full length of the hypotenuse. Dividing by a number that approaches zero causes the ratio to grow without bound — it tends toward positive infinity. At exactly 90°, the adjacent side would have zero length, making the denominator exactly zero, which produces a mathematically undefined expression. This is why tan(90°) is undefined, why the tangent function has a vertical asymptote at 90° (and at all odd multiples of 90°), and why right triangles by definition cannot have a right angle and an angle of 90° at the same vertex simultaneously.
How is the right triangle calculator useful for construction projects?
Right triangle calculations appear at virtually every stage of construction work. Calculating rafter lengths requires solving for the hypotenuse given the run and rise. Confirming square corners uses the 3-4-5 (or any Pythagorean triple) method. Designing ramps, stairs, and sloped surfaces requires finding the angle from the rise-to-run ratio using arctan. Setting out foundation corners, cutting bracing angles, and determining the length of diagonal runs in plumbing or electrical conduit all reduce to right triangle problems. The calculator eliminates the need for manual computation with trigonometric tables or the iterative approximation methods used before digital tools, delivering precise results in seconds.
What are co-function identities and how do they apply to right triangles?
Co-function identities state that the trigonometric function of an angle equals the co-function of its complementary angle (90° minus that angle). For a right triangle with acute angles A and B where A + B = 90°: sin(A) = cos(B), cos(A) = sin(B), and tan(A) = cot(B). This makes intuitive geometric sense: the side that is “opposite” angle A is simultaneously “adjacent” to angle B, since the two angles share the same triangle. Co-function identities are useful for simplifying expressions, converting between perspectives within the same triangle, and understanding why sine and cosine have identical values at 45° — since 45° is its own complement.
Can this calculator be used for 3D geometry problems involving right triangles?
The right triangle calculator handles two-dimensional right triangles directly. For 3D geometry, most problems are solved by identifying the right triangles embedded within the three-dimensional figure — for example, the space diagonal of a rectangular box can be found by applying the Pythagorean Theorem twice: first to find the face diagonal, then to treat that diagonal and the height as the two legs of a second right triangle whose hypotenuse is the space diagonal. The formula for the three-dimensional distance is d = √(x² + y² + z²), which is obtained by applying the Pythagorean Theorem in exactly this two-step fashion. So while the calculator itself handles the 2D case, it can be applied twice sequentially to solve many practical 3D distance problems.
How do I verify that my right triangle solution is correct?
There are three quick verification methods. First, check the Pythagorean Theorem: square the two legs and confirm they sum to the square of the hypotenuse (allow for small rounding differences). Second, check that the two acute angles sum to exactly 90°. Third, re-derive one of the known inputs from your calculated values: if you were given a side length, recalculate it from your computed angle and another side, and confirm you get the original value back within rounding tolerance. If any of these checks fail, at least one input or calculation step contains an error. A complete, consistent solution will always pass all three checks simultaneously.
Summary: Everything You Need to Solve Any Right Triangle
A right triangle is fully defined by its one fixed 90-degree angle plus any two additional pieces of information — whether those are two side lengths, one side and one acute angle, or in the special cases of 30-60-90 and 45-45-90 triangles, even just one side. The Pythagorean Theorem (a² + b² = c²) handles all pure side-finding tasks. SOH-CAH-TOA connects sides and angles through the three primary trigonometric ratios. Inverse trig functions (arcsin, arccos, arctan) reverse those ratios to recover angles from side-length ratios.
The 30-60-90 triangle (side ratio 1 : √3 : 2) and the 45-45-90 triangle (side ratio 1 : 1 : √2) are the two special cases worth memorizing — they appear constantly in standard exam problems, architectural geometry, and electrical engineering. For all other triangles, the general formulas from the table in Section 4 above provide the complete toolkit. Real-world applications span construction, navigation, physics, computer graphics, and everything in between.
Use the free calculator at the top of this page for instant, accurate results with any valid input combination. And explore the full library of free tools at WalDev to find the right calculator for every step of your mathematical, scientific, or professional workflow.
SOH-CAH-TOA
Sin = Opp/Hyp · Cos = Adj/Hyp · Tan = Opp/Adj — the complete trig toolkit for right triangles.
Special Triangles
30-60-90 (1 : √3 : 2) and 45-45-90 (1 : 1 : √2) — solve instantly without a calculator.
Universal Application
Construction, navigation, physics, GPS, game engines — right triangles are everywhere.
