Right Triangle Calculator

A right triangle has one angle that measures exactly 90°, and that single constraint unlocks a whole toolkit of formulas. Whether you're working through a geometry problem, checking your homework, or figuring out a real-world measurement, a right triangle calculator lets you punch in whatever values you already know and get the rest back instantly. This page walks you through every piece of the puzzle: sides, angles, area, perimeter, and the special triangles that show up constantly in math and engineering. You'll also find the key formulas explained so the numbers actually make sense, not just appear out of nowhere.

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How to Use the Right Triangle Calculator

Using the calculator is straightforward. You need at least two known values to solve a right triangle completely. That can be two sides, one side and one angle, or any combination as long as one of the angles is the right angle (which is already given).

Enter your known values into the appropriate fields. The calculator will figure out which formula applies and compute the remaining sides and angles automatically. Results typically include all three sides, both non-right angles, the area, and the perimeter.

A few things to keep in mind:

Angles can usually be entered in degrees or radians. Double-check which mode is selected before you calculate.
Side lengths must be positive numbers. The calculator doesn't care about units, but make sure you're consistent (all in inches, all in meters, etc.).
If you enter values that can't form a valid triangle, you'll get an error rather than a nonsense answer.

Solve a Right Triangle from Two Known Values

"Solving" a triangle means finding every unknown side and angle. For a right triangle, you already know one angle is 90°, so you really need just two more pieces of information to nail down the shape completely.

Here's what each combination gives you:

Two sides known: Use the Pythagorean theorem to find the third side, then use inverse trig to find the missing angles.
One side and one acute angle known: Use SOH CAH TOA to find the other sides, then subtract from 180° to get the remaining angle (since the three angles must sum to 180°).
Two angles known (plus the right angle): You can find the third angle, but you need at least one side length to determine the actual size of the triangle. Angles alone only tell you the shape, not the scale.

The right angle is always the largest angle and sits opposite the longest side, which is called the hypotenuse. Keeping that picture in your head helps you sanity-check answers as you go.

Find the Hypotenuse

The hypotenuse is the side directly across from the 90° angle. It's always the longest side of a right triangle, and it comes up in almost every right triangle calculation you'll ever do.

There are two main ways to find it depending on what you already know.

Pythagorean Theorem

The Pythagorean theorem is the classic approach when you know both legs of the triangle. The two shorter sides (legs) are usually labeled a and b, and the hypotenuse is c.

The formula is: a² + b² = c²

To find the hypotenuse, rearrange it: c = √(a² + b²)

So if one leg is 3 and the other is 4, the hypotenuse is √(9 + 16) = √25 = 5. That 3-4-5 combo is probably the most famous right triangle in existence, and it's a good one to have memorized.

Hypotenuse Formula

When you know one leg and one of the acute angles instead of both legs, the trig approach is faster. If you know angle A and the adjacent leg (b), the hypotenuse is:

c = b / cos(A)

If you know angle A and the opposite leg (a), use:

c = a / sin(A)

Both formulas come straight from the definitions of sine and cosine, so they're not separate rules to memorize. Once you understand SOH CAH TOA, these just fall out naturally.

Find Missing Sides and Angles

After the hypotenuse, you might need to find a leg or one of the acute angles. Trigonometric ratios handle both jobs cleanly, and they work any time you have a right triangle with at least one side and one acute angle known.

Sine, Cosine, and Tangent (SOH CAH TOA)

SOH CAH TOA is the mnemonic that ties three ratios to an angle in a right triangle. For a given acute angle A:

Sine (SOH): sin(A) = Opposite / Hypotenuse
Cosine (CAH): cos(A) = Adjacent / Hypotenuse
Tangent (TOA): tan(A) = Opposite / Adjacent

"Opposite" is the side directly across from angle A. "Adjacent" is the leg that touches angle A (but isn't the hypotenuse). Mixing those two up is the most common mistake, so take a second to label the triangle before you plug in numbers.

For example, if angle A is 35° and the hypotenuse is 10, the side opposite A is 10 × sin(35°) ≈ 5.74, and the adjacent side is 10 × cos(35°) ≈ 8.19.

Inverse Trigonometric Functions

Inverse trig functions do the opposite job: given a ratio of sides, they give you back the angle. On a calculator you'll see them written as sin⁻¹, cos⁻¹, and tan⁻¹ (also called arcsin, arccos, and arctan).

Say you know the opposite side is 6 and the hypotenuse is 10. The angle A is:

A = sin⁻¹(6 / 10) = sin⁻¹(0.6) ≈ 36.87°

Once you have one acute angle, the other is just 90° minus that value, since the three angles of any triangle add up to 180°. That's a nice shortcut that saves you a second calculation.

Right Triangle Area and Perimeter Calculator

Area and perimeter are straightforward once you have all three sides, but the formulas are simple enough to use any time.

Area of a right triangle uses the two legs as the base and height (they meet at the right angle, so they're already perpendicular):

Area = (1/2) × a × b

Perimeter is just the sum of all three sides:

Perimeter = a + b + c

If you only know one leg and the hypotenuse, find the missing leg first using the Pythagorean theorem, then plug into the perimeter formula. Same idea if you have an angle and one side: solve for the other sides with trig, then add them all up. The calculator handles that chain of steps automatically, but knowing the underlying formulas helps you catch any input mistakes.

Special Right Triangles

Two right triangles show up so often in geometry and trigonometry that they've earned their own names. Their side ratios are fixed, which means you can solve them without a calculator if you recognize the pattern. They also appear constantly in standardized tests, physics problems, and construction math.

30°–60°–90° Triangle

This triangle has angles of 30°, 60°, and 90°. Its sides always follow the ratio 1 : √3 : 2, where the shortest side is opposite the 30° angle and the hypotenuse is twice that shortest side.

Angle	Opposite Side	Ratio
30°	Short leg	1x
60°	Long leg	√3 x (≈ 1.732x)
90°	Hypotenuse	2x

So if the hypotenuse is 8, the short leg is 4 and the long leg is 4√3 ≈ 6.93. This triangle comes directly from cutting an equilateral triangle in half, which is a handy way to remember the ratios without memorizing them cold.

45°–45°–90° Triangle

This one is an isosceles right triangle: two equal legs and angles of 45°, 45°, and 90°. The side ratio is 1 : 1 : √2. The hypotenuse is always the leg length times √2.

Angle	Opposite Side	Ratio
45°	Leg a	1x
45°	Leg b	1x
90°	Hypotenuse	√2 x (≈ 1.414x)

If each leg is 5, the hypotenuse is 5√2 ≈ 7.07. This triangle is what you get when you cut a square diagonally, which explains why it pops up in architecture, tile work, and coordinate geometry so often.

Right Triangle Formulas

Here's a quick reference for every formula covered on this page, collected in one place.

Formula	Equation	Use When…
Pythagorean Theorem	a² + b² = c²	You know two sides, need the third
Hypotenuse (from angle + leg)	c = a / sin(A) or c = b / cos(A)	You know one angle and one leg
Missing leg	a = c × sin(A) or a = c × cos(A)	You know hypotenuse and an angle
Tangent ratio	tan(A) = a / b	You know both legs, need an angle
Inverse sine	A = sin⁻¹(a / c)	You know opposite leg and hypotenuse
Inverse cosine	A = cos⁻¹(b / c)	You know adjacent leg and hypotenuse
Inverse tangent	A = tan⁻¹(a / b)	You know both legs, want the angle
Area	(1/2) × a × b	Always (uses the two legs)
Perimeter	a + b + c	Always (sum of all sides)

Keep this table handy when you're working through problems. Once you've used each formula a few times, you'll stop needing to look them up.

Step-by-Step Right Triangle Examples

Walking through real examples is the fastest way to see how these formulas connect.

Example 1: Find the hypotenuse given two legs

Legs: a = 5, b = 12
Apply the Pythagorean theorem: c = √(5² + 12²) = √(25 + 144) = √169 = 13
The hypotenuse is 13.

Example 2: Find a missing leg and both angles given a leg and hypotenuse

Known: a = 7, c = 25
Find angle A: A = sin⁻¹(7 / 25) = sin⁻¹(0.28) ≈ 16.26°
Find angle B: B = 90° – 16.26° = 73.74°
Find leg b: b = √(25² – 7²) = √(625 – 49) = √576 = 24
Area = (1/2) × 7 × 24 = 84 square units. Perimeter = 7 + 24 + 25 = 56 units.

Example 3: Solve from one leg and one angle

Known: angle A = 40°, adjacent leg b = 9
Find hypotenuse: c = 9 / cos(40°) ≈ 9 / 0.766 ≈ 11.75
Find opposite leg: a = 9 × tan(40°) ≈ 9 × 0.839 ≈ 7.55
Find angle B: B = 90° – 40° = 50°

Notice how each example uses a different entry point but follows the same basic process: identify what you have, pick the right formula, solve, then verify the answer looks reasonable given the triangle's shape.