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Right Triangle Calculator

Solve right triangles from any two known values. Computes all sides, angles, area, and perimeter using the Pythagorean theorem and basic trigonometry.

Back to Triangle Calculator

Solve Mode

Select the combination of known values

Details: 2 sides + included angle

Known Values

Enter two sides and the angle between them

Triangle Solver Tips

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Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Classic 3-4-5 Right Triangle

The simplest Pythagorean triple — legs 3 and 4, hypotenuse 5.

Key values: a = 3 · b = 4 · c = 5

Roof Rafter Calculation

A roof with a horizontal run of 5 m and vertical rise of 2.5 m — what is the rafter length?

Key values: run = 5 m · rise = 2.5 m · rafter ≈ 5.59 m

Documentation

Right Triangle Properties

A right triangle has one angle of exactly 90°. The side opposite the right angle is called the hypotenuse (always the longest side), and the other two sides are called legs.

The Pythagorean theorem governs right triangles:

a2+b2=c2a^2 + b^2 = c^2

where cc is the hypotenuse and a,ba, b are the legs. This calculator pre-sets the right angle at C = 90°, so you only need to provide two sides or one side and one acute angle.

Trigonometric Ratios (SOH-CAH-TOA)

The six trigonometric functions are defined using the sides of a right triangle relative to an acute angle θ\theta:

SOH

sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

CAH

cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

TOA

tan⁡θ=oppositeadjacent\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

The reciprocal ratios are cosecant (csc⁡=1/sin⁡\csc = 1/\sin), secant (sec⁡=1/cos⁡\sec = 1/\cos), and cotangent (cot⁡=1/tan⁡\cot = 1/\tan).

Special Right Triangles

Two right triangles have exact side ratios that appear constantly in geometry, trigonometry, and standardized tests:

45-45-90 Triangle

1:1:21 : 1 : \sqrt{2}

The isosceles right triangle. Both legs are equal, and the hypotenuse is 2\sqrt{2} times a leg. Formed by cutting a square along its diagonal.

30-60-90 Triangle

1:3:21 : \sqrt{3} : 2

Formed by bisecting an equilateral triangle. The short leg (opposite 30°) is half the hypotenuse. The long leg (opposite 60°) is 3\sqrt{3} times the short leg.

Solving Right Triangles

Given any two measurements (besides the right angle), you can find everything else:

  • Two legs known: Use Pythagorean theorem for hypotenuse, inverse trig for angles: θ=arctan⁡(a/b)\theta = \arctan(a/b)
  • Leg + hypotenuse: Second leg via b=c2−a2b = \sqrt{c^2 - a^2}, angles via inverse trig
  • One side + one acute angle: Use trig ratios to find the other sides, then third angle=90°−θ\text{third angle} = 90° - \theta

The area of a right triangle is particularly simple:

A=12×a×bA = \frac{1}{2} \times a \times b

where aa and bb are the legs (they form the base and height automatically).

Worked Examples

Ladder Against a Wall

A 13-foot ladder leans against a wall, with its base 5 feet from the wall. How high does it reach?

h=132−52=169−25=144=12 fth = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \text{ ft}

Angle with the ground: θ=arccos⁡(5/13)≈67.4°\theta = \arccos(5/13) \approx 67.4°.

Surveying a River Width

Standing 50 m from a river bank, you sight a tree on the far side at a 32° angle. The river width is:

w=50×tan⁡(32°)≈50×0.6249≈31.2 mw = 50 \times \tan(32°) \approx 50 \times 0.6249 \approx 31.2 \text{ m}

Frequently Asked Questions

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, a2+b2=c2a^2 + b^2 = c^2, where cc is the hypotenuse (the side opposite the right angle) and aa and bb are the two legs. It is the fundamental relationship governing all right triangles.

What is SOH-CAH-TOA?

SOH-CAH-TOA is a mnemonic for the three basic trigonometric ratios: sin⁡θ=opposite/hypotenuse\sin\theta = \text{opposite}/\text{hypotenuse} (SOH), cos⁡θ=adjacent/hypotenuse\cos\theta = \text{adjacent}/\text{hypotenuse} (CAH), and tan⁡θ=opposite/adjacent\tan\theta = \text{opposite}/\text{adjacent} (TOA). These ratios relate the sides of a right triangle to its acute angles.

What are the 30-60-90 and 45-45-90 special triangles?

These are right triangles with exact side ratios. A 45-45-90 triangle has sides in the ratio 1:1:21 : 1 : \sqrt{2} (formed by cutting a square diagonally). A 30-60-90 triangle has sides in the ratio 1:3:21 : \sqrt{3} : 2 (formed by bisecting an equilateral triangle). These ratios are used extensively in geometry and trigonometry.

How do I find a missing side of a right triangle?

If you know two legs, find the hypotenuse with c=a2+b2c = \sqrt{a^2 + b^2}. If you know the hypotenuse and one leg, find the other leg with b=c2−a2b = \sqrt{c^2 - a^2}. If you know one side and one acute angle, use trigonometric ratios (sine, cosine, or tangent) to find the missing sides.

What are Pythagorean triples?

Pythagorean triples are sets of three positive integers that satisfy a2+b2=c2a^2 + b^2 = c^2. Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a triple is also a triple, so (6, 8, 10) works because it is twice (3, 4, 5).

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