What is a right triangle?

A right triangle (or right-angled triangle) is a triangle in which one interior angle is exactly 90 degrees. The side opposite the right angle is the longest side and is called the hypotenuse. The other two sides are called legs (or catheti). Because the three angles must sum to 180 degrees, the two acute angles always add up to exactly 90 degrees.

How does the calculator find the hypotenuse?

It uses the Pythagorean theorem: c = sqrt(a^2 + b^2), where a and b are the two legs. For example, legs 3 and 4 give c = sqrt(9 + 16) = sqrt(25) = 5. This is the classic 3-4-5 right triangle, the simplest integer-sided example.

How are the angles found from two side lengths?

If both legs are known, angle A is found with the inverse tangent (arctan) of a divided by b. If one leg and the hypotenuse are known, the inverse sine (arcsin) of the leg divided by the hypotenuse gives the opposite angle. The second acute angle is then simply 90 degrees minus the first, because the two acute angles are complementary.

SOH CAH TOA is a memory aid for the three basic trig ratios in a right triangle. SOH means sin = Opposite over Hypotenuse. CAH means cos = Adjacent over Hypotenuse. TOA means tan = Opposite over Adjacent. This calculator applies all six ratios (sin, cos, tan for each acute angle) and shows them in the trig ratio table.

Can I use this for non-right triangles?

No. This tool is designed specifically for right triangles. For a general triangle where no angle is known to be 90 degrees, use a general triangle calculator that applies the Law of Sines and Law of Cosines. However, any triangle can be split into two right triangles by dropping a perpendicular from one vertex to the opposite side.

What units does the calculator use?

The calculator is unit-agnostic. Enter lengths in any unit (millimetres, centimetres, metres, inches, feet) and all results are in the same unit. Angles are always in decimal degrees. Area is in the square of your length unit and perimeter is in the same length unit.

Right Triangle Calculator

A right triangle calculator that fully solves the triangle from any two known values — whether you have both legs, one leg and the hypotenuse, one leg and an acute angle, or the hypotenuse and an acute angle. Every solve gives you all three sides, both acute angles, area, perimeter, a live SVG diagram, and a complete trig ratio table for both angles.

How it works

A right triangle has exactly one 90-degree angle (labelled C in this tool). That leaves three sides (legs a, b and hypotenuse c) and two unknown angles (A and B). Because the angles must sum to 180 degrees, A + B = 90, so knowing one acute angle instantly gives the other.

Depending on which two values you provide, the calculator applies one of four solution paths:

Both legs known (a, b)

c = sqrt(a^2 + b^2) — Pythagorean theorem

A = arctan(a / b), B = 90 - A

Leg and hypotenuse known (a, c)

b = sqrt(c^2 - a^2)

A = arcsin(a / c), B = 90 - A

Leg and opposite angle known (a, A)

B = 90 - A

c = a / sin(A), b = a / tan(A)

Hypotenuse and angle known (c, A)

B = 90 - A

a = c * sin(A), b = c * cos(A)

Once all three sides are known, area and perimeter follow directly:

Area = a * b / 2, Perimeter = a + b + c

The trig ratio table is then derived from the solved sides, giving sin, cos and tan for both angle A and angle B — matching the SOH CAH TOA rules taught at GCSE, A-level and in most university engineering courses.

Worked example

Suppose a ramp rises 1.2 m vertically over a horizontal run of 5 m. What angle does it make with the ground, how long is the ramp surface, and what is the enclosed area?

We know: leg a = 1.2 (vertical), leg b = 5 (horizontal).
Hypotenuse: c = sqrt(1.2^2 + 5^2) = sqrt(1.44 + 25) = sqrt(26.44) ≈ 5.142 m
Angle A (elevation angle) = arctan(1.2 / 5) = arctan(0.24) ≈ 13.50°
Angle B = 90 - 13.50 = 76.50°
Area = 1.2 * 5 / 2 = 3.0 m²

Now plug the same numbers into “Both legs” mode and the calculator confirms every figure in under a second, with all steps shown.

Known values	Leg a	Leg b	Hypotenuse c	Angle A	Angle B
a = 3, b = 4	3	4	5	36.87°	53.13°
a = 5, c = 13	5	12	13	22.62°	67.38°
a = 1, A = 45°	1	1	1.4142	45°	45°
c = 10, A = 30°	5	8.660	10	30°	60°

Formula reference

The six trigonometric ratios for angle A (where a is opposite, b is adjacent, c is hypotenuse):

sin A = a / c (SOH)
cos A = b / c (CAH)
tan A = a / b (TOA)
csc A = c / a, sec A = c / b, cot A = b / a

The Pythagorean identity sin^2(A) + cos^2(A) = 1 always holds. The complementary angle identities mean sin(A) = cos(B) and cos(A) = sin(B), which the trig table illustrates directly.

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