Right Triangle Calculator

What this calculator does

The Right Triangle Calculator takes any two sides of a right triangle and finds the third side. It also reports the two acute angles, the area, the perimeter, and the altitude drawn to the hypotenuse.

A right triangle has one 90-degree angle. The side opposite that angle is the hypotenuse, conventionally called c. The other two sides are the legs, called a and b. Right triangles are central to trigonometry, construction layout, navigation, and computer graphics because their side ratios define the sine, cosine, and tangent functions.

How to use this calculator

1. Enter exactly two of the three sides: side a, side b, and hypotenuse c.
2. Leave the side you want to calculate blank.
3. Click Calculate.
4. Read the missing side, angles, area, perimeter, and altitude.

The default example enters side a as 3 and hypotenuse c as 5, leaving side b blank. The result is a 3-4-5 triangle: side b is 4, angle α is about 36.87 degrees, angle β is about 53.13 degrees, the area is 6 square units, and the perimeter is 12 units.

Right triangle formulas

Hypotenuse: c = √(a² + b²)
Leg:        a = √(c² − b²)
Leg:        b = √(c² − a²)
Angle α:    α = arcsin(a / c)
Angle β:    β = 90° − α
Area:       A = ½ab
Perimeter:  P = a + b + c
Altitude to c: h = ab / c

Angle α is opposite side a, and angle β is opposite side b. The two acute angles always add up to 90 degrees.

What the results mean

The highlighted result is the side that was calculated. The supporting rows show all three sides, both acute angles, the area, the perimeter, and the altitude to the hypotenuse. The altitude is the perpendicular distance from the right-angle vertex to the hypotenuse.

The hypotenuse must be longer than either leg. If the entered hypotenuse is equal to or shorter than a leg, the sides cannot form a right triangle, and the calculator reports an error instead of an imaginary result.

Assumptions and limitations

• The triangle contains a 90-degree angle, and side c is the hypotenuse.
• Exactly two sides must be provided. Entering one side or all three produces an error message.
• Every side must be greater than zero.
• All sides must use the same unit. No unit conversion is performed.
• Angles are reported in degrees and rounded for display.
• The calculator does not accept an angle as the second known value. It solves from side lengths only.
• Results are for an ideal planar triangle, not a triangle drawn on a curved surface.

Frequently asked questions

Which side is the hypotenuse?

The hypotenuse is the longest side and the side opposite the right angle. In this calculator it is always labeled c. The legs are a and b. If you are measuring a physical object, identify the right angle first, then locate the side across from it.

Why must I leave one side blank?

The calculator uses the blank field to decide which value to solve for. If you enter all three sides, the inputs may contradict each other, and there is nothing left to calculate. If you enter only one side, there are infinitely many right triangles with that side length.

How are the angles calculated?

The sine of an acute angle equals the opposite side divided by the hypotenuse. The calculator uses the inverse sine to find angle α from a and c, then subtracts that value from 90 degrees to find angle β. Because the triangle is right-angled, the third angle is known to be 90 degrees.

Reference tables

Angle formulas

The two acute angles sum to 90°. The altitude to the hypotenuse is h = ab ÷ c, and the area is ½ab.

Source

Reference: Calculator.net — Right Triangle Calculator.

This page is for general educational and estimation purposes. Verify critical dimensions before cutting, surveying, or building.