How to Solve a Right Triangle: Sides, Angles, Area and Perimeter

 

A right triangle has one angle of 90 degrees. Its two perpendicular sides are the legs, and the side opposite the right angle is the hypotenuse. To solve the triangle, identify which measurements you know before selecting a formula. Confusing a leg with the hypotenuse or using the wrong acute angle can produce an incorrect result even when the arithmetic is correct.

The Right Triangle Calculator accepts two legs, a leg and the hypotenuse, or a leg and its opposite acute angle. It returns the sides, both acute angles, area and perimeter. It does not solve every possible triangle or provide a structural design.

Label the sides and angles first

In the calculator's diagram, leg a is vertical on the left, leg b is the horizontal base, and hypotenuse c is the sloping side. Angle A is opposite leg a, while angle B is opposite leg b. The right angle lies where a and b meet.

The diagram is schematic rather than drawn to your measurements. Rotating or reflecting a triangle does not change its geometry, but you must keep the labels consistent when using the formulas. The hypotenuse is always the longest side of a valid right triangle.

Find the hypotenuse from two legs

The Pythagorean theorem states that a squared + b squared = c squared. Therefore c is the square root of the sum of the squared legs. For legs of 3 ft and 4 ft, c = square root of (9 + 16) = 5 ft.

The area is a x b / 2, giving 6 square feet. The perimeter is a + b + c, giving 12 feet. Notice the different units: area is measured in square feet, while sides and perimeter are measured in feet.

Find a missing leg from the hypotenuse

If leg a and hypotenuse c are known, rearrange the theorem to b = square root of (c squared - a squared). With a = 3 ft and c = 5 ft, b = square root of (25 - 9) = 4 ft.

The hypotenuse must be longer than the known leg. Equal values would make the other leg zero, creating a degenerate shape rather than a triangle. A shorter hypotenuse is incompatible with the chosen side labels, so the calculator rejects it.

Use an acute angle with the correct opposite side

When leg a and angle A are known, sine gives sin(A) = a / c, so c = a / sin(A). Tangent gives tan(A) = a / b, so b = a / tan(A). Enter angle A in degrees, strictly between zero and ninety.

For a = 3 ft and A = 30 degrees, c = 6 ft and b is approximately 5.196152 ft. The other acute angle is B = 90 - A = 60 degrees. If the angle you know is adjacent to leg a, it is B in this diagram; subtract it from ninety before entering A.

Check the angles and numerical precision

The two acute angles sum to ninety degrees. For the 3-4-5 triangle, angle A is approximately 36.869898 degrees and angle B is approximately 53.130102 degrees. Displayed values are rounded, so small rounding differences do not necessarily indicate a geometric inconsistency.

Very small angles and nearly equal leg/hypotenuse measurements can amplify measurement errors. The calculator has numerical input limits and rejects derived sides outside its supported range. Those limits do not establish practical construction tolerances or measurement accuracy.

Keep geometry separate from construction requirements

Use one length unit for both sides. Changing the input method or unit clears measurements to avoid silently reusing an angle as a side. For flat-shape areas, use the Area Calculator. For finished rise and going, use the Stair Calculator. A hypotenuse alone is not a stair-stringer cutting plan, load calculation or code check. Calculations run locally without saving or transmitting your entries.