A right triangle has one 90° angle, two legs that form it and a hypotenuse across from it. Because the right angle is already known, two more facts, at least one of them a length, fix the whole triangle. This calculator accepts six combinations, from two legs to the area plus one leg, and returns every side, both acute angles, the area, the perimeter, the height to the hypotenuse and the radii of the inscribed and circumscribed circles.
How to use the right triangle calculator
- In I know the, choose what you have: two legs, one leg and the hypotenuse, leg a and the angle opposite it (A), leg a and the angle next to it (B), the hypotenuse and one acute angle (A), or the area and leg a.
- Fill in the fields that appear. Acute angles must be greater than 0° and less than 90°.
- Choose the Units for lengths and the Angle unit (degrees by default, or radians).
- Read the tape. Angle C is always the right angle, and Show the work names the ratio used for each side.
Labeling matters: leg a is opposite angle A, leg b is opposite angle B, and c is the hypotenuse. Angle B sits where leg a meets the hypotenuse.
Right triangle formulas (SOH-CAH-TOA)
The three trig ratios link an acute angle to two sides:
SOH-CAH-TOA is the reminder: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. Two more rules close every gap:
| Measure | Formula |
|---|---|
| Area | ½ab |
| Height to hypotenuse | ab ÷ c |
| Inradius | (a + b − c) ÷ 2 |
| Circumradius | c ÷ 2 |
To turn a ratio back into an angle, use an inverse function, for example A = arctan(a ÷ b); the inverse trig functions calculator covers all three.
Worked example: setting a ladder
A 20 ft extension ladder should lean at about 75° to the ground. How high does it reach, and how far out does the foot go? Choose Hypotenuse c and one acute angle (A), enter c = 20 ft and A = 75°. Here A is the angle at the foot of the ladder, opposite the wall height a.
- Height reached: a = c sin A = 20 × sin 75° ≈ 19.32 ft
- Distance from the wall: b = c cos A = 20 × cos 75° ≈ 5.18 ft
- Angle between ladder and wall: B = 90° − 75° = 15°
OSHA’s construction ladder rule sets the foot about one quarter of the working length from the wall, which is 5 ft for this ladder. Choose one leg and the hypotenuse, enter leg a = 5 ft (the ground distance) and c = 20 ft, and angle B, between the ground and the ladder, comes out to 75.52°. That is where the familiar “about 75 degrees” comes from.
Roof pitch, ramps and slopes
Roof pitch is the rise per 12 inches of horizontal run, so each pitch is a right triangle with leg b = 12. Enter the rise as leg a to get the roof angle (A) and the sloped length per foot of run (c).
| Pitch | Roof angle | Rafter length per 12 in of run |
|---|---|---|
| 3/12 | 14.04° | 12.37 in |
| 4/12 | 18.43° | 12.65 in |
| 6/12 | 26.57° | 13.42 in |
| 8/12 | 33.69° | 14.42 in |
| 12/12 | 45.00° | 16.97 in |
Multiply the last column by the run in feet to estimate a rafter’s length before adding the overhang.
Wheelchair ramps use the same geometry. The 2010 ADA Standards for Accessible Design limit a ramp’s running slope to 1:12, so a 30-inch rise needs 360 inches of run. With legs of 30 and 360, the ramp surface is 361.25 in (about 30.1 ft) and its angle is just 4.76°. For stair layouts use the stair calculator, and to express a slope as a ratio or percent grade, try the slope calculator.
Tips for accurate answers
- Do not enter 90° as an acute angle. The right angle is C and is already assumed; A or B at 90° is rejected.
- Check the hypotenuse. If c is not longer than leg a, no right triangle exists, and the calculator says so.
- Carry full precision. Results show up to six decimals. If you reuse one in another calculation, copy the full value rather than a rounded one.
- Need only the third side? The Pythagorean theorem calculator is the quickest route and gives exact radical answers. For a triangle without a right angle, use the triangle calculator.
Frequently asked questions
What does SOH-CAH-TOA stand for?
It is a memory aid for the three basic trig ratios in a right triangle: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse and Tangent = Opposite ÷ Adjacent. Opposite and adjacent are always judged from the angle you are working with.
How many values do I need to solve a right triangle?
Two, and at least one of them must be a length. Two angles alone fix the shape but not the size: every right triangle with a 30° angle is a scaled copy of every other one.
How do I convert a roof pitch to degrees?
Take the arctangent of rise ÷ run. A 6/12 pitch is arctan(6 ÷ 12) ≈ 26.57°. In this calculator, choose two legs, enter the rise as leg a and 12 as leg b, and read angle A.
Why is the circumradius half the hypotenuse?
By Thales' theorem, the hypotenuse of a right triangle is always a diameter of the circle through its three corners. The circle's radius is therefore c ÷ 2, and its center sits at the midpoint of the hypotenuse.
Can I enter angles in radians?
Yes. Set Angle unit to Radians; an acute angle must then be between 0 and π/2, about 1.5708. Each angle result is shown in your chosen unit with the other unit underneath.