What this calculator does
Find every side and angle of a right triangle from just two things you know. A right triangle has one square (90°) corner. So once you know any two separate measurements, the rest of the triangle is fixed. Two sides settle it through the Pythagorean theorem. A side and one of the two sharp (acute) angles settle it through sine, cosine and tangent. Those are ratios of a triangle's sides, explained below. This calculator handles both kinds of starting point. It also works backward from the area, or from the height dropped onto the hypotenuse (the longest side).
Pick what you know and enter the two values. The calculator returns all three sides, both acute angles, the area and the perimeter. It also gives the height to the hypotenuse and the radii of the circles that fit inside and around the triangle. When the angles are 30°-60°-90° or 45°-45°-90°, it points out the special ratio as well. Each step shows which trig ratio was used and the numbers that went into it.
Where it's used
Why this math matters outside the classroom:
- Ladder angle. A 20 ft ladder set at 75° to the ground reaches 20 × sin 75°, about 19.3 ft, up the wall.
- Roof pitch. A roof that rises 6 in for every 12 in of run slopes at tan⁻¹(6 ÷ 12), about 26.6°.
- Ramp length. A ramp sloping at 4.8° that has to rise 30 in needs a run of 30 ÷ tan 4.8°, about 357 in or 29.8 ft.
- Measuring heights. A surveyor 50 m from a tower sights its top at 32°, so it stands 50 × tan 32°, about 31.2 m, above eye level.
- Navigation. A boat that sails 10 km on a course 30° east of north ends up 10 × sin 30° = 5 km east and about 8.66 km north.
Grade level and standards: Grades 9–10; Common Core State Standards for Mathematics HSG.SRT.C.8. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose the pair of values you already know from the list, for example a leg and the angle opposite it.
- Enter those two values. Angles are in degrees and must be between 0° and 90°. Lengths can use any unit, as long as they all use the same one.
- Read the headline side, then the remaining sides, both angles and the other measurements below it.
- Open the working to see SOH CAH TOA or the Pythagorean theorem applied with your numbers.
What the inputs mean
- What you know
- The letters follow the usual pattern. The legs (the two sides that meet at the square corner) are a and b, and c is the hypotenuse. Angle A sits opposite leg a. So leg b runs along angle A, and the hypotenuse is its other side.
- Leg a
- The leg opposite angle A.
- Leg b
- The leg next to angle A.
- Hypotenuse c
- The longest side, opposite the right angle.
- Angle A (degrees)
- Enter one of the two acute angles in degrees. The other is always 90° minus this one, so knowing either is enough.
- Height to the hypotenuse (h)
- The height to the hypotenuse is the line from the square corner that meets the long side at a right angle. It splits the triangle into two smaller right triangles. Each one has the same shape as the original.
Formula and method
sin A = a ÷ c, cos A = b ÷ c, tan A = a ÷ b c² = a² + b² B = 90° − A area = a × b ÷ 2, perimeter = a + b + c height to the hypotenuse h = a × b ÷ c inradius r = (a + b − c) ÷ 2, circumradius R = c ÷ 2
Here a and b are the legs, c is the hypotenuse, and A and B are the two acute angles, with A opposite a. h is the height to the hypotenuse. r is the radius of the circle that fits inside the triangle (the inradius). R is the radius of the circle through all three corners (the circumradius).
The first line is the memory aid SOH CAH TOA. Sine is opposite over hypotenuse, cosine is adjacent (next to) over hypotenuse, and tangent is opposite over adjacent. Rearranging one of them turns a known side and angle into a missing side, for example a = c × sin A or c = a ÷ sin A.
When two sides are known instead, the Pythagorean theorem gives the third. An inverse ratio, which turns a ratio back into an angle, gives the angle, such as A = tan⁻¹(a ÷ b). The angles of any triangle add up to 180°, and one of them is already 90°. So the two acute angles share the remaining 90°.
The legs meet at the right angle, so either one can be the base and the other the height. That's why the area is half their product. The midpoint of the hypotenuse is equally far from all three corners. So the circle around the triangle has the hypotenuse as its diameter.
Worked example
A kite flies on 150 feet of tight string that makes a 40° angle with the ground. How high is the kite above the hand holding the string, and how far away is it along the ground?
- The string is the hypotenuse (c = 150) and the 40° angle is at the hand. So the height is the leg opposite that angle, and the distance along the ground is the leg next to it.
- Height: a = c × sin 40° = 150 × 0.64278761 = 96.4181 feet.
- Horizontal distance: b = c × cos 40° = 150 × 0.76604444 = 114.9067 feet.
- The other acute angle, at the kite, is 90° − 40° = 50°.
Result: The kite is about 96.4181 feet above the hand and 114.9067 feet away along the ground. The triangle it forms has an area of about 5,539.5436 square feet and a perimeter of about 361.3248 feet.
Understanding the result
The hypotenuse is always the longest side, and every leg is shorter than it. A larger angle A makes the opposite leg a longer and the adjacent leg b shorter. The hypotenuse stays the same.
Special triangles give exact ratios. A 45-45-90 triangle has equal legs and a hypotenuse √2 times a leg. A 30-60-90 triangle has a hypotenuse twice the short leg, and a long leg √3 times the short one. Most other angles give sides that are endless, non-repeating decimals. So the decimals are rounded to the precision you choose.
Assumptions and limitations
- The triangle has an exact 90° angle, and angle A is opposite leg a.
- Angles are in degrees, and all lengths share one unit. The area is in that unit squared.
- This tool solves right triangles only. For a triangle with no right angle, use the law of sines or cosines in the triangle solver.
- Measured inputs carry their own error, and small angles make it bigger. A 1° mistake on a 5° angle changes the sides far more than the same mistake on a 45° angle.
Frequently asked questions
What does SOH CAH TOA mean?
It's a reminder of the three basic ratios in a right triangle: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent. Opposite and adjacent are judged from the angle you're working with.
How do I find an angle when I know two sides?
Use the inverse ratio that matches the sides you have. With both legs, A = tan⁻¹(a ÷ b). With a leg and the hypotenuse, A = sin⁻¹(a ÷ c). For legs of 5 and 12, the angle opposite the 5 is tan⁻¹(5 ÷ 12), about 22.6199°.
What are the side ratios of a 30-60-90 triangle?
The sides are in the ratio 1 : √3 : 2. The shortest side is opposite the 30° angle, and the hypotenuse is twice that. The remaining leg is the short side times √3, about 1.732 times it.
Why can't angle A be 90° or more?
The right angle already uses 90° of the triangle's 180°. So the other two angles must each be less than 90° and add up to exactly 90°. An angle of 90° or more would leave nothing for the third corner.
Can I solve a right triangle from one side only?
No. One side fixes the size but not the shape. You need a second value: another side, an acute angle, the area or the height to the hypotenuse. The only exception is when you already know it's a special triangle, which gives you the angle.
How this calculator is checked
Method: Right-triangle trigonometric ratio definitions (sine, cosine, tangent), the Pythagorean theorem, the angle sum of a triangle, and standard inradius and circumradius formulas for right triangles. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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