Math
Right Triangle Calculator
A right triangle is fixed by any two independent measurements.
Pick the pair you know — two sides, a side and an acute angle, or the area or perimeter with a side or angle — and this calculator solves the rest: all three sides, both acute angles, area, perimeter, the altitude to the hypotenuse, inradius, circumradius, and the trig ratios, with the work shown. For sine, cosine, and tangent from two sides only, the right triangle trig calculator is a quicker page.
A side next to the right angle. If you know leg b instead, enter it here — the triangle is the same.
The other side next to the right angle.
Try an example
Result
Sides
—
- Angle A
- —
- Angle B
- —
- Area
- —
- Perimeter
- —
- Altitude to hypotenuse
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What this right triangle calculator solves
Legs a and b meet at the right angle C; the The longest side of a right triangle, always opposite the right angle. c is opposite it. Angle A is opposite leg a and angle B is opposite leg b, so A + B = 90°. Any two of the values below determine the whole triangle, and the calculator picks the right formula for each pair.
| You know | Key relationship | Notes |
|---|---|---|
| Two legs | c = √(a² + b²) | Pythagorean theorem |
| Leg and hypotenuse | b = √(c² − a²) | c must be longer than the leg |
| Leg and angle | a = c·sin A, b = c·cos A, a = b·tan A | SOH-CAH-TOA |
| Hypotenuse and angle | a = c·sin A, b = c·cos A | |
| Leg and area | b = 2·Area / a | |
| Hypotenuse and area | (a ± b)² = c² ± 4·Area | Needs Area ≤ c²/4 |
| Leg and perimeter | c − b = a² / (b + c) | Perimeter must exceed 2a |
| Hypotenuse and perimeter | a + b = P − c, ab = ((P − c)² − c²)/2 | Perimeter between 2c and c(1 + √2) |
| Angle and area | b = √(2·Area / tan A) | |
| Angle and perimeter | c = P / (1 + sin A + cos A) | |
| Area and perimeter | c = (P² − 4·Area) / (2P) | Then a, b from a quadratic |
Formulas
How to use it
- Choose the pair of values you know from the list.
- Enter the two values. Angles are in degrees; sides, area, and perimeter share one unit.
- Read the three sides and two acute angles in the panel, then the derived values below the drawing.
- Open Show the work for the formula path used for your pair.
How to read the answer
- The hypotenuse is listed with its exact A root expression written with the √ symbol, such as a square root. when one exists (for legs 1 and 1, c = √2).
- The dashed line in the drawing is the altitude to the hypotenuse; its foot splits c into the two segments shown.
- The inradius is the radius of the largest circle that fits inside the triangle; the circumradius is half the hypotenuse because the hypotenuse is a diameter of the circumscribed circle.
- When you start from area or perimeter, the two legs are interchangeable — the calculator lists the shorter leg as a.
Common mistakes and edge cases
- An acute angle must be strictly between 0° and 90°. Entering 90° makes the triangle collapse.
- The hypotenuse must be longer than any leg. c = 3 with a = 5 is impossible.
- A hypotenuse c limits the area to c²/4 (reached by the 45-45-90 triangle). Larger areas are rejected with the limit shown.
- Perimeter limits: with leg a the perimeter must exceed 2a; with hypotenuse c it must be between 2c and c(1 + √2).
- Area and perimeter together can still be impossible. The largest area for a given perimeter P is P²(3 − 2√2)/2, again the 45-45-90 case.
Worked examples
Legs 3 and 4
The 3-4-5 triangle
Sides
a = 3, b = 4, c = 5
Leg 5, hypotenuse 13
Other leg is 12
Sides
a = 5, b = 12, c = 13
Leg 10 with opposite angle 30°
Hypotenuse 20, other leg 10√3
Sides
a = 10, b = 17.3205, c = 20
Leg 7 with adjacent angle 45°
Isosceles right triangle
Sides
a = 7, b = 7, c = 9.8995
Hypotenuse 10 and angle 60°
a = 10·sin 60°
Sides
a = 8.6603, b = 5, c = 10
Leg 6 and area 24
Other leg = 48 / 6 = 8
Sides
a = 6, b = 8, c = 10
Hypotenuse 5 and area 6
Recovers the 3-4-5 triangle
Sides
a = 3, b = 4, c = 5
Leg 3 and perimeter 12
b + c = 9, c − b = 1
Sides
a = 3, b = 4, c = 5
Hypotenuse 5 and perimeter 12
Legs 3 and 4
Sides
a = 3, b = 4, c = 5
Angle 30° and area 8
b = √(16 / tan 30°)
Sides
a = 3.0393, b = 5.2643, c = 6.0787
Angle 45° and perimeter 10
c = 10 / (1 + √2)
Sides
a = 2.9289, b = 2.9289, c = 4.1421
Area 6 and perimeter 12
c = (144 − 24) / 24 = 5
Sides
a = 3, b = 4, c = 5
Hypotenuse shorter than leg
Impossible
Sides
—
Area too big for hypotenuse
c = 5 allows area at most 6.25
Sides
—
Frequently asked questions
How do you solve a right triangle with one side and one angle?+
Use SOH-CAH-TOA. If you know leg a and angle A opposite it, the hypotenuse is a / sin A and the other leg is a / tan A. The other acute angle is 90° − A.
Can I solve a right triangle from area and perimeter alone?+
Yes. The hypotenuse is c = (P² − 4·Area) / (2P); then a + b = P − c and ab = 2·Area give the legs from a quadratic. Not every pair is possible — the calculator tells you when it is not.
What is the altitude to the hypotenuse?+
The perpendicular from the right angle to the hypotenuse. Its length is ab / c, and it splits the hypotenuse into pieces a²/c and b²/c. It also creates two smaller triangles similar to the original.
Why is the circumradius half the hypotenuse?+
Thales' theorem: an angle inscribed in a semicircle is a right angle, so the hypotenuse of a right triangle is a diameter of its circumscribed circle.
Which angle is A?+
Angle A is opposite leg a (across from it). Angle B is opposite leg b. The right angle is C. If your diagram labels things differently, just match sides to their opposite angles.
How is this different from the right triangle trig calculator?+
That page focuses on sine, cosine, and tangent from two sides. This page solves the whole triangle from any pair of values, including area and perimeter, and adds altitude, inradius, and circumradius.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026