REFERENCE SHEET

Right Triangle Formulas

Every formula you need for right-triangle geometry and trigonometry, in one place — with the reasoning behind each one.

How to use this reference

Every formula on this page uses the same labeling convention as the rest of this site: vertices A, B, and C with the right angle at C; leg a (side BC) opposite angle α at vertex A; leg b (side AC) opposite angle β at vertex B; hypotenuse c (side AB); and altitude h drawn from C perpendicular to the hypotenuse. If you'd rather plug in numbers than work through formulas by hand, the right triangle calculator applies every formula below automatically and shows its own step-by-step substitution for whichever two values you provide.

The formulas are grouped into six families: the Pythagorean theorem itself; area and perimeter; the three SOH-CAH-TOA trigonometric ratios; the pair of complementary acute angles; the altitude-on-hypotenuse relations (including the lesser-known geometric-mean identities); and the inscribed and circumscribed circle radii. Together they cover everything needed to fully solve any right triangle from any two known values.

Pythagorean theorem

a² + b² = c²

The sum of the squares of the two legs equals the square of the hypotenuse. Solve for any one side given the other two: c = √(a²+b²), a = √(c²−b²), b = √(c²−a²).

Area of a right triangle formula

Area from the legs

Area = (1/2)·a·b

Because the two legs are perpendicular, either one can serve as the "base" while the other is the "height" — half their product is the area.

Area from the hypotenuse and its altitude

Area = (1/2)·c·h

Equivalent to the standard base-times-height formula using the hypotenuse as the base and the altitude drawn to it as the height.

Area from the hypotenuse and an angle

Area = (1/2)·c²·sin α·cos α

Each leg is the hypotenuse times the sine or cosine of α (a = c·sin α, b = c·cos α), so half their product gives the area. Equivalently, Area = (1/4)·c²·sin 2α.

Area from one leg and an angle

Area = a² / (2·tan α)

With leg a opposite angle α, the other leg is b = a / tan α. Substituting into (1/2)·a·b gives the area from a single leg and one acute angle.

Perimeter

P = a + b + c

Simply the sum of all three side lengths — the two legs and the hypotenuse.

Trigonometric ratios (SOH-CAH-TOA)

Sine (SOH)

sin(α) = opposite / hypotenuse = a / c

For angle α at vertex A, the "opposite" side is a (side BC) and the hypotenuse is c. Rearranged: a = c·sin(α), c = a / sin(α).

Cosine (CAH)

cos(α) = adjacent / hypotenuse = b / c

For angle α, the "adjacent" leg is b (side AC). Rearranged: b = c·cos(α), c = b / cos(α).

Tangent (TOA)

tan(α) = opposite / adjacent = a / b

The tangent of an acute angle is the ratio of the opposite leg to the adjacent leg — useful for finding an angle when only the two legs are known: α = atan(a/b).

Angles

Complementary acute angles

α + β = 90°

The two non-right angles of any right triangle always sum to 90°, so knowing one immediately gives the other.

Altitude-on-hypotenuse relations

Altitude as a geometric mean

h = √(p·q)

The altitude drawn from the right angle to the hypotenuse splits it into two segments p and q. The altitude is the geometric mean of those segments — a direct consequence of the three triangles formed all being similar.

Altitude from the legs and hypotenuse

h = (a·b) / c

Derived by equating the two area formulas (1/2)ab = (1/2)ch and solving for h.

Each leg as a geometric mean

a² = p·c, b² = q·c

Each leg is the geometric mean of the whole hypotenuse and the hypotenuse segment adjacent to that leg — the basis of the "geometric mean (leg) theorem".

Inscribed and circumscribed circles

Inradius

r = (a + b − c) / 2

The radius of the circle inscribed in a right triangle, tangent to all three sides. A useful identity for a quick sanity check on any solved triangle.

Circumradius

R = c / 2

The hypotenuse of a right triangle is always a diameter of its circumscribed circle (Thales’s theorem), so the circumradius is simply half the hypotenuse.

Why the altitude relations work

The altitude-on-hypotenuse identities look like a separate set of rules, but they all fall out of one observation: dropping the altitude from the right angle to the hypotenuse creates two smaller right triangles, and both of them are similar to the original triangle (and to each other), because all three share the same set of angles — just relabeled. Once you know the three triangles are similar, their corresponding side ratios must match, and matching those ratios directly produces h² = p·q, a² = p·c, and b² = q·c, where p and q are the two segments the altitude splits the hypotenuse into (p + q = c). This is sometimes called the "geometric mean theorem" or "right triangle altitude theorem," and it's a favorite proof exercise in introductory geometry courses precisely because a single similarity argument unlocks three separate-looking formulas at once.

Putting it together: a quick numeric check

Take the well-known 3-4-5 right triangle: legs a = 3, b = 4, hypotenuse c = 5. The Pythagorean theorem confirms 3² + 4² = 9 + 16 = 25 = 5². The area is (1/2)(3)(4) = 6, and the perimeter is 3 + 4 + 5 = 12. The altitude to the hypotenuse is h = (3 × 4)/5 = 2.4, which splits the hypotenuse into segments p = a²/c = 9/5 = 1.8 and q = b²/c = 16/5 = 3.2 — and sure enough, p + q = 1.8 + 3.2 = 5 = c, and h² = p·q checks out as 2.4² = 5.76 = 1.8 × 3.2. Finally, the inradius is (3 + 4 − 5)/2 = 1 and the circumradius is 5/2 = 2.5. Every formula on this page is internally consistent — which is exactly why they're used together throughout this calculator's step-by-step solutions.

Right triangle calculator

Find the area, sides, and angles of any right triangle from any 2 known values.

Pythagorean theorem calculator

Find the missing side of a right triangle, or check if 3 sides form one.

Special right triangle calculator

Solve 30-60-90 and 45-45-90 triangles from a single known side.

30-60-90 triangle calculator

Solve a 30-60-90 triangle from a single known side.

Isosceles right triangle calculator

Solve a 45-45-90 isosceles right triangle from a single known side.

Right triangle FAQ

Answers to common right-triangle questions.