Right Triangle Formulas

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

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Area of a right triangle formula

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Perimeter

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Trigonometric ratios (SOH-CAH-TOA)

Try it: Hypotenuse calculator

Angles

Try it: Special right triangle calculator

Altitude-on-hypotenuse relations

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Inscribed and circumscribed circles

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30-60-90 triangle

A 30-60-90 triangle is a right triangle whose acute angles are 30° and 60°. Its sides are always in the ratio 1 : √3 : 2: if the short leg, opposite the 30° angle, is x, then the long leg, opposite the 60° angle, is x√3 and the hypotenuse is 2x. A short leg of 5, for example, gives a long leg of 5√3 ≈ 8.66 and a hypotenuse of 10. To solve one from any side, use the 30-60-90 triangle calculator.

<figure>
  <img src="https://righttrianglecalculators.com/images/30-60-90-triangle.png" width="1200" height="800" style="max-width: 100%; height: auto;" alt="A 30-60-90 triangle: the short leg x faces the 30° angle, the long leg is x√3 and the hypotenuse is 2x.">
  <figcaption><a href="https://righttrianglecalculators.com/right-triangle-formulas">Right triangle diagram by Right Triangle Calculators</a></figcaption>
</figure>

45-45-90 triangle

A 45-45-90 triangle, also called an isosceles right triangle, has two 45° angles and two equal legs. Its sides are always in the ratio 1 : 1 : √2: if each leg is x, the hypotenuse is x√2. Legs of 6, for example, give a hypotenuse of 6√2 ≈ 8.49. To solve one from any side, use the 45-45-90 triangle calculator.

<figure>
  <img src="https://righttrianglecalculators.com/images/45-45-90-triangle.png" width="1200" height="800" style="max-width: 100%; height: auto;" alt="A 45-45-90 triangle: two equal legs x and the hypotenuse x√2, with both acute angles 45°.">
  <figcaption><a href="https://righttrianglecalculators.com/right-triangle-formulas">Right triangle diagram by Right Triangle Calculators</a></figcaption>
</figure>

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.

References

The formulas and definitions on this page match these standard references.

1. Weisstein, Eric W. “Right Triangle.” MathWorld, Wolfram Research.
2. Weisstein, Eric W. “Pythagorean Theorem.” MathWorld, Wolfram Research.
3. “§4.14 Definitions and Periodicity (Trigonometric Functions).” NIST Digital Library of Mathematical Functions, National Institute of Standards and Technology.