Right Triangle Calculator

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Part of:Geometry Guide: Area, Volume & Surface Area Formulas for Every Shape (2026)

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|Prof. Emily Rodriguez, Ph.D. Mathematics|Updated June 1, 2026Our Standards →

Last updated: June 21, 2026

Calculate right triangle sides, angles, area & perimeter using Pythagorean theorem and trigonometry. Free geometry triangle solver with step-by-step solutions.

Triangle Calculator

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Comprehensive triangle calculator solving right triangles and general triangles using SSS, SAS, ASA, AAS methods. Calculate area, perimeter, angles, and advanced properties.

Details

Triangle Type

Angle Unit

Measurements

Side a (opposite A)

Angle A

Side b (opposite B)

Angle B

Side c (opposite C)

Angle C

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📚 In-Depth Guide

This calculator is part of a comprehensive guide

Geometry Guide: Area, Volume & Surface Area Formulas for Every Shape (2026)

Full Guide

Complete 2026 geometry reference covering 2D area formulas, 3D volume and surface area calculations, triangles, circles, coordinate geometry, and composite shapes. Includes free area calculator, volume calculator, surface area calculator, and right triangle solver. NIST-aligned, with 25+ shapes, 40+ formulas, and real-world applications for flooring, landscaping, construction, and engineering.

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Geometry & Trigonometry

Right Triangle Calculator — Complete Trig & Geometry Guide

Solve any right triangle using the Pythagorean theorem, trig ratios, and special angle properties. Calculate sides, angles, area, altitude, circumradius, and inradius instantly.

Pythagorean Theorem

a² + b² = c²

Area

(1/2) × a × b

Sine Rule

sin A = a / c

Circumradius

R = c / 2

✓ Reviewed by the CalculatorApp Mathematics & Geometry Team

What Is a Right Triangle?

A right triangle has exactly one 90° angle. The side opposite the right angle is the hypotenuse (longest side); the other two sides are called legs. Right triangles are the foundation of trigonometry: every angle has unique sine, cosine, and tangent ratios linking it to the side lengths.

Right triangle geometry underlies architecture, surveying, GPS triangulation, structural engineering, computer graphics (rasterization), and satellite navigation. The Pythagorean theorem — perhaps the most famous equation in mathematics — applies exclusively to right triangles.

Key Formulas

Hypotenuse

c = √(a² + b²)

Given both legs

Missing Leg

a = √(c² − b²)

Given hyp + one leg

Altitude to c

h = (a × b) / c

Creates two similar Δs

Inradius

r = (a + b − c) / 2

Incircle radius

Area

A = (1/2) × a × b

Product of legs ÷ 2

Circumradius

R = c / 2

Hypotenuse is diameter

sin A

a / c

Opposite / hypotenuse

tan A

a / b

Opposite / adjacent

Special Right Triangles

Name	Angles	Side Ratio	Key Property
30-60-90	30°, 60°, 90°	1 : √3 : 2	Equilateral triangle bisected
45-45-90 (Isosceles rt)	45°, 45°, 90°	1 : 1 : √2	Diagonal of a unit square
3-4-5 Pythagorean	~37°, ~53°, 90°	3 : 4 : 5	Most-used in construction
5-12-13 Pythagorean	~22.6°, ~67.4°, 90°	5 : 12 : 13	Next simplest triple
8-15-17 Pythagorean	~25.4°, ~64.6°, 90°	8 : 15 : 17	Popular in tiling problems

History of Triangle Geometry

~2000 BC — Babylonian

Plimpton 322 tablet lists Pythagorean triples — predating Pythagoras by over 1,000 years.

~600 BC — Indian Vedic

Baudhayana Sulba Sutra states the Pythagorean theorem geometrically for altar construction.

~570–495 BC — Greek

Pythagoras (reportedly) proves the general theorem: a² + b² = c² for all right triangles.

~300 BC — Euclid

Formalizes triangle geometry in Elements (Books I and VI), including Pythagorean proof.

~150 AD — Ptolemy

Develops chord tables equivalent to modern sine tables for astronomical calculations.

Modern era — Applied

Right triangle geometry underpins GPS, computer graphics, FEA mesh generation, and satellite navigation.

Key Resources & Research

External link

NIST Mathematics Reference

Authoritative mathematical constants and geometry reference data for science and engineering.

External link

Khan Academy — Trigonometry

Free trig and geometry lessons covering SOHCAHTOA, unit circle, and triangle solving.

External link

Wolfram MathWorld — Right Triangle

Deep mathematical treatment including proofs, formulas, and special triangle properties.

External link

OSHA — Slope & Angle Safety

Construction safety standards applying right-triangle slope calculations to ladder safety.

External link

NASA — Geometry in Space

How spacecraft trajectory calculations rely on spherical geometry and inverse trig.

External link

MIT OpenCourseWare — Geometry

Free lecture notes and problem sets on Euclidean geometry and trigonometry.

Myths vs Facts

❌ Myth: The Pythagorean theorem works for all triangles.

✅ Fact: a²+b²=c² applies ONLY to right (90°) triangles. For others, use the law of cosines: c²=a²+b²−2ab·cos(C).

❌ Myth: You always need two sides and an angle to solve a triangle.

✅ Fact: Three sides (SSS) alone are sufficient — use the law of cosines to find all angles.

❌ Myth: 3-4-5 only works in specific units.

✅ Fact: The ratio 3:4:5 works in any consistent unit (cm, m, ft, in) — unit consistency is all that matters.

❌ Myth: Sine and cosine are only for right triangles.

✅ Fact: The law of sines (a/sin A = b/sin B) and cosines extend trigonometry to all triangle types.

Frequently Asked Questions (12)

What is the Pythagorean theorem?+

For a right triangle with legs a, b and hypotenuse c: a² + b² = c². It applies ONLY to right (90°) triangles. Any triangle satisfying this equation is guaranteed to be a right triangle (converse theorem). Example: 3² + 4² = 9 + 16 = 25 = 5², confirming 3-4-5 is a right triangle.

How do I find a missing angle in a right triangle?+

Use inverse trig: A = arcsin(opposite/hypotenuse), arccos(adjacent/hypotenuse), or arctan(opposite/adjacent). Since one angle is already 90°, the third angle = 90° − A. Example: if a=3, c=5, then A = arcsin(3/5) = arcsin(0.6) ≈ 36.87°, so B ≈ 53.13°.

What are the 30-60-90 triangle ratios?+

Sides are in ratio 1 : √3 : 2. If the short leg (opposite 30°) = n, the long leg (opposite 60°) = n√3, and hypotenuse = 2n. Derived by bisecting an equilateral triangle. Used in architecture, roof framing, and tiling. Example: short leg = 5 → long leg ≈ 8.66, hypotenuse = 10.

What are the 45-45-90 triangle ratios?+

Sides are in ratio 1 : 1 : √2. Both legs equal; the hypotenuse = leg × √2. Appears when cutting a square diagonally. If the leg = 5 cm, hypotenuse = 5√2 ≈ 7.07 cm. Both acute angles are exactly 45°.

What is the area of a right triangle?+

Area = (1/2) × a × b, where a and b are the two legs (the sides forming the right angle). This is a special case of A = (1/2) × base × height, because in a right triangle, the two legs serve directly as base and height. Example: legs 6 and 8 → Area = (1/2)(6)(8) = 24 sq units.

Can I solve a triangle knowing only three sides (SSS)?+

Yes. With all three sides, compute angles using inverse cosine: A = arccos((b²+c²−a²)/(2bc)). For a right triangle, the hypotenuse is opposite the 90° angle. Verify first: if the longest side² equals the sum of squares of the other two, you have a right triangle.

What is the law of sines?+

For any triangle: a/sin(A) = b/sin(B) = c/sin(C). Useful for AAS and ASA cases. For right triangles, since sin(90°) = 1, the law simplifies to a/sin(A) = c, confirming sin(A) = a/c (SOHCAHTOA). The law of sines cannot uniquely solve SSA configurations (the ambiguous case).

What does SOHCAHTOA mean?+

SOHCAHTOA is a memory aid for right-triangle trig ratios: SOH = Sine × Opposite/Hypotenuse; CAH = Cosine × Adjacent/Hypotenuse; TOA = Tangent × Opposite/Adjacent. Example: angle A, opposite=4, adjacent=3, hypotenuse=5 → sin A = 4/5 = 0.8, cos A = 3/5 = 0.6, tan A = 4/3 ≈ 1.333.

What is a Pythagorean triple?+

Three positive integers (a, b, c) satisfying a² + b² = c². Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29. All multiples also work: 6-8-10, 9-12-15. In construction, a 3-4-5 string triangle is used to verify right angles without a protractor (measure 3 units, 4 units, and confirm the diagonal is 5 units).

How is the circumradius of a right triangle calculated?+

R = c/2, where c is the hypotenuse. By Thales' theorem, the hypotenuse is always the diameter of the circumscribed circle. If hypotenuse = 10 cm, then R = 5 cm. The right-angle vertex always lies on the circumscribed circle. This property is used in circle geometry proofs and engineering design.

What is the inradius of a right triangle?+

r = (a + b − c) / 2, where a and b are legs and c is the hypotenuse. The incircle touches all three sides. Example: 3-4-5 triangle → r = (3 + 4 − 5)/2 = 1 unit. The inradius is used in structural design and pipe-fitting problems (fitting a circular pipe inside a triangular space).

What is the altitude to the hypotenuse?+

h = (a × b) / c, where a and b are the legs and c the hypotenuse. The altitude from the right-angle vertex to the hypotenuse divides the triangle into two smaller triangles, each similar to the original. Geometric mean relationship: h² = p × q, where p and q are the two hypotenuse segments. Used in lens design and optical path calculations.

References & Further Reading

1. Wolfram MathWorld — Right Triangle
2. Khan Academy — Trigonometry
3. NIST — Mathematics Reference Data
4. MIT OpenCourseWare — Geometry & Trigonometry
5. NASA — Geometry in Aerospace
6. Euclid. Elements, Books I and VI (~300 BC). Standard reference for Euclidean triangle geometry.

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Right Triangle Calculator — Quick Reference

Calculate sides, angles, area, and perimeter of a right triangle using the Pythagorean theorem.

Formula: Pythagorean Theorem

c² = a² + b²

a = Side a

b = Side b

c = Hypotenuse

Example Calculation

A 3-4-5 right triangle has hypotenuse 5 and area 6.

Key Facts

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides.

Sources & Validation

Euclidean geometry

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