Triangle Calculator

Math

Solve any triangle given any 3 values: SSS (3 sides), SAS (2 sides and included angle), SSA (with ambiguous case detection), ASA, and AAS. Computes sides, angles, area, perimeter, altitudes, medians, inradius, and circumradius with interactive diagrams.

Presets:

Solved Triangle Scalene · Acute

Primary Metrics

Area (A) 26.8328

Perimeter (P) 24

Inradius (r) 2.2361

Circumradius (R) 4.6957

Interior Angles

Angle α (A) 48.19° 0.8411 rad

Angle β (B) 58.41° 1.0195 rad

Angle γ (C) 73.40° 1.2810 rad

Altitudes & Medians

Altitudes (h_a, h_b, h_c) 7.67, 6.71, 5.96

Medians (m_a, m_b, m_c) 7.76, 7.00, 6.02

Trigonometric & Algebraic Derivations

1 1. Law of Cosines for Angle α (A)

cos(α) = (b² + c² - a²) / (2bc) = (8² + 9² - 7²) / (2 × 8 × 9) = 0.6667 → α = 48.19°

Determines the first angle opposite to side a.

2 2. Law of Sines / Cosines for Angle β (B)

cos(β) = (a² + c² - b²) / (2ac) → β = 58.41°

Determines the second angle opposite to side b.

3 3. Angle Sum Property for Angle γ (C)

γ = 180° - (α + β) = 180° - (48.19° + 58.41°) = 73.40°

The sum of all interior angles of a Euclidean triangle is always 180°.

4 4. Heron's Formula for Area

s = (a + b + c) / 2 = 12.00, Area = √[s(s - a)(s - b)(s - c)] = 26.8328

Calculates the area directly from semi-perimeter and side lengths.

How This Calculator Works

A triangle is a fundamental polygon with three edges and three vertices. By trigonometry, knowing any three independent pieces of information (except AAA, which only determines shape) is sufficient to solve for all remaining sides, angles, area, and circumcircles.

📐 Fundamental Trigonometric Laws of Triangles

Law of Cosines: c² = a² + b² - 2ab·cos(γ) | Law of Sines: a/sin(α) = b/sin(β) = c/sin(γ)

Heron’s Formula: Area = √[s(s - a)(s - b)(s - c)] where s = (a + b + c) / 2 is the semi-perimeter

Inradius (r): Radius of incircle inscribed inside the triangle: r = Area / s

Circumradius (R): Radius of circumscribed circle passing through all 3 vertices: R = abc / (4 × Area)

Ambiguous SSA Case: When given two sides and a non-included acute angle with a < b, either 0, 1, or 2 distinct triangles can exist

Key Mathematical Rules & Properties

SSS (Three Sides)

Apply the Law of Cosines to solve for angle α, then β, and subtract from 180° for γ.

SAS (Side-Angle-Side)

Find the third side using the Law of Cosines, then find the smaller remaining angle using Law of Sines.

ASA & AAS

Subtract the two known angles from 180° to find the third angle, then solve unknown sides via Law of Sines.

Understanding your calculation

Method, example and limitations

How the calculation works

Triangle calculations use the supplied sides or angles with relationships such as the angle sum, Pythagorean theorem, sine rule and cosine rule.

Example to check

A triangle with sides 3, 4 and 5 has perimeter 12 and area 6, and the angle opposite side 5 is 90°.

Before using the result

Sides must satisfy the triangle inequality. Some side-angle combinations admit two triangles; rounded measurements may not describe an exact triangle.

Calculation methodology · Report a result that needs review