How to Calculate Triangle Area, Angles & Perimeter: Complete Guide

Three side lengths. That’s all you need to derive every meaningful property of a triangle — area, all three angles, heights from each vertex, and whether it’s right, acute, or obtuse. Here’s how the math works, with worked examples you can follow through.

What three sides give you

Given sides a, b, c, you can calculate:

• Area (using Heron’s formula)
• Perimeter (a + b + c)
• All three interior angles (using the Law of Cosines)
• Height from each vertex (area / base × 2)
• Triangle type by sides: equilateral, isosceles, or scalene
• Triangle type by angles: right, acute, or obtuse

Heron’s formula: area from three sides

Heron’s formula computes the area of any triangle without needing to know the height:

s = (a + b + c) / 2        (semi-perimeter)
area = √(s(s−a)(s−b)(s−c))

Example — the 3-4-5 right triangle:

• s = (3+4+5)/2 = 6
• area = √(6 × 3 × 2 × 1) = √36 = 6

You can verify this the easy way: a right triangle with legs 3 and 4 has area = ½ × 3 × 4 = 6 ✓

The Law of Cosines: angles from sides

Once you have all three sides, each angle follows from:

cos(A) = (b² + c² − a²) / (2bc)
cos(B) = (a² + c² − b²) / (2ac)
cos(C) = (a² + b² − c²) / (2ab)

Example — the same 3-4-5 triangle:

cos(C) = (3² + 4² − 5²) / (2 × 3 × 4) = (9 + 16 − 25) / 24 = 0/24 = 0
C = arccos(0) = 90°

The other two angles:

• cos(A) = (4² + 5² − 3²) / (2 × 4 × 5) = 32/40 = 0.8 → A ≈ 36.87°
• cos(B) = (3² + 5² − 4²) / (2 × 3 × 5) = 18/30 = 0.6 → B ≈ 53.13°
• Check: 36.87 + 53.13 + 90 = 180° ✓

The triangle inequality

Not every combination of three positive numbers forms a valid triangle. The rule is simple: the sum of any two sides must be strictly greater than the third.

• a + b > c
• a + c > b
• b + c > a

Sides 1, 2, 10 fail this check — 1 + 2 = 3, which is less than 10, so no triangle exists. The calculator checks this before computing and shows an error if the input is invalid.

Triangle classification

By sides:

Type	Condition
Equilateral	a = b = c
Isosceles	Exactly two sides equal
Scalene	All sides different

By angles:

Type	Condition
Right	One angle = 90°
Acute	All angles < 90°
Obtuse	One angle > 90°

A few things that follow from this: an equilateral triangle is always acute (all angles are 60°). A right triangle can never be equilateral. An obtuse triangle can be isosceles or scalene, but not equilateral.

Where this comes up in practice

Carpenters and builders use the 3-4-5 rule constantly — if the diagonal of a corner measures 5 when the two legs are 3 and 4, the corner is square. No protractor needed.

Surveyors use triangulation to measure distances they can’t walk. GPS systems do something similar. In 3D graphics, mesh normals are calculated from triangle vertex positions, which is why understanding triangle geometry matters for rendering.

Try the Triangle Calculator — enter three side lengths and get area, angles, heights, and triangle type instantly, free and private.