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

Whether you’re a student solving geometry homework, an engineer checking a structural calculation, or a designer working with precise shapes, the ability to calculate a triangle’s properties from its side lengths is a fundamental skill. Here’s how the math works and how to get the numbers instantly with a free online calculator.

What can we calculate from three side lengths?

Given three sides a, b, c, we can derive every property of a triangle:

• Area (using Heron’s formula)
• Perimeter (trivially: 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, scalene)
• Triangle type by angles (right, acute, 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: a right triangle with legs 3 and 4 has area = ½ × 3 × 4 = 6 ✓

The Law of Cosines: angles from sides

Once you know all three sides, you can find each angle:

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

Example — the 3-4-5 right triangle:

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

The third angle (C) is 90°, confirming it’s a right triangle.

The angles at A and B:

• 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° ✓

Triangle inequality

Not every combination of three positive numbers forms a valid triangle. The triangle inequality requires:

The sum of any two sides must be strictly greater than the third side.

• a + b > c
• a + c > b
• b + c > a

If any of these conditions fails — say sides 1, 2, 10 — no triangle can exist. Our calculator checks this before computing and shows a clear 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°

An equilateral triangle is always acute (all angles = 60°). A right triangle is never equilateral. An obtuse triangle can be isosceles or scalene but never equilateral.

Practical uses

• Construction and carpentry: checking if a corner is square by measuring the diagonal (3-4-5 or 5-12-13 right triangles)
• Navigation and surveying: triangulation uses the Law of Cosines extensively
• 3D graphics: mesh normals are calculated from triangle vertex positions
• Physics and statics: resolving forces using vector triangles

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