Triangle Area Calculator: Heron's formula

Heron's formula lets you compute the area of any triangle knowing only the lengths of its three sides, without needing the height or any angle.

Heron's formula

Given a triangle with sides a, b and c:

• Semi-perimeter: s = (a + b + c) / 2
• Area: A = √[s × (s − a) × (s − b) × (s − c)]

The semi-perimeter is half the total perimeter. The formula works for equilateral, isosceles and scalene triangles.

Example 1: 3-4-5 triangle

Problem: A triangle has sides a = 3 cm, b = 4 cm, c = 5 cm.

1. Semi-perimeter:
   • s = (3 + 4 + 5) / 2 = 6 cm.
2. Area:
   • A = √[6 × (6−3) × (6−4) × (6−5)] = √[6 × 3 × 2 × 1] = √36 = 6 cm².

Answer: A = 6 cm².

Example 2: scalene triangle

Problem: A triangle has sides a = 7 m, b = 8 m, c = 9 m.

1. Semi-perimeter:
   • s = (7 + 8 + 9) / 2 = 12 m.
2. Area:
   • A = √[12 × (12−7) × (12−8) × (12−9)] = √[12 × 5 × 4 × 3] = √720 ≈ 26.83 m².

Answer: A ≈ 26.83 m².

Usi comuni

• Computing areas of triangular land plots in surveying.
• Determining surface areas of triangular structures in engineering.
• Solving geometry problems when the height is unknown.
• Verifying whether three lengths can form a valid triangle.
• Computing areas in graphic design and 3D modeling.
• Estimating materials for triangular coverings in construction.

Common mistakes with Heron's formula

• Using sides that do not form a valid triangle (the sum of two sides must exceed the third).
• Miscalculating the semi-perimeter by forgetting to divide by 2.
• Getting a negative value under the square root, which indicates invalid sides.
• Mixing units among the three sides.

Consiglio professionale

Before applying Heron's formula, verify the triangle inequality: a + b > c, a + c > b and b + c > a. If any fails, those lengths cannot form a triangle.