Heron's Formula Calculator

About Heron's Formula Calculator

Heron's formula calculates triangle area using only the three side lengths, without needing to know any angles.

How to use this calculator

1. Enter the three side lengths of the triangle as \(a\), \(b\), and \(c\).
2. Check that the side lengths can actually form a triangle.
3. Find the semiperimeter with \(s = \frac{a+b+c}{2}\).
4. Compute the area using \(A = \sqrt{s(s-a)(s-b)(s-c)}\).

The formula explained

The formula computes the area of any triangle from its three side lengths. The quantity \(s\) is the semiperimeter, which is half of the triangle's perimeter.

• A = area of the triangle
• a = length of one side
• b = length of the second side
• c = length of the third side
• s = semiperimeter, or half the perimeter

Step by step method

1. Add the three side lengths and divide by 2 to get the semiperimeter.
2. Subtract each side length from \(s\).
3. Multiply \(s(s-a)(s-b)(s-c)\), then take the square root to find the area.

Worked example

Problem. Find the area of a triangle with side lengths 5 cm, 6 cm, and 7 cm.

1. Compute the semiperimeter: \(s = \frac{5+6+7}{2} = 9\).
2. Substitute into Heron's formula: \(A = \sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9\cdot4\cdot3\cdot2}\).
3. Simplify: \(A = \sqrt{216} \approx 14.7\), so the area is about \(14.7\text{ cm}^2\).

Tips and common mistakes

• Make sure the side lengths satisfy the triangle inequality, because the calculator only works for real triangles.
• Keep your units consistent, and remember that area is always in square units like \(\text{cm}^2\) or \(\text{m}^2\).

Frequently asked questions