Heron's formula gives the area of a triangle when you know only the three side lengths. This is useful because many triangle area formulas require a height, which may not be given or easy to measure. With Heron's formula, the sides a, b, and c are enough to compute the area.
It connects perimeter, geometry, and square roots in one practical formula.
The key idea is to first find the semi-perimeter, which is half the total distance around the triangle. Then the area is found by multiplying the semi-perimeter by three related differences: s - a, s - b, and s - c. The formula works for any triangle as long as the side lengths can actually form a triangle.
It is often used in surveying, construction, navigation, and coordinate geometry when side lengths are easier to find than heights.
Key Facts
- Semi-perimeter: s = (a + b + c) / 2
- Heron's formula: A = sqrt(s(s - a)(s - b)(s - c))
- The side lengths must satisfy the triangle inequality: a + b > c, a + c > b, and b + c > a
- For sides 13, 14, and 15, s = (13 + 14 + 15) / 2 = 21
- For sides 13, 14, and 15, A = sqrt(21(8)(7)(6)) = sqrt(7056) = 84 square units
- Area units are always square units, such as cm^2, m^2, or in^2
Vocabulary
- Heron's formula
- A formula for finding the area of a triangle using only its three side lengths.
- Semi-perimeter
- Half of a triangle's perimeter, written as s = (a + b + c) / 2.
- Perimeter
- The total distance around a polygon, found for a triangle by adding its three side lengths.
- Triangle inequality
- The rule that the sum of any two side lengths of a triangle must be greater than the third side length.
- Area
- The amount of two-dimensional space inside a shape, measured in square units.
Common Mistakes to Avoid
- Using the full perimeter instead of the semi-perimeter is wrong because Heron's formula requires s, not a + b + c.
- Forgetting parentheses in s = (a + b + c) / 2 is wrong because only dividing one side length by 2 gives an incorrect semi-perimeter.
- Applying the formula to impossible side lengths is wrong because numbers such as 2, 3, and 8 do not form a triangle.
- Leaving the answer in regular units is wrong because area must be reported in square units, such as cm^2 or m^2.
Practice Questions
- 1 A triangle has side lengths 5 cm, 6 cm, and 7 cm. Use Heron's formula to find its area.
- 2 A triangular garden has sides 10 m, 13 m, and 13 m. Find its semi-perimeter and area.
- 3 Two students are given a triangle with sides 4, 9, and 15. One student starts using Heron's formula immediately. Explain why this is not valid.