Heron's formula: the area of a field from its sides
A farmer can measure the three sides of a triangular field but not its height. Change the sides, find the semi-perimeter and the area with Heron's formula, and see when three lengths cannot make a triangle.
Core concept
The area of a triangle with sides a, b and c is √(s(s − a)(s − b)(s − c)), where s = (a + b + c) ÷ 2 is the semi-perimeter; any two sides together must be longer than the third
Topic: Mensuration
The usual formula, area: How much flat space a shape covers, measured in square units like cm². All key words = ½ × base × height, needs the height, which is hard to measure in a field. Heron of Alexandria found a way to use only the three sides a, b and c. First find the semi-perimeter: The total distance around the edge of a shape. All key words, half the distance round: s = (a + b + c) ÷ 2. Then area = √(s(s − a)(s − b)(s − c)). For a field with sides 13 m, 14 m and 15 m, s = 21 and the area is √(21 × 8 × 7 × 6) = √7056 = 84 m². Three lengths only make a triangle if any two together are longer than the third; otherwise one of s − a, s − b or s − c is negative and there is no area.
What you will learn
- Find the semi-perimeter of a triangle.
- Calculate the area of a triangle from its three sides with Heron's formula.
- Check whether three lengths can form a triangle.
Before you start
- Know how to find a square root.
Materials
- A long rope or measuring tape, and three sticks to mark corners
Equipment
- A calculator
Procedure
- Record a field with sides 13 m, 14 m and 15 m.
- Record a field with sides 5 m, 12 m and 13 m.
- Record a field with sides 6 m, 8 m and 10 m.
- Change the third side to 15 m and record.
How this simulation is simplified
- The field is flat and its sides are straight.