Measurement
Heron's formula
Find the area of a triangle from its three sides, without knowing its height.
Half the base times the height is fine — as long as somebody has told you the height. Often nobody has. A triangular field measured with a tape has three side lengths and no height anywhere, and dropping a perpendicular to find one is real work. Heron of Alexandria found a way round it: three sides in, area out, no height needed at any point.
The formula
First find the semi-perimeter: s = (a + b + c) ÷ 2, which is just half the way round. Then the area is the square root of s × (s - a) × (s - b) × (s - c). Four numbers multiplied, one square root taken, and you are done.
Worked example
Find the area of a triangle with sides 13, 14 and 15 units.
Add the sides: 13 + 14 + 15 = 42. Halve it: s = 21.
Everything else is built from s, so an error here spoils all four factors at once.
Try it together
Find the area of a triangle with sides 5, 12 and 13 units.
Take it in the same four moves: total, halve, differences, root.
1.Add the three sides: what is 5 + 12 + 13?
Watch out
The number you get after multiplying the four factors is not the area — the square root of it is. And if one of the differences comes out negative, no triangle with those three sides exists in the first place.
Have a go
Have a go on your own: a triangle has sides of 8, 15 and 17 cm. What is its area, in square centimetres?
Hint: The semi-perimeter is 20, and the three differences are 12, 5 and 3.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.