Four sides on one circle
Four rods joined at the ends can be tilted into many shapes.
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- A circle.
- A four-sided shape with all four corners on it.
- Its sides: 3, 4, 5 and 6.
- Its area: √360, about 18.97.
The rule
Four rods of 5 cm make a square. Its area is 5 × 5 = 25 cm². Tilt it until one corner is 30°. The height drops to 2.5 cm. So the area falls to 5 × 2.5 = 12.5 cm². The sides did not change. So four sides alone do not fix the area.
Put all four corners on one circle. Then four sides do fix the area.
Cyclic quadrilateral — a four-sided shape whose corners all lie on one circle.
A bracket is half the perimeter minus one side.
Rule: Half the perimeter. Take each of the four sides from it. Multiply the four brackets. Then find the number that squares to the product.
Take sides 3, 4, 5, 6. Half the perimeter is (3 + 4 + 5 + 6) ÷ 2 = 9. The brackets are 9 − 3 = 6 and 9 − 4 = 5. Also 9 − 5 = 4 and 9 − 6 = 3. Their product is 6 × 5 × 4 × 3 = 360. The area is √360, about 18.97.
Brahmagupta found this rule, in 628 CE. It gives the area from the four sides alone. No diagonal is needed.
| Sides | Half-perimeter | Brackets | Area |
|---|---|---|---|
| 3, 4, 5, 6 | 9 | 6, 5, 4, 3 | √360 ≈ 18.97 |
| 3, 4, 5, 0 | 6 | 3, 2, 1, 6 | 6 |
Shrink the fourth side to 0. The shape becomes a triangle. The fourth bracket is 6 − 0 = 6, the half-perimeter. So the product is 6 × 3 × 2 × 1 = 36. The area is √36 = 6. That is Heron's rule for the triangle 3, 4, 5.
A rectangle is cyclic. Its four corners are equally far from its centre. Take a 5 by 12 rectangle. The half-perimeter is 17. The brackets are 12, 5, 12, 5. The product is 3600, and √3600 = 60 = 5 × 12.