How do you find the area of a shape with no straight edges? People were stuck on it for thousands of years.
Egypt: close, but not exact
Around 1650 BC, an Egyptian scribe called Ahmes wrote down a trick in the Rhind Papyrus: turn the circle into a square whose side is 8/9 of the circle's width. It works like using π ≈ 3.16. Close, but not exact.
Greece: Archimedes cracks it
Around 250 BC, Archimedes proved that a circle's area equals a triangle with the radius as its height and the circumference as its base. He also pinned π between 3 10/71 and 3 1/7.
See it for yourself: slice the circle
Cut a circle into slices, like a pizza. Line them up, point up, point down. The thinner the slices, the more the row becomes a rectangle.
- Its height is the radius, r.
- Its length is half the crust: half of 2πr, which is πr.
So the area is πr × r = πr².
Kepler and the wine barrels
In 1615, Johannes Kepler used the same slicing idea in a book about measuring wine barrels. He wrote it after buying wine for his own wedding and doubting how the merchants measured it.
Worked example
A pizza with a radius of 10 cm: π × 10² = 100π, which is about 314 cm².
What's the area of a circle with radius 3?
Show the answer
π × 3² = 9π, which is about 28.3 square units.
Save this for your next test
- Slice a circle and line the slices up.
- It becomes a rectangle r tall and πr long.
- So the area is πr².
- Square the radius first, then multiply by π.
Read the video transcript
- How do you find the area of a shape with no straight edges? People were stuck on it for four thousand years.
- In ancient Egypt, a scribe called Ahmes had a trick: turn the circle into a square, a bit narrower than the circle. Close, but not exact.
- Then, around 250 BC, the Greek genius Archimedes cracked it.
- Here's the idea. Cut the circle into slices, like a pizza.
- Line them up, point up, point down.
- The thinner the slices, the more it becomes a rectangle.
- Its height is the radius. Its length is half the crust: pi times r.
- So the area is pi r times r. Pi r squared.
- Two thousand years later, Johannes Kepler used the same slicing idea to measure wine barrels for his own wedding.
- Let's use it. A pizza with a radius of 10 centimetres: pi times 10 squared. About 314 square centimetres.
- Now you: what's the area of a circle with radius 3? Tell me in the comments.
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