Area by Cutting and Sliding

Lesson 7 of 10 · 7 min · Free, no account needed

What you will learn

Cutting a shape and moving the pieces keeps its area: a parallelogram is base × height.
π ≈ 3.14 is how many diameters one turn of a wheel rolls out.
Thin slices of a circle zip into a rectangle π r by r, so a circle's area is π r².

Why is the area of a circle πr²? Not "because the formula says". First, a bet.
How many 1 by 1 squares does this circle cover?
Warm up with a slanted shape.
The left corner of this parallelogram is cut off. Slide it to the other side.
Build a wheel and roll it one full turn.
Pick your wheel's radius. Then drag its center along the ruler until the red dot touches down again.
Now cut your wheel into slices and zip them together.
Slide to make more, thinner slices. What shape does the strip become?
Area by cutting, and where π r² comes from.
Area by dissection: cut a shape into pieces and move them. The area stays the same.
π is one turn of a wheel ÷ its diameter, about 3.14. So half the way round is π r.
The zipped slices make a rectangle π r wide and r tall: area of a circle = π r². Radius 4: about 50.3 squares.
Real rooms aren't rectangles. Cut them into shapes you know.
Tap dashed lines to cut, until every piece is a rectangle or a triangle.
Game: find π with random darts.
Programs estimate areas by throwing random points. Get within 0.03 of π using as few darts as you can.
A round table has radius 10 cm. About how many square cm is its area?
Area = π r² = 3.14 × 10 × 10 ≈ 314 cm².
Pizza night. Same price: one 16 inch pizza, or two 10 inch?
Tap each offer to slice it into a strip and see its area.

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