Depth · one concept from first principles · step 2 of 4

Draw a line to split it into straight and curved pieces

The figure at the right shows parts of circles overlapping inside a square with sides of $12\ \text{cm}$ . What is the area of the shaded part, in $\text{cm}^2$ ?

Geometry

Your answer

How to solve

Strategy Visualize Spatial Relationships — Draw the line between the two crossing points and the lens falls into two identical curved pieces. Each is a quarter circle with a right triangle cut out of it, so one calculation covers both.

1STEP 1

Cut the lens along the line between the crossings

The chord from the bottom-left corner to the centre of the square splits the lens into two pieces bulging opposite ways, and they match.

2STEP 2

Recognise each piece

In one circle the radii to the crossing points meet at a right angle, so the piece is a quarter circle minus the triangle between them.

6 × 6 × 3.14 ÷ 4 = 28.26

3STEP 3

Take the triangle off

The triangle's two sides are radii meeting at a right angle, so it is 6 by 6 halved.

6 × 6 ÷ 2 = 18, 28.26 - 18 = 10.26

4STEP 4

Double it

The lens is two such pieces.

10.26 × 2 = 20.52

Answer

20.52 cm²

Sanity-check the size: the lens sits inside a 6 by 6 square of area 36 cm² and is clearly a good deal thinner than it, so about 20 cm² is the right order.

Takeaway

A lens between two equal arcs splits into two curved pieces, each a sector with a triangle taken out.

Cut the lens along the line between the crossings
Recognise each piece
Take the triangle off
Double it

Where next?

Another one like thissuggested

12 practice problems

Same look, different logicsuggested

Slide pieces so the shaded part becomes one lump

Step back and solidifysuggested

One full roll moves it its own circumference A part's share of a circle is its central angle's share

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▶ Practice — 12 problems