Problem
Draw a line to split it into straight and curved pieces
Two arcs inside a square cross, and the lens between them is shaded. We need its area.
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, one bulging each way, and they are the same shape.
Two matching curved pieces.
6.G.A.1Visualize Spatial Relationships2STEP 2
Recognise each piece
For one circle, the two radii to the crossing points meet at a right angle, so the piece is a quarter circle with the triangle between those radii removed.
6 × 6 × 3.14 ÷ 4 = 28.26
A quarter circle of 28.26 cm² to start from.
7.G.B.4Draw A DiagramEach half of the lens is a quarter circle with a right triangle taken out.
Why?
The two radii drawn from one arc's centre to the crossing points are equal and meet at a right angle, so they cut off exactly a quarter of that circle.
🧱A circle's radius is constantEvery point on a circle sits exactly one radius from the center, so all radii of a circle are equal.
Why?
The triangle between those two radii sits inside that quarter, so taking it away leaves the curved piece and nothing else.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
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
Each curved piece is 10.26 cm².
6.G.A.1Change Focus Count The Complement4STEP 4
Double it
The lens is two such pieces.
10.26 × 2 = 20.52
The shaded area is 20.52 cm².
7.G.B.4Draw A DiagramAnswer
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