Problem
A circle's snug square has the diameter as its side
Four equal circles are packed two by two inside a square, each touching the square and its neighbours. We need the area left empty.
Geometry
Your answer
How to solve
Strategy Change Focus / Count the Complement — The empty space has an awkward shape, so measure everything else instead and subtract. Both the square and the circles come from one number -- the radius -- which the circumference gives back.
1STEP 1
Recover the radius from the circumference
The circumference is the radius times two times 3.14, so divide it back out.
43.96 ÷ 3.14 = 14, 14 ÷ 2 = 7
Each circle has a radius of 7 cm.
7.G.B.4Work Backwards2STEP 2
Find the square's side
Two circles sit side by side along each edge, so the side is two diameters.
7 × 2 × 2 = 28
The square is 28 cm on a side.
6.G.A.1Draw A DiagramThe square's side is two diameters.
Why?
Two circles sit side by side along each edge, touching each other and touching the square, so the side runs from one circle's far edge to the other's without any gap.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Each circle contributes its own diameter to that run, because a circle's width across the centre is the same in every direction.
🧱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.
3STEP 3
Measure the square and the circles
The square is its side multiplied by itself; each circle is its radius twice times 3.14, and there are four of them.
28 × 28 = 784, 7 × 7 × 3.14 × 4 = 615.44
784 against 615.44 cm².
7.G.B.4Draw A Diagram4STEP 4
Subtract
What is empty is the square with the four circles taken out.
784 - 615.44 = 168.56
The empty space is 168.56 cm².
7.G.B.4Change Focus Count The ComplementAnswer
168.56 cm²
Do it for one circle in its own square: a 14 by 14 square holds a circle of 153.86 cm², leaving 42.14 cm², and four of those corners give 168.56 cm².
Takeaway
When a leftover shape is awkward, measure everything else and subtract.
- Recover the radius from the circumference
- Find the square's side
- Measure the square and the circles
- Subtract