Problem
A part's share of a circle is its central angle's share
A circle is divided into twelve equal parts and a region bounded by two chords and two arcs is shaded. We need its area.
GeometryRatios
Your answer
How to solve
Strategy Identify Subproblems — The shape has no formula of its own, so cut it at the centre. Drawing the four radii to A, B, C and D splits it into two triangles and two sectors, and every piece is then something with a rule.
1STEP 1
Work out the four central angles
Each step around the circle is 30 degrees. From A to D is three steps and from B to C is three; from A round to B is two and from C round to D is four.
360 ÷ 12 = 30
Angles of 90, 60, 90 and 120 degrees.
7.G.B.5Visualize Spatial Relationships2STEP 2
Measure the two triangles
A central angle of 90 degrees makes a right triangle whose two legs are radii, so each has legs 12 and 12.
12 × 12 ÷ 2 = 72
Each triangle is 72 cm².
7.G.B.5Draw A Diagram3STEP 3
Measure the two sectors
A sector is the share of the whole circle its angle takes: 60 degrees is a sixth and 120 degrees is a third.
12 × 12 × 3.14 = 452.16
The sectors are 75.36 and 150.72 cm².
7.G.B.4Identify SubproblemsA -degree sector is one sixth of the circle.
Why?
Six equal -degree angles at the centre fill one full turn, and they cut the circle into six pieces with no gap and no overlap.
🧱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?
Those six pieces are all the same size, because turning the circle about its centre carries any one of them exactly onto the next.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
4STEP 4
Add the four pieces
The two triangles and the two sectors together fill the shaded region exactly.
72 + 75.36 + 72 + 150.72 = 370.08
The shaded area is 370.08 cm².
7.G.B.4Identify SubproblemsAnswer
370.08 cm²
Check what is left out: the whole circle is 452.16 cm², so the two white pieces come to 82.08 cm² -- and each is a 90-degree sector minus a triangle, 113.04 - 72 = 41.04 cm².
Takeaway
A shape with no formula can be cut at the centre into sectors and triangles, which do have one.
- Work out the four central angles
- Measure the two triangles
- Measure the two sectors
- Add the four pieces