Problem
Angles adding past 360° cover more than one circle
A sector of the same radius is drawn at each vertex of a pentagon and shaded. We need the total shaded area.
Geometry
Your answer
How to solve
Strategy Change Focus / Count the Complement — Because every sector has the same radius, they can be slid together into one shape whose angle is the sum of theirs. That sum is the pentagon's interior angles, which is more than a full turn -- so the answer is more than one circle.
1STEP 1
Add the sectors' angles
The angles are the pentagon's five interior angles, and a pentagon splits into three triangles, each with 180 degrees.
180 × 3 = 540
The angles come to 540 degrees.
8.G.A.5Change Focus Count The Complement2STEP 2
See how much of a circle that is
A full circle is 360 degrees, so 540 degrees is one and a half circles.
540 ÷ 360 = 1.5
The pieces make one and a half circles.
8.G.A.5Identify SubproblemsSectors of the same radius add up by their angles alone.
Why?
Every point of each sector is the same from its own corner, so all five sectors are pieces of circles of exactly one size.
🧱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?
Pieces of the same circle can be slid together without changing their size, and a -degree turn's worth of them makes one whole circle.
Why?
Sliding or turning a piece lays it exactly onto a copy of itself, so it still covers the same amount.
🧱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).
3STEP 3
Work out one whole circle
Every sector has a radius of 6 cm, so the circle they belong to does too.
6 × 6 × 3.14 = 113.04
One circle is 113.04 cm².
7.G.B.4Draw A Diagram4STEP 4
Take one and a half of it
Multiply the circle's area by how many circles the sectors make.
113.04 × 1.5 = 169.56
The shaded parts come to 169.56 cm².
7.G.B.4Identify SubproblemsAnswer
169.56 cm²
Check with a regular pentagon: each angle is 540 ÷ 5 = 108 degrees, and five sectors of 108 degrees give 113.04 × 108 ÷ 360 × 5, which is 169.56 cm² again.
Takeaway
Sectors with the same radius add up by their angles alone -- and the total can pass a full circle.
- Add the sectors' angles
- See how much of a circle that is
- Work out one whole circle
- Take one and a half of it