Depth · one concept from first principles · step 2 of 4
In the figure at the right, a part of a circle with radius is drawn at each vertex of a pentagon and shaded. What is the total area of the shaded parts, in ?
The angles are the pentagon's five interior angles, and a pentagon splits into three triangles, each with 180 degrees.
The angles come to 540 degrees.
8.G.A.5Change Focus Count The ComplementA full circle is 360 degrees, so 540 degrees is one and a half circles.
The pieces make one and a half circles.
8.G.A.5Identify SubproblemsSectors of the same radius add up by their angles alone.
Every point of each sector is the same from its own corner, so all five sectors are pieces of circles of exactly one size.
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.
Sliding or turning a piece lays it exactly onto a copy of itself, so it still covers the same amount.
Every sector has a radius of 6 cm, so the circle they belong to does too.
One circle is 113.04 cm².
7.G.B.4Draw A DiagramMultiply the circle's area by how many circles the sectors make.
The shaded parts come to 169.56 cm².
7.G.B.4Identify SubproblemsSectors with the same radius add up by their angles alone -- and the total can pass a full circle.
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