Circled Corners
Three circles have been drawn at the vertices of this triangle. What is the area of the inner shaded area?
Age
14 to 16
Challenge level
Problem
The diagram shows a triangle and three circles whose centres are at the vertices of the triangle. The area of the triangle is $80 cm^2$ and each of the circles has radius $2cm$. What is the area, in $cm^2$, of the shaded area?
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Student Solutions
The three angles of the triangle add to $180^{\circ}$, so the combined area of the three sectors of the circles that are inside the triangle add up to half a circle with area:
$\frac{1}{2} \times \pi \times 2^2 = \frac {4\pi}{2}= 2\pi$.
So the grey area is $(80- 2\pi) cm^2$.
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