Indigo Interior
The diagram shows 8 shaded squares inside a circle. What is the shaded area?
Age
14 to 16
Challenge level
Problem
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The diagram shows a pattern of eight equal shaded squares inside a circle of area $\pi$ square units.
What is the area (in square units) of the shaded region?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
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As the circle has area $\pi$ square units, its radius is $1$ unit. So $OB$ is $1$ unit long.
Let the length of the side of each of the shaded squares be $x$ units.
By Pythagoras's Theorem, $OB^2 = OA^2 + AB^2$, that is $1^2 = (2x)^2 + x^2$.
So $5x^2=1$. Now the total shaded area is $8x^2 = 8 \times \frac{1}{5} = 1 \frac{3}{5}$ square units.