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Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

Griddy Region

Stage: 4 Short Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

$$\frac{11}{12} \; \text{square units}$$


We want to find out how far the points $A$ and $B$ (the points on both the triangle and square) are from $Y$. The triangle $OQT$ is $3$ units across by $2$ units down. The triangle $OPA$ is similar to $OQT$, and is half its size (since it is one unit down rather than two). So the point $A$ is $\frac{3}{2}$ units from $O$, so $\frac{3}{2}-1=\frac{1}{2}$ unit from $X$, so $1-\frac{1}{2}=\frac{1}{2}$ unit from $Y$

Similarly, the triangle $BST$ is $\frac{1}{3}$ the size of $OQT$, so the point $B$ is $\frac{2}{3}$ unit from $Z$, so $1-\frac{2}{3}=\frac{1}{3}$ unit from $Y$. So the triangle $AYB$ has area $$\frac{1}{2}\times \frac{1}{2}\times \frac{1}{3} \; \text{square units}= \frac{1}{12}\; \text{square units}$$ so the area of the overlap is $$1-\frac{1}{12}\; \text{square units}=\frac{11}{12}\; \text{square units}$$

This problem is taken from the UKMT Mathematical Challenges.