HardMathematicsHigh School (9–10)Mensuration › Areas and PerimetersNumericalreasoningBoard Exam SAT

figure: ar-a3-leaf.png ABCD is a square of side 28 cm…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A square ABCD with side a, with A at the bottom left and C at the top right. One arc of radius a is drawn with its centre at A and runs from B to D. Another arc of radius a is drawn with its centre at C and also runs from B to D. The leaf-shaped region between the two arcs, which lies along the diagonal BD, is shaded orange and labelled leaf.

$ABCD$ is a square of side $28\ \text{cm}$ . A quarter circle of radius $28\ \text{cm}$ is drawn with its centre at $A$ , and another with its centre at $C$ . Find the area of the leaf that is common to both. Take $\pi = \dfrac{22}{7}$ .

Need a reminder? The key ideaShow

A square with all corners on a circle of radius $r$ has the area $2r^2$ . An equilateral triangle with all corners on the circle has the side $r\sqrt{3}$ and the area $\dfrac{3\sqrt{3}}{4}r^2$ . The leaf of two quarter circles of radius $a$ in a square of side $a$ is $\left(\dfrac{\pi}{2} - 1\right)a^2$ .

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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