Area of a Leaf

This post is to outline step by step solution on how to find the area of the leaf made by intersecting semicircles. This is a classical geometry problem. It might be easy for some but if this is your first time to see this problem, I doubt if you can solve it for 3 minutes.

In the figure, a leaf is made by intersecting 4 semicircles with radius 1.What is the area of  the leaf made? leaf2

Solution:

Label the figure accordingly and locate the point of intersection of the semicircles.

leaf 3

 

Locate the Center of Diameter \overline{AD} label it F. Draw the square AFEG. Shade the figure like shown below.

 

leaf 7

 

Shade this part of the figure. The area of the leaf is 8 times of the area of the shaded region.

Area of shaded region = Area of Quarter Circle AFE – Area of triangle AFE

Solving area of Area of Quarter Circle AFE:

A=\displaystyle\frac{1}{4}\pi(r^2)=\displaystyle\frac{1}{4}\pi (1)=\displaystyle\frac{\pi}{4}

Solving for Area of triangle AFE:

A=\displaystyle\frac{1}{2}bh =\displaystyle\frac{1}{2}(1)(1)=\displaystyle\frac{1}{2}

Area of shaded region= Area of Quarter Circle AFE – Area of triangle AFE

Area of shaded region=\displaystyle\frac{\pi}{4}-\displaystyle\frac{1}{2}

Solving for area of leaf:

Area of leaf = 8(Area of shaded region)

Area of leaf =8(\displaystyle\frac{\pi}{4}-\displaystyle\frac{1}{2})

Area of leaf =2\pi -4

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