Notes
five circles in a rectangle in a semi-circle solution

Five Circles in a Rectangle in a Semi-Circle

Each of the small circles has area $1$ . What’s the area of the semicircle?

Solution by Pythagoras' Theorem and Circle Area

Five circles in a rectangle in a semi-circle labelled

Let $r$ be the radius of the smaller circles, so $\pi r^2 = 1$ , and let $R$ be the radius of the semi-circle. Then with the points labelled as in the diagram above, the length of $A B$ is $2 r$ , of $O A$ is $5 r$ , and of $O B$ is $R$ . Applying Pythagoras' theorem to triangle $O A B$ shows that:

R^2 = (2 r)^2 + (5 r)^2 = 29 r^2

The area of the semi-circle is therefore:

\frac{1}{2} \pi R^2 = \frac{29}{2} \pi r^2 = \frac{29}{2}

Created on August 12, 2021 08:35:50 by Andrew Stacey

Notes five circles in a rectangle in a semi-circle solution

Five Circles in a Rectangle in a Semi-Circle

Solution by Pythagoras' Theorem and Circle Area

Notes
five circles in a rectangle in a semi-circle solution