# Semicircular Design

Weekly Problem 9 - 2016

The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?

The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?

## Problem

Image

The diagram to the right shows a logo made from semi-circular arcs, each with a radius of $2$cm, $4$cm or $8$cm.

What fraction of the logo is shaded?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

## Student Solutions

**Answer**: $\frac14$

**Using areas of semicircles**

Shaded part:

Image

Image

Semicircle radius $4$ cm has area $\frac12\pi\times4^2 = 8\pi$ cm$^2$

Whole design:

Image

Image

Semicircle radius $8$ cm has area $\frac12\pi\times8^2 = 32\pi$ cm$^2$

$\frac{8\pi}{32\pi} = \frac{1}{4}$ of the shape is shaded.

**Using scale factors**

Shaded part:

Image

Image

Whole design:

Image

Shaded part is similar to whole design, length scale factor 2

Length scale factor 2 $\Rightarrow$ area scale factor 4

$\frac14$ of the shape is shaded