Using just a ruler and a compass, can you divide a circle into 8 regions of equal area? You are NOT allowed to “slice” it into wedges!

**Answer**

Start with a circle and divide its diameter into 8 equal segments.

Build semicircles as shown below. Together, two corresponding semicircles constructed in this way will represent one of the 8 regions of equal area.

The circle with all the 8 regions of equal area will look like this:

Beautiful, isn’t it?! Can it be divided into any number of regions? Yes! The most famous division of the circle is the Chinese symbol for yin and yang.

Now let’s check that the regions we constructed have the same area.

We can start with the first region we drew. We can show that it is ^{1}/_{8} of the area of the circle. Feel free to use the same steps to show that the area of the other regions is also ^{1}/_{8} of the area of the circle.

The first region comprises areas 1 and 2.

Let r be the radius of the circle.

Region_{1} = Area_{1} + Area_{2}

……………….Area_{1} = (^{1}/_{2}) π (^{r}/_{8})^{2} = (^{1}/_{128}) π r^{2 }

……………….Area_{2} = (^{1}/_{2}) Area circle – Area_{3}

…………………………….Area circle = π r^{2}

…………………………….Area_{3} = (^{1}/_{2}) π (^{7r}/_{8})^{2} = (^{49}/_{128}) π r^{2}

……………….Area_{2} = (^{1}/_{2}) π r^{2} – (^{49}/_{128}) π r^{2} = (^{15}/_{128}) π r^{2}

Region_{1} = (^{1}/_{128}) π r^{2} + (^{15}/_{128}) π r^{2} = (^{16}/_{128}) π r^{2} = (^{1}/_{8}) π r^{2}

Region_{1} = (^{1}/_{8}) Area circle