Circles with squares and triangles

A square inside a circle

Draw a circle, and then draw a square with all four corners on the circle. The diagonal of the square passes through the centre. So the diagonal is a diameter, $2r$.

If the side is $s$, then the diagonal is $s\sqrt{2}$ (Pythagoras). So $s\sqrt{2} = 2r$, and $s = r\sqrt{2}$. The area of the square is

Example 1. A circle has a radius of $7\ \text{cm}$. A square has all four corners on it. Find the area between the circle and the square. Take $\pi = \dfrac{22}{7}$.

1. The circle. $\dfrac{22}{7} \times 49 = 154\ \text{cm}^2$.
2. The square. $2r^2 = 2 \times 49 = 98\ \text{cm}^2$.
3. The shaded part. $154 - 98 = 56\ \text{cm}^2$. The exact form is $(49\pi - 98)\ \text{cm}^2$.

An equilateral triangle inside a circle

Now draw an equilateral triangle with all three corners on a circle of radius $r$. The centre divides each median in the ratio $2 : 1$, so the distance from the centre to a side is $\dfrac{r}{2}$. A chord with $120^\circ$ at the centre has the length $2r\sin 60^\circ$, so the side of the triangle is

The area of an equilateral triangle of side $s$ is $\dfrac{\sqrt{3}}{4}s^2$. So with $s = r\sqrt{3}$,

Example 2. An equilateral triangle has all its corners on a circle of radius $6\ \text{cm}$. Find the area between the circle and the triangle. Give the exact value, and then a number with $\pi = 3.14$ and $\sqrt{3} = 1.73$.

1. The circle. $\pi \times 36 = 36\pi\ \text{cm}^2$.
2. The triangle. $\dfrac{3\sqrt{3}}{4} \times 36 = 27\sqrt{3}\ \text{cm}^2$.
3. The shaded part. $(36\pi - 27\sqrt{3})\ \text{cm}^2$ is the exact form.
4. The number. $36 \times 3.14 = 113.04$ and $27 \times 1.73 = 46.71$. The area is $113.04 - 46.71 = 66.33\ \text{cm}^2$.

The simulation shows a general rule. A regular polygon with $n$ corners on a circle of radius $r$ is made of $n$ thin triangles, each with area $\dfrac{1}{2}r^2\sin\dfrac{360^\circ}{n}$. So the polygon has the area $\dfrac{n}{2}r^2\sin\dfrac{360^\circ}{n}$. Check it for $n = 4$: $2r^2$. For $n = 3$: $\dfrac{3}{2}r^2 \times \dfrac{\sqrt{3}}{2} = \dfrac{3\sqrt{3}}{4}r^2$. Both agree with the results above.

Quarter circles in a square

Example 3, the middle region. A square has a side of $28\ \text{cm}$. A quarter circle of radius $14\ \text{cm}$ is drawn at each corner with its centre at the corner. The quarter circles touch each other along the sides. Find the area of the middle region that is not covered. Take $\pi = \dfrac{22}{7}$.

1. The four quarters. Four quarters of radius $14\ \text{cm}$ make one whole circle: $\dfrac{22}{7} \times 14 \times 14 = 616\ \text{cm}^2$.
2. The square. $28 \times 28 = 784\ \text{cm}^2$.
3. The middle region. $784 - 616 = 168\ \text{cm}^2$.

The leaf

Example 4. Draw a square of side $a$. With the corner $A$ as centre, draw a quarter circle of radius $a$ through the two neighbouring corners. With the opposite corner $C$ as centre, draw another one. The two quarter circles overlap in a leaf-shaped region.

Each quarter circle covers the whole square except a piece at the far corner. Together, the two quarter circles cover the square, and they cover the leaf twice. So

For $a = 14\ \text{cm}$ and $\pi = \dfrac{22}{7}$: leaf $= 2 \times 154 - 196 = 112\ \text{cm}^2$.

Practise with new numbers

Try it in Python

This program finds the area of a regular polygon on a circle of radius $1$, for several numbers of corners, and how much of the circle is left over.

Common mistakes

• Using the radius as the side of the inscribed square. The side is $r\sqrt{2}$.
• Using the area of the square as $r^2$. It is $2r^2$.
• Forgetting the leaf is counted twice. Two quarter circles minus the square gives the leaf once.
• Mixing the exact value and the decimal. Say which values of $\pi$ and $\sqrt{3}$ you used.

Remember these

• Square on a circle of radius $r$: side $r\sqrt{2}$ and area $2r^2$.
• Equilateral triangle on a circle of radius $r$: side $r\sqrt{3}$ and area $\dfrac{3\sqrt{3}}{4}r^2$.
• Four quarter circles of radius $\dfrac{a}{2}$ in a square of side $a$ leave $a^2 - \dfrac{\pi a^2}{4}$.
• The leaf of two quarter circles of radius $a$ in a square of side $a$ is $\left(\dfrac{\pi}{2} - 1\right)a^2$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A square has all four corners on a circle of radius $10\ \text{cm}$. Write the area between the circle and the square in terms of $\pi$.

2. An equilateral triangle has all three corners on a circle of radius $4\ \text{cm}$. What is the length of its side?

3. A square of side $8\ \text{cm}$ has a quarter circle of radius $4\ \text{cm}$ at each corner. Write the area of the middle region in terms of $\pi$.

4. A square has a side of $7\ \text{cm}$. Two quarter circles of radius $7\ \text{cm}$ are drawn at opposite corners. Take $\pi = \dfrac{22}{7}$. Find the area of the leaf.

5. Why does a square with all corners on a circle of radius $r$ not have the side $r$?