Flower beds, windows and running tracks

A window with a round top

A window is a rectangle with a semicircle on the top side. The diameter of the semicircle is the width of the window.

Example 1. A window is $70\ \text{cm}$ wide and the rectangular part is $120\ \text{cm}$ high. Take $\pi = \dfrac{22}{7}$. Find the area of the glass, and the length of the frame, if the frame goes round the whole edge.

1. The rectangle. $70 \times 120 = 8400\ \text{cm}^2$.
2. The semicircle. The radius is $35\ \text{cm}$. The area is $\dfrac{1}{2} \times \dfrac{22}{7} \times 35 \times 35 = 1925\ \text{cm}^2$.
3. The glass. $8400 + 1925 = 10\,325\ \text{cm}^2$.
4. The frame. The bottom is $70$ and the two sides are $120$ each. The curved top is half a circle: $\dfrac{22}{7} \times 35 = 110\ \text{cm}$. The frame is $70 + 240 + 110 = 420\ \text{cm}$.

The side of the rectangle under the semicircle is not part of the frame, because it is inside the window.

A running track

A running track has two straight sides and two semicircular ends. The two semicircles together make one circle. So the length of the inner edge is

where $L$ is each straight side and $r$ is the radius of the ends.

Example 2. The inner edge of a track has two straight sides of $90\ \text{m}$ and ends of radius $28\ \text{m}$. The track is $7\ \text{m}$ wide. Take $\pi = \dfrac{22}{7}$.

1. The inner edge. $2 \times 90 + 2 \times \dfrac{22}{7} \times 28 = 180 + 176 = 356\ \text{m}$.
2. The outer edge. The outer radius is $28 + 7 = 35\ \text{m}$. The length is $180 + 2 \times \dfrac{22}{7} \times 35 = 180 + 220 = 400\ \text{m}$.
3. The difference. $400 - 356 = 44\ \text{m}$. This is $2\pi \times 7$, so it depends only on the width.
4. The area of the track. The two straight strips are $2 \times 90 \times 7 = 1260\ \text{m}^2$. The two ends make a ring: $\dfrac{22}{7}\,(35^2 - 28^2) = 1386\ \text{m}^2$. The total is $1260 + 1386 = 2646\ \text{m}^2$.

Because the difference between two lanes is $2\pi \times (\text{lane width})$, runners in outer lanes start further ahead in a one-lap race. This is called the stagger.

Three circles that touch each other

Three equal circles of radius $r$ touch each other. Their centres form an equilateral triangle of side $2r$. The curved triangle in the middle is that triangle minus three sectors of $60^\circ$ at its corners.

Example 3. The three circles have a radius of $7\ \text{cm}$. Take $\pi = \dfrac{22}{7}$ and $\sqrt{3} = 1.73$. Find the area of the curved triangle between them.

1. The triangle of centres. The side is $14\ \text{cm}$, and the area is $\dfrac{\sqrt{3}}{4} \times 14^2 = 49\sqrt{3} = 84.77\ \text{cm}^2$.
2. The three sectors. Each sector has an angle of $60^\circ$, so the three together make half a circle: $\dfrac{1}{2} \times \dfrac{22}{7} \times 49 = 77\ \text{cm}^2$.
3. The curved triangle. $84.77 - 77 = 7.77\ \text{cm}^2$. The exact form is $\left(49\sqrt{3} - \dfrac{49\pi}{2}\right)\ \text{cm}^2$.

Practise with new numbers

Try it in Python

This program finds the inner and outer edge and the area of the track of Example 2. It uses $\pi = \dfrac{22}{7}$ because the question says so.

Common mistakes

• Counting the joined side in the edge of a window. The side under the semicircle is inside the figure.
• Using one semicircle for the two ends of a track. The two ends together make a full circle.
• Using the width as the outer radius. The outer radius is the inner radius plus the width.
• Taking the sectors of the three circles as a full circle. They are three sectors of $60^\circ$, which is half a circle.

Remember these

• Window: rectangle $+$ semicircle. The edge leaves out the joined side.
• Track: the inner edge is $2L + 2\pi r$. The outer edge is longer by $2\pi w$.
• The area of a track is two straight strips plus a ring.
• Three touching circles of radius $r$ enclose $\left(\sqrt{3} - \dfrac{\pi}{2}\right)r^2$.

Check yourself

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

1. A window is a rectangle $60\ \text{cm}$ wide and $80\ \text{cm}$ high with a semicircle on top. Write its area in terms of $\pi$.

2. A track has two straight sides of $100\ \text{m}$ and ends of radius $20\ \text{m}$. Write the length of its inner edge in terms of $\pi$.

3. The track in question 2 is $5\ \text{m}$ wide. By how much is its outer edge longer than the inner edge, in terms of $\pi$?

4. Three circles of radius $2\ \text{cm}$ touch each other. What is the side of the triangle formed by their centres?

5. Why is the area of the three sectors in the curved triangle equal to half a circle?