Adding and subtracting areas

Plain shapes inside a picture

A tile has a circle painted in the middle. A lawn has a round flower bed. A window has a half circle on top. Each picture is made of shapes you know: squares, rectangles, triangles, circles, semicircles and sectors. To find the area of a shaded region, follow five steps.

1. Name the shapes. Find the plain shapes inside the picture.
2. Decide: add or subtract? If the shaded region is the pieces put together, add. If it is a big shape with a smaller shape cut out of it, subtract.
3. Find each area. Keep $\pi$ as a letter if you can.
4. Add or subtract. Put in the value of $\pi$ only at the end.
5. Check. The answer must have the right unit, and it must be smaller than the big shape that holds the region.

Subtracting: a square with a circle

Example 1. A square has a side of $14\ \text{cm}$. The largest circle that fits in it touches all four sides. The four corners are shaded. Find the area of the shaded corners. Take $\pi = \dfrac{22}{7}$.

1. The shapes. A square and a circle. The shaded region is the square without the circle.
2. The circle. The circle touches opposite sides, so its diameter is $14\ \text{cm}$ and its radius is $7\ \text{cm}$. Its area is $\dfrac{22}{7} \times 7 \times 7 = 154\ \text{cm}^2$.
3. The square. $14 \times 14 = 196\ \text{cm}^2$.
4. The corners. $196 - 154 = 42\ \text{cm}^2$.
5. The exact form. $196 - 49\pi\ \text{cm}^2$.

The sliders show something surprising. The corners are always about $21.5\%$ of the square. The exact fraction is $1 - \dfrac{\pi}{4}$, and it does not depend on the side. This is the rule from the first course: when every length grows $k$ times, every area grows $k^2$ times, so the fraction stays the same.

Example 2, a lawn with a round bed. A lawn is a rectangle $20\ \text{m}$ long and $14\ \text{m}$ wide. A round flower bed of radius $3.5\ \text{m}$ is in the middle. Take $\pi = \dfrac{22}{7}$. The lawn without the bed has an area of $20 \times 14 - \dfrac{22}{7} \times 3.5 \times 3.5 = 280 - 38.5 = 241.5\ \text{m}^2$.

Adding: a square with a semicircle

Example 3. A square of side $14\ \text{cm}$ has a semicircle added on one side, using that side as its diameter. Take $\pi = \dfrac{22}{7}$.

1. The square. $196\ \text{cm}^2$.
2. The semicircle. $\dfrac{1}{2} \times \dfrac{22}{7} \times 7 \times 7 = 77\ \text{cm}^2$.
3. The whole figure. $196 + 77 = 273\ \text{cm}^2$.

The edge is not the area

Many questions ask for the perimeter of the figure, which is the length of its outer edge. The perimeter counts only the edge you could walk along. It does not count a side that is joined to another shape.

In Example 3, the edge is three sides of the square and the curved edge of the semicircle. It is $3 \times 14 + \dfrac{22}{7} \times 7 = 42 + 22 = 64\ \text{cm}$. The side of the square under the semicircle is inside the figure, so it is not counted.

Words you will meet

• A circle is inscribed in a square if it touches all four sides. Its diameter is the side of the square.
• A circle is circumscribed about a square if it passes through the four corners. Its diameter is the diagonal of the square.
• A shape that fits inside another with all its corners on the boundary is also called inscribed.

Practise with new numbers

Try it in Python

This program finds the corners of a square with an inscribed circle for four sides, and also the fraction of the square that the corners fill.

Common mistakes

• Adding when you should subtract. Ask whether the shaded part is the whole thing or what is left after a part is taken out.
• Using the side as the radius. For an inscribed circle the side is the diameter.
• Counting a joined side in the perimeter.
• Rounding $\pi$ at the start. Keep $\pi$ as a letter until the last step.

Remember these

• Shaded region $=$ the sum of the pieces, or a big shape minus a smaller shape.
• A circle inscribed in a square has a diameter equal to the side.
• The perimeter of a combined figure is only its outer edge.
• Keep $\pi$ exact until the last step, and say which value of $\pi$ you used.

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 a side of $20\ \text{cm}$ and a circle is inscribed in it. Write the area of the four corners in terms of $\pi$.

2. A circle of radius $10\ \text{cm}$ has a square hole of side $10\ \text{cm}$ cut out of it. Write the area of the rest in terms of $\pi$.

3. A rectangle $12\ \text{cm}$ by $9\ \text{cm}$ has a semicircle added on one of the $12\ \text{cm}$ sides. Write the area of the whole figure in terms of $\pi$.

4. What is the perimeter of the figure in question 3? Write it in terms of $\pi$.

5. Why is the area of the corners always about $21.5\%$ of the square, for any side?