Shaded Regions and Combined Figures: Quick Revision

A brief note on Shaded Regions and Combined Figures. Read it before a test. Try each question at the end before you open its answer.

The method

• Name the plain shapes. Decide: add the pieces, or subtract a part from a big shape.
• Keep $\pi$ exact until the last step. Say which value of $\pi$ and $\sqrt{3}$ you used.
• The perimeter counts only the outer edge. A side that is joined to another shape is not counted.
• A circle inscribed in a square has its diameter equal to the side. A circle through the corners of a square has its diameter equal to the diagonal.

Shapes on a circle

• Square on a circle of radius $r$: side $r\sqrt{2}$, area $2r^2$.
• Equilateral triangle on a circle of radius $r$: side $r\sqrt{3}$, area $\dfrac{3\sqrt{3}}{4}r^2$.
• Regular polygon with $n$ corners: area $\dfrac{n}{2}r^2\sin\dfrac{360^\circ}{n}$.
• Square with four quarter circles of radius $\dfrac{a}{2}$ at the corners: middle region $a^2 - \dfrac{\pi a^2}{4}$.
• Leaf of two quarter circles of radius $a$ in a square of side $a$: $\left(\dfrac{\pi}{2} - 1\right)a^2$.

Windows, tracks and circles that touch

• Window: rectangle plus semicircle. Its frame leaves out the joined side.
• Track: inner edge $2L + 2\pi r$. Each lane further out adds $2\pi w$.
• Three touching circles of radius $r$ enclose $\left(\sqrt{3} - \dfrac{\pi}{2}\right)r^2$.

A worked example

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. Find the area of the middle region. Take $\pi = \dfrac{22}{7}$.

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

Practice with new numbers

A program to try

The program finds the middle region of the worked example, and the leaf of two quarter circles of radius $28\ \text{cm}$ in the same square.

Mistakes to avoid

• Adding when you should subtract.
• Using the side as the radius of an inscribed circle. The side is the diameter.
• Counting a joined side in a perimeter.
• Rounding $\pi$ or $\sqrt{3}$ early.

Check yourself

1. A square of side $10\ \text{cm}$ has a circle inscribed. Write the area of the four corners in terms of $\pi$.

2. A square has all four corners on a circle of radius $6\ \text{cm}$. What is the area of the square?

3. A track has straight sides of $50\ \text{m}$ and ends of radius $10\ \text{m}$. Write the inner edge in terms of $\pi$.

4. What is the area of a window that is a square of side $10\ \text{cm}$ with a semicircle on top, in terms of $\pi$?