nce around the outside of a full circle. The **area** is the space inside the circle. For a circle with radius $r$: $$ \begin{aligned} \text{area} &= \pi r^2 \\ \text{circumference} &= 2\pi r \end{aligned} $$ 1. A circle has radius 6 cm. For the circumference, substitute $r=6$: $$ C=2\pi \times 6=12\pi $$ 2. For the area, substitute $r=6$: $$ A=\pi \times 6^2=36\pi $$ 3. So the circumference is $12\pi$ cm and the area is $36\pi$ cm². ## 2. What is a sector? A **sector** is a slice of a circle, made from two radii and the curved edge between them. - An **arc** is part of the circumference of a circle. - The **central angle** is the angle at the centre of the circle. - A **minor sector** is less than half a circle. - A **major sector** is more than half a circle. A full turn is 360°. So a sector with angle $\theta$ is this fraction of the full circle: $$ \frac{\theta}{360} $$ The same fraction, $\frac{\theta}{360}$, is used for both sector area and arc length. 1. A sector has angle 120°. Write the angle over 360: $$ \frac{120}{360} $$ 2. Simplify the fraction: $$ \frac{120}{360}=\frac{1}{3} $$ 3. So a 120° sector is one third of the full circle. ## 3. Area of a sector To find the area of a sector, take the same fraction of the full circle area. $$ \text{sector area}=\frac{\theta}{360}\times \pi r^2 $$ Here, $r$ is the radius and $\theta$ is the angle at the centre. 1. A sector has radius 8 cm and angle 135°. Use the formula: $$ A=\frac{135}{360}\times \pi \times 8^2 $$ 2. Simplify and calculate: $$ A=24\pi \approx 75.398 $$ 3. Round to 3 significant figures. 4. The area is 75.4 cm². If the sector angle is 220°, it is a major sector. Use $\frac{220}{360}$ unless the question asks for the smaller remaining part. ## 4. Arc length The **arc length** is the length of the curved edge of the sector. To find it, take the same fraction of the full circumference. $$ \text{arc length}=\frac{\theta}{360}\times 2\pi r $$ 1. A sector has radius 10 cm and angle 108°. Substitute into the arc length formula: $$ L=\frac{108}{360}\times 2\pi \times 10 $$ 2. Simplify: $$ L=\frac{3}{10}\times 20\pi=6\pi $$ 3. The arc length is $6\pi$ cm. If the question asks for an answer “in terms of π”, leave π in your answer instead of using a decimal. ## 5. Perimeter of a sector The **perimeter** is the total distance around the outside of a shape. For a sector, the outside is made from: - one arc - two radii So: $$ \text{perimeter}=\text{arc length}+2r $$ 1. A sector has radius 4.8 cm and angle 75°. First find the arc length: $$ L=\frac{75}{360}\times 2\pi \times 4.8=2\pi $$ 2. Add the two radii: $$ P=2\pi+4.8+4.8 $$ 3. Calculate the decimal value: $$ P\approx 15.883 $$ 4. The perimeter is 15.9 cm to 3 significant figures. For perimeter questions, do not stop after finding the arc length. You must add both straight sides as well. ## 6. Shaded regions with semicircles A **semicircle** is half a circle. If you are given its diameter, halve it to get the radius. For shaded area questions, use: $$ \text{shaded area}=\text{total area}-\text{unshaded area} $$ 1. A rectangle is 14 cm by 10 cm. Its area is: $$ 14\times 10=140 $$ 2. A semicircle inside it has diameter 10 cm, so its radius is 5 cm. Find the semicircle area: $$ \frac