124e. Sectors and Composite Areas

Learning Intentions

• To know what a sector is
• To understand that a sector’s area can be found by taking a fraction of the area of a circle with the same radius
• find the area of a sector given its radius and the angle at the centre
• find the area of composite shapes involving sectors

Pre-requisite Summary

• Know that a circle has a centre, radius and diameter
• Know that the area of a circle is $A=\pi r^2$
• Understand that angles at the centre of a circle are measured in degrees
• Know that a full turn is ${360}^{\circ }$
• Be able to find a fraction of a quantity
• Understand that a sector is part of a circle
• Be able to add or subtract areas when working with composite shapes

Worked Examples

Worked Example 1

State what a sector is and identify the fraction of the circle represented by each angle:
a) ${90}^{\circ }$
b) ${180}^{\circ }$
c) ${45}^{\circ }$

Worked Example 2

Find the area of each sector:
a) radius $=6\text{ cm}$, angle at centre $={90}^{\circ }$
b) radius $=10\text{ cm}$, angle at centre $={180}^{\circ }$

Worked Example 3

Find the area of each sector:
a) radius $=8\text{ m}$, angle at centre $={120}^{\circ }$
b) radius $=5\text{ cm}$, angle at centre $={45}^{\circ }$

Worked Example 4

A circle has radius $7\text{ cm}$.
a) Find the area of the whole circle
b) Find the area of a sector with angle ${60}^{\circ }$

Worked Example 5

Find the area of the composite shape:
a) a rectangle of area $40{\text{ cm}}^2$ joined to a sector of area $12{\text{ cm}}^2$
b) a circle of radius $6\text{ cm}$ with a ${90}^{\circ }$ sector removed

Worked Example 6

Find the area of the composite shape involving sectors:
a) two identical ${60}^{\circ }$ sectors of radius $9\text{ cm}$
b) a semicircle of radius $4\text{ cm}$ joined to a quadrant of radius $4\text{ cm}$

Problems

Problem 1

State what a sector is and identify the fraction of the circle represented by each angle:
a) ${60}^{\circ }$
b) ${180}^{\circ }$
c) ${30}^{\circ }$

Problem 2

Find the area of each sector:
a) radius $=4\text{ cm}$, angle at centre $={90}^{\circ }$
b) radius $=12\text{ cm}$, angle at centre $={180}^{\circ }$

Problem 3

Find the area of each sector:
a) radius $=9\text{ m}$, angle at centre $={120}^{\circ }$
b) radius $=6\text{ cm}$, angle at centre $={45}^{\circ }$

Problem 4

A circle has radius $5\text{ cm}$.
a) Find the area of the whole circle
b) Find the area of a sector with angle ${72}^{\circ }$

Problem 5

Find the area of the composite shape:
a) a rectangle of area $36{\text{ cm}}^2$ joined to a sector of area $18{\text{ cm}}^2$
b) a circle of radius $8\text{ cm}$ with a ${90}^{\circ }$ sector removed

Problem 6

Find the area of the composite shape involving sectors:
a) two identical ${45}^{\circ }$ sectors of radius $10\text{ cm}$
b) a semicircle of radius $3\text{ cm}$ joined to a quadrant of radius $3\text{ cm}$

Exercises

Understanding and Fluency

1. Complete each statement:
   a) A sector is a ______ of a circle
   b) The area of a sector is found by taking a ______ of the area of the whole circle
   c) A full circle has angle ______ at the centre
   d) A semicircle is a sector with angle ______

2. State the fraction of the circle for each central angle:
   a) ${90}^{\circ }$
   b) ${180}^{\circ }$
   c) ${120}^{\circ }$
   d) ${30}^{\circ }$

3. Find the area of each sector:
   a) radius $=5\text{ cm}$, angle $={90}^{\circ }$
   b) radius $=7\text{ cm}$, angle $={180}^{\circ }$
   c) radius $=6\text{ cm}$, angle $={60}^{\circ }$

4. Find the area of each sector:
   a) radius $=10\text{ cm}$, angle $={45}^{\circ }$
   b) radius $=8\text{ m}$, angle $={120}^{\circ }$
   c) radius $=9\text{ mm}$, angle $={270}^{\circ }$

5. A circle has the given radius. Find the area of the whole circle and then the sector area:
   a) $r=4\text{ cm}$, sector angle $={180}^{\circ }$
   b) $r=12\text{ cm}$, sector angle $={90}^{\circ }$
   c) $r=7\text{ cm}$, sector angle $={60}^{\circ }$

6. Find the area of each composite shape:
   a) a square of area $25{\text{ cm}}^2$ joined to a sector of area $10{\text{ cm}}^2$
   b) a circle of radius $5\text{ cm}$ with a ${90}^{\circ }$ sector removed
   c) a rectangle of area $48{\text{ cm}}^2$ with a semicircle of radius $3\text{ cm}$ attached

7. Find the area of each composite shape involving sectors:
   a) two ${90}^{\circ }$ sectors of radius $6\text{ cm}$
   b) a semicircle and a quadrant, each with radius $8\text{ cm}$
   c) a circle of radius $10\text{ cm}$ with a ${120}^{\circ }$ sector removed

8. Use the sector formula to find each missing value:
   a) sector angle $={90}^{\circ }$, so multiply the circle area by ______
   b) sector angle $={60}^{\circ }$, so multiply the circle area by ______
   c) sector angle $={270}^{\circ }$, so multiply the circle area by ______

Reasoning

9. Explain why the area of a ${90}^{\circ }$ sector is one quarter of the area of the whole circle.

10. A student says that the area of a sector with angle ${60}^{\circ }$ is found by multiplying the circle area by ${\displaystyle \frac{60}{100}}$. Explain the mistake.

11. Noah says that a semicircle is not a sector because it is “too big”. Is he correct? Explain.

12. Explain why the angle at the centre must be compared with ${360}^{\circ }$ when finding sector area.

13. A student finds the area of a composite shape by adding the outside edge lengths instead of adding or subtracting the areas. Describe the error.

Problem-solving

14. A sector has radius $9\text{ cm}$ and angle at the centre ${120}^{\circ }$. Find its area.

15. A pizza has radius $14\text{ cm}$. One slice is a ${45}^{\circ }$ sector. Find the area of the slice.

16. A circle of radius $6\text{ m}$ has a ${90}^{\circ }$ sector removed. Find the area that remains.

17. A garden design is made from a rectangle of area $60{\text{ m}}^2$ and a semicircle of radius $4\text{ m}$. Find the total area.

18. A quadrant has radius $10\text{ cm}$. Find its area.

19. A composite logo is made from two identical ${60}^{\circ }$ sectors of radius $12\text{ cm}$ and a square of area $36{\text{ cm}}^2$. Find the total area.

Potential Misunderstandings

• Students may think a sector is the same as a segment or a semicircle only
• Students may forget that a sector is a fraction of a full circle
• Students may compare the central angle with ${180}^{\circ }$ instead of ${360}^{\circ }$
• Students may use the diameter instead of the radius in the circle area formula
• Students may forget to square the radius when finding the area of the whole circle
• Students may use the wrong fraction for the sector, such as ${\displaystyle \frac{90}{100}}$ instead of ${\displaystyle \frac{90}{360}}$
• Students may confuse area of a sector with circumference or arc length
• Students may add or subtract side lengths instead of areas in composite shapes
• Students may forget to use square units in the final answer