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
• Solve 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 . See 123. Area of Circles and Parts of Circles
• Understand that angles at the centre of a circle are measured in degrees. See 087e. Arc Length and Sector Perimeter
• Know that a full turn is . See 057. Angle Relationships and Angle Sums
• Be able to find a fraction of a quantity. See 032. Ratios, Fractions and Proportions
• Understand that a sector is part of a circle
• Be able to add or subtract areas when working with composite shapes. See 091e. Area of Composite Shapes

Worked Examples

Worked Example 1

State what a sector is and Identify the fraction of the circle represented by each angle:

a)

b)

c)

Worked Example 2

Find the area of each sector:

a) radius , angle at centre

b) radius , angle at centre

Worked Example 3

Find the area of each sector:

a) radius , angle at centre

b) radius , angle at centre

Worked Example 4

A circle has radius .

a) Find the area of the whole circle

b) Find the area of a sector with angle

Worked Example 5

Find the area of the composite shape:

a) a rectangle of area joined to a sector of area

b) a circle of radius with a sector removed

Worked Example 6

Find the area of the composite shape involving sectors:

a) two identical sectors of radius

b) a semicircle of radius joined to a quadrant of radius

Problems

Problem 1

State what a sector is and identify the fraction of the circle represented by each angle:

a)

b)

c)

Problem 2

Find the area of each sector:

a) radius , angle at centre

b) radius , angle at centre

Problem 3

Find the area of each sector:

a) radius , angle at centre

b) radius , angle at centre

Problem 4

A circle has radius .

a) Find the area of the whole circle

b) Find the area of a sector with angle

Problem 5

Find the area of the composite shape:

a) a rectangle of area joined to a sector of area

b) a circle of radius with a sector removed

Problem 6

Find the area of the composite shape involving sectors:

a) two identical sectors of radius

b) a semicircle of radius joined to a quadrant of radius

Exercises

Understanding and Fluency

Exercise 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 ______

Exercise 2.

State the fraction of the circle for each central angle:

a)

b)

c)

d)

Exercise 3.

Find the area of each sector:

a) radius , angle

b) radius , angle

c) radius , angle

Exercise 4.

Find the area of each sector:

a) radius , angle

b) radius , angle

c) radius , angle

Exercise 5.

A circle has the given radius. Find the area of the whole circle and then the sector area:

a) , sector angle

b) , sector angle

c) , sector angle

Exercise 6.

Find the area of each composite shape:

a) a square of area joined to a sector of area

b) a circle of radius with a sector removed

c) a rectangle of area with a semicircle of radius attached

Exercise 7.

Find the area of each composite shape involving sectors:

a) two sectors of radius

b) a semicircle and a quadrant, each with radius

c) a circle of radius with a sector removed

Exercise 8.

Use the sector formula to find each missing value:

a) sector angle , so multiply the circle area by ______

b) sector angle , so multiply the circle area by ______

c) sector angle , so multiply the circle area by ______

Reasoning

Exercise 9.

Explain why the area of a sector is one quarter of the area of the whole circle.

Exercise 10.

A student says that the area of a sector with angle is found by multiplying the circle area by . Explain the mistake.

Exercise 11.

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

Exercise 12.

Explain why the angle at the centre must be compared with when finding sector area.

Exercise 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

Exercise 14.

A sector has radius and angle at the centre . Find its area.

Exercise 15.

A pizza has radius . One slice is a sector. Find the area of the slice.

Exercise 16.

A circle of radius has a sector removed. Find the area that remains.

Exercise 17.

A garden design is made from a rectangle of area and a semicircle of radius . Find the total area.

Exercise 18.

A quadrant has radius . Find its area.

Exercise 19.

A composite logo is made from two identical sectors of radius and a square of area . 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 instead of
• 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 instead of
• 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

Next: 125e. Nets and Surface Area of Prisms