Areas Related to Circles · Learner pages

Lesson 1 · Find a sector area as a fraction

Use the central angle fraction for area and arc length.

Name or initials: ____________   Date: ____________

A full turn is 360°. A full circle has area πr² and circumference 2πr.

1. Identify the fraction of a turn. A 90° sector is 90/360 = 1/4 of a circle.
2. Use the circle area. For radius 14 cm, full area is 196π cm². One quarter is 49π cm².
3. Keep arc and area separate. Its arc length is one quarter of 2π × 14 = 28π, hence 7π cm. Area uses square units; arc length uses length units.
4. Evaluate only when requested. If π = 22/7 is specified, area is 154 cm² and arc length is 22 cm. Otherwise keep the exact π forms.

Worked example: A 60° sector of radius 12 cm has area (1/6) × 144π = 24π cm² and arc length (1/6) × 24π = 4π cm. Its perimeter is 24 + 4π cm.

1. A circle has radius 6 cm. What is the exact area of a 90° sector?

   1. 9π cm² · 2. 10π cm² · 3. 36π cm²

2. A circle has radius 8 cm. What is the exact area of a 90° sector?

   1. 17π cm² · 2. 64π cm² · 3. 16π cm²

3. A circle has radius 12 cm. What is the exact area of a 60° sector?

   1. 144π cm² · 2. 24π cm² · 3. 25π cm²

Discuss: Choose a radius and compare 60°, 120° and 180° sectors. Predict the area ratio before calculating.

Lesson 2 · Subtract a triangle to get a segment

Distinguish a circular segment from a sector.

Name or initials: ____________   Date: ____________

A chord joins two points on a circle. A segment lies between a chord and its arc.

1. Identify the boundaries. A 90° sector of radius 14 contains a right triangle formed by the two radii and the chord.
2. Find the sector area. The sector is one quarter of the circle: area = 49π cm².
3. Find the triangle area. The two perpendicular radii are legs of length 14, so triangle area = ½ × 14 × 14 = 98 cm².
4. Subtract the triangle. Minor segment area = 49π − 98 cm². With π = 22/7, it is 154 − 98 = 56 cm².

Worked example: For radius 6 and a 90° central angle, sector area = 9π and triangle area = 18. The segment area is 9π − 18 square units, which is positive because π > 2.

1. Find the minor segment area cut off by a 90° central angle in a circle of radius 2.

   1. 1π − 2 · 2. 1π + 2 · 3. 1π

2. Find the minor segment area cut off by a 90° central angle in a circle of radius 4.

   1. 4π + 8 · 2. 4π · 3. 4π − 8

3. Find the minor segment area cut off by a 90° central angle in a circle of radius 6.

   1. 9π · 2. 9π − 18 · 3. 9π + 18

Discuss: On the same circle draw the sector, its triangle and its segment using different shading. Explain how their areas fit together.

Lesson 3 · Combine circular regions without double counting

Choose addition or subtraction from a clearly partitioned diagram.

Name or initials: ____________   Date: ____________

The area of disjoint parts adds; overlapping area must not be counted twice.

1. Identify the outer boundary. A circular path lies between radii 7 m and 5 m with the same centre. The outer disk has area 49π.
2. Identify what is removed. The inner disk, area 25π, is not part of the path.
3. Subtract the areas. Path area = 49π − 25π = 24π m². Squaring the width 2 m would not give the ring area.
4. Check a different partition. For a rectangle with a semicircle attached along one side, add the rectangle area and half a disk only when their interiors do not overlap.

Worked example: A 10 m by 6 m rectangle has a semicircle attached externally along its 6 m side. Radius = 3 m. Total area = 60 + ½π×9 = 60 + 4.5π m².

1. Two concentric circles have radii 5 cm and 3 cm. Find the ring area.

   1. 16π cm² · 2. 4π cm² · 3. 34π cm²

2. Two concentric circles have radii 7 cm and 4 cm. Find the ring area.

   1. 9π cm² · 2. 65π cm² · 3. 33π cm²

3. Two concentric circles have radii 10 cm and 6 cm. Find the ring area.

   1. 136π cm² · 2. 64π cm² · 3. 16π cm²

Discuss: Draw a playground consisting of a rectangle and two external semicircles on opposite ends. Express its area using the rectangle dimensions.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. A circle has radius 9 cm. What is the exact area of a 120° sector?

   28π cm² / 81π cm² / 27π cm²

2. A circle has radius 14 cm. What is the exact area of a 90° sector?

   196π cm² / 49π cm² / 50π cm²

3. Find the minor segment area cut off by a 90° central angle in a circle of radius 10.

   25π + 50 / 25π / 25π − 50

4. Find the minor segment area cut off by a 90° central angle in a circle of radius 12.

   36π / 36π − 72 / 36π + 72

5. Two concentric circles have radii 8 cm and 5 cm. Find the ring area.

   9π cm² / 89π cm² / 39π cm²

6. Two concentric circles have radii 9 cm and 7 cm. Find the ring area.

   130π cm² / 32π cm² / 4π cm²

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. A circle has radius 15 cm. What is the exact area of a 120° sector?

   75π cm² / 76π cm² / 225π cm²

2. A circle has radius 4 cm. What is the exact area of a 180° sector?

   9π cm² / 16π cm² / 8π cm²

3. Find the minor segment area cut off by a 90° central angle in a circle of radius 14.

   49π − 98 / 49π + 98 / 49π

4. Find the minor segment area cut off by a 90° central angle in a circle of radius 16.

   64π + 128 / 64π / 64π − 128

5. Two concentric circles have radii 12 cm and 8 cm. Find the ring area.

   80π cm² / 16π cm² / 208π cm²

6. Two concentric circles have radii 11 cm and 9 cm. Find the ring area.

   4π cm² / 202π cm² / 40π cm²