Areas Related to Circles · Answer guide

Lesson 1 · Find a sector area as a fraction

Use the central angle fraction for area and arc length.

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

   Answer: 9π cm². Sector area = (90/360) × π × 6² = 9π cm².

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

   Answer: 16π cm². Sector area = (90/360) × π × 8² = 16π cm².

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

   Answer: 24π cm². Sector area = (60/360) × π × 12² = 24π cm².

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

Teaching note: The sector perimeter includes two radii as well as the arc. Arc length alone is not the whole perimeter. Open responses need discussion; they are not automatically marked.

Lesson 2 · Subtract a triangle to get a segment

Distinguish a circular segment from a sector.

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

   Answer: 1π − 2. Segment = sector − triangle. The quarter-circle sector is 1π and the right triangle area is ½×2×2 = 2.

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

   Answer: 4π − 8. Segment = sector − triangle. The quarter-circle sector is 4π and the right triangle area is ½×4×4 = 8.

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

   Answer: 9π − 18. Segment = sector − triangle. The quarter-circle sector is 9π and the right triangle area is ½×6×6 = 18.

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

Teaching note: A segment is not the same region as a sector. Draw the chord before deciding what to subtract. Open responses need discussion; they are not automatically marked.

Lesson 3 · Combine circular regions without double counting

Choose addition or subtraction from a clearly partitioned diagram.

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

   Answer: 16π cm². Ring area = πR² − πr² = π(25 − 9) = 16π cm².

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

   Answer: 33π cm². Ring area = πR² − πr² = π(49 − 16) = 33π cm².

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

   Answer: 64π cm². Ring area = πR² − πr² = π(100 − 36) = 64π cm².

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

Teaching note: Area of a ring is π(R² − r²), not π(R − r)². A radius is half a diameter. Open responses need discussion; they are not automatically marked.

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?

   27π cm². Sector area = (120/360) × π × 9² = 27π cm².

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

   49π cm². Sector area = (90/360) × π × 14² = 49π cm².

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

   25π − 50. Segment = sector − triangle. The quarter-circle sector is 25π and the right triangle area is ½×10×10 = 50.

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

   36π − 72. Segment = sector − triangle. The quarter-circle sector is 36π and the right triangle area is ½×12×12 = 72.

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

   39π cm². Ring area = πR² − πr² = π(64 − 25) = 39π cm².

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

   32π cm². Ring area = πR² − πr² = π(81 − 49) = 32π 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. Sector area = (120/360) × π × 15² = 75π cm².

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

   8π cm². Sector area = (180/360) × π × 4² = 8π cm².

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

   49π − 98. Segment = sector − triangle. The quarter-circle sector is 49π and the right triangle area is ½×14×14 = 98.

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

   64π − 128. Segment = sector − triangle. The quarter-circle sector is 64π and the right triangle area is ½×16×16 = 128.

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

   80π cm². Ring area = πR² − πr² = π(144 − 64) = 80π cm².

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

   40π cm². Ring area = πR² − πr² = π(121 − 81) = 40π cm².