Year 7 Maths: Parts of a circle

Introduction

What is a Circle?

A circle is a two-dimensional shape where all points are equidistant from a central point, known as the centre. Circles are important in geometry and have several key components. Understanding the different parts of a circle is crucial for solving problems related to circles, such as calculating areas, perimeters, and angles.

Key Parts of a Circle

1. Radius ($r$): The distance from the centre of the circle to any point on the circumference.
2. Diameter ($d$): The distance across the circle, passing through the centre. It is twice the radius:
   $$d = 2r$$
3. Circumference: The distance around the edge of the circle. It is calculated using the formula:
   $$C = 2\pi r$$
4. Arc: A part of the circumference of the circle.
5. Sector: The area of a circle that looks like a ‘slice of pie,’ bounded by two radii and an arc.
6. Chord: A straight line joining two points on the circumference of a circle, but not passing through the centre.
7. Tangent: A straight line that touches the circle at exactly one point, called the point of tangency.
8. Segment: A region inside the circle, created by a chord and the corresponding arc.

Question Sets

Easy Level Questions

Q1-Q10: Basic Definitions and Identifications

1. What is the radius of a circle with a diameter of 10 cm?
2. If the radius of a circle is 5 cm, what is its diameter?
3. Identify the centre of the circle in the diagram.
4. What part of a circle is the distance around its edge?
5. What do you call a line segment that passes through the centre of the circle and connects two points on the circumference?
6. How many radii make up a diameter?
7. Name the part of a circle that is a straight line touching the circle at exactly one point.
8. If a circle has a radius of 7 cm, what is its circumference?
9. What do you call a portion of the circumference of a circle?
10. What is the relationship between the diameter and the radius of a circle?

Q11-Q20: Simple Calculations and Measurements

11. Calculate the diameter of a circle with a radius of 8 cm.
12. What is the radius of a circle if its diameter is 16 cm?
13. If the diameter of a circle is 20 cm, what is the radius?
14. What is the circumference of a circle with a radius of 6 cm?
15. A chord is drawn in a circle. What do you call the area between the chord and the arc it cuts off?
16. Calculate the circumference of a circle with a diameter of 12 cm.
17. A sector has an angle of 90°. What fraction of the circle does the sector represent?
18. Identify the arc in the diagram that is less than half the circumference.
19. How many degrees are in a full circle?
20. What is the name of the region inside a circle that is enclosed by two radii and an arc?

Medium Level Questions

Q1-Q10: Intermediate Concepts and Calculations

1. Calculate the area of a circle with a radius of 7 cm.
2. What is the length of the arc subtended by an angle of 60° in a circle with a radius of 10 cm?
3. Find the radius of a circle with an area of 78.5 cm².
4. What is the length of the diameter of a circle with an area of 154 cm²?
5. Calculate the area of a sector with an angle of 45° in a circle with a radius of 9 cm.
6. A circle has a circumference of 62.8 cm. What is its radius?
7. What is the area of a segment formed by a chord in a circle with a radius of 10 cm and a central angle of 120°?
8. Calculate the length of a chord that is 8 cm away from the centre of a circle with a radius of 10 cm.
9. Find the circumference of a circle with an area of 314 cm².
10. A circle has a radius of 12 cm. Calculate the area of the circle and the length of an arc subtended by an angle of 90°.

Q11-Q20: Application and Problem Solving

11. A sector in a circle has an angle of 60°. If the radius of the circle is 8 cm, calculate the area of the sector.
12. Calculate the length of a tangent to a circle from a point 10 cm away from the centre, if the radius of the circle is 6 cm.
13. A chord is drawn 5 cm away from the centre of a circle with a radius of 13 cm. Calculate the length of the chord.
14. A segment is formed by a chord that subtends an angle of 120° at the centre of a circle with a radius of 15 cm. Calculate the area of the segment.
15. A circle has a circumference of 31.4 cm. What is the radius?
16. Calculate the area of a sector with a radius of 10 cm and an angle of 120°.
17. A chord is 6 cm away from the centre of a circle with a radius of 10 cm. Calculate the length of the chord.
18. What is the area of a circle with a circumference of 44 cm?
19. If the diameter of a circle is doubled, how does its area change?
20. Find the length of an arc in a circle with a radius of 14 cm and a central angle of 120°.

Hard Level Questions

Q1-Q10: Advanced Applications and Problem Solving

1. The radius of a circle is 9 cm. Calculate the area of a sector with an angle of 60°.
2. A circle has a diameter of 20 cm. Calculate the area of a segment formed by a chord that subtends an angle of 90° at the centre.
3. Calculate the length of a tangent drawn from a point 12 cm away from the centre of a circle with a radius of 7 cm.
4. What is the area of a segment formed by a chord in a circle with a radius of 12 cm and a central angle of 150°?
5. Calculate the length of the arc subtended by an angle of 75° in a circle with a radius of 14 cm.
6. A circle has an area of 201 cm². Calculate its diameter.
7. The radius of a circle is increased by 50%. How does this affect the area of the circle?
8. Find the area of a sector with an angle of 135° in a circle with a radius of 10 cm.
9. Calculate the length of the chord in a circle with a radius of 16 cm, where the chord is 9 cm away from the centre.
10. A chord subtends an angle of 140° at the centre of a circle with a radius of 18 cm. Calculate the area of the segment formed.

Q11-Q20: Challenging Calculations

11. Calculate the length of the arc subtended by an angle of 80° in a circle with a diameter of 30 cm.
12. A sector has an angle of 150° in a circle with a radius of 8 cm. Find the area of the sector.
13. A circle has a circumference of 75.36 cm. Calculate its radius and area.
14. Find the area of a segment formed by a chord that subtends an angle of 100° in a circle with a radius of 20 cm.
15. A chord is 8 cm away from the centre of a circle with a radius of 12 cm. Calculate the length of the chord.
16. Calculate the area of a sector with a radius of 14 cm and an angle of 75°.
17. Find the radius of a circle with an area of 500 cm².
18. A circle has a diameter of 24 cm. Calculate the length of an arc that subtends an angle of 110° at the centre.
19. If the radius of a circle is doubled, by what factor does its circumference increase?
20. A chord in a circle with a radius of 25 cm is 10 cm away from the centre. Calculate the length of the chord.

Answers and Explanations

Easy Level

Q1-Q10: Basic Definitions and Identifications

1. Answer:
   $$r = 5 \, \text{cm}$$
   Explanation: The radius is half of the diameter.
   $$r = \frac{d}{2} = \frac{10}{2} = 5 \, \text{cm}$$ .
2. Answer:
   $$d = 10 \, \text{cm}$$
   Explanation: The diameter is twice the radius.
   $$d = 2r = 2 \times 5 = 10 \, \text{cm}$$ .
3. Answer:
   The centre is the point at the exact middle of the circle.
   Explanation: The centre is equidistant from all points on the circumference of the circle.
4. Answer:
   The circumference is the distance around the edge of the circle.
   Explanation: The circumference is the perimeter of the circle.
5. Answer:
   Diameter
   Explanation: The diameter is a straight line passing through the centre of the circle and connecting two points on the circumference.
6. Answer:
   Two radii make up a diameter.
   Explanation: $$d = 2r$$ , meaning the diameter is twice the radius.
7. Answer:
   Tangent
   Explanation: A tangent is a straight line that touches the circle at exactly one point.
8. Answer:
   $$C = 2\pi \times 7 = 43.96 \, \text{cm}$$
   Explanation: The formula for the circumference of a circle is $$C = 2\pi r$$ . Substituting the radius as 7 cm gives the answer.
9. Answer:
   Arc
   Explanation: An arc is part of the circumference of the circle, which spans between two points on the edge of the circle.
10. Answer:
    The diameter is twice the radius.
    $$d = 2r$$

Q11-Q20: Simple Calculations and Measurements

11. Answer:
    $$d = 16 \, \text{cm}$$
    Explanation: The diameter is twice the radius.
    $$d = 2 \times 8 = 16 \, \text{cm}$$ .
12. Answer:
    $$r = 8 \, \text{cm}$$
    Explanation: The radius is half of the diameter.
    $$r = \frac{16}{2} = 8 \, \text{cm}$$ .
13. Answer:
    $$r = 10 \, \text{cm}$$
    Explanation: The radius is half of the diameter.
    $$r = \frac{20}{2} = 10 \, \text{cm}$$ .
14. Answer:
    $$C = 2\pi \times 6 = 37.7 \, \text{cm}$$
    Explanation: The formula for circumference is $$C = 2\pi r$$ .
15. Answer:
    Segment
    Explanation: A segment is the area between a chord and the arc it encloses.
16. Answer:
    $$C = \pi \times d = \pi \times 12 = 37.7 \, \text{cm}$$
    Explanation: The formula for circumference using the diameter is $$C = \pi d$$ .
17. Answer:
    The sector represents a quarter of the circle.
    Explanation: A 90° sector represents 1/4th of the total circle (360°).
18. Answer:
    Minor Arc
    Explanation: An arc that is less than half of the circle’s circumference is called a minor arc.
19. Answer:
    $$360°$$
    Explanation: A full circle always measures 360°.
20. Answer:
    Sector
    Explanation: A sector is the area enclosed by two radii and the arc between them.

Medium Level

Q1-Q10: Intermediate Concepts and Calculations

1. Answer:
   $$A = \pi r^2 = \pi \times 7^2 = 153.94 \, \text{cm}^2$$
   Explanation: The formula for the area of a circle is $$A = \pi r^2$$ .
2. Answer:
   $$L = \frac{\theta}{360} \times 2\pi r = \frac{60}{360} \times 2\pi \times 10 = 10.47 \, \text{cm}$$
   Explanation: The length of an arc is given by the formula $$L = \frac{\theta}{360} \times 2\pi r$$ .
3. Answer:
   $$r = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{78.5}{\pi}} = 5 \, \text{cm}$$
   Explanation: Rearranging the area formula to find the radius:
   $$r = \sqrt{\frac{A}{\pi}}$$ .
4. Answer:
   $$d = 14 \, \text{cm}$$
   Explanation: The formula for the area of a circle is $$A = \pi r^2$$ . Solving for $$r$$ , then doubling it gives the diameter.
5. Answer:
   $$A_{\text{sector}} = \frac{45}{360} \times \pi r^2 = \frac{45}{360} \times \pi \times 9^2 = 31.8 \, \text{cm}^2$$
   Explanation: The area of a sector is given by $$A_{\text{sector}} = \frac{\theta}{360} \times \pi r^2$$ .
6. Answer:
   $$C = 2\pi r = \frac{62.8}{2\pi} = 10 \, \text{cm}$$
   Explanation: Rearranging the circumference formula to solve for $$r$$ .
7. Answer:
   Complex formula for a segment
   Explanation: Finding the area of a segment requires using the formula for the area of a sector minus the area of the triangular portion.
8. Answer:
   Pythagoras’ Theorem
   Explanation: Use the Pythagorean Theorem to solve for the chord length:
   $$c = 2\sqrt{r^2 – d^2}$$ , where $$d$$ is the distance from the centre.
9. Answer:
   $$C = 2\pi r = 2\pi \times \sqrt{\frac{314}{\pi}} = 62.8 \, \text{cm}$$
   Explanation: First, find the radius from the area, then use it to calculate the circumference.
10. Answer:
    $$A = \pi r^2 = \pi \times 12^2 = 452.4 \, \text{cm}^2$$
    $$L_{\text{arc}} = \frac{90}{360} \times 2\pi r = \frac{90}{360} \times 2\pi \times 12 = 18.85 \, \text{cm}$$
    Explanation: First, calculate the area, then the arc length using the formula for a sector.

Q11-Q20: Application and Problem Solving

11. Answer:
    $$A_{\text{sector}} = \frac{60}{360} \times \pi r^2 = \frac{60}{360} \times \pi \times 8^2 = 33.51 \, \text{cm}^2$$
    Explanation: The area of a sector is given by $$A_{\text{sector}} = \frac{\theta}{360} \times \pi r^2$$ .
12. Answer:
    $$L = \sqrt{10^2 – 6^2} = 8 \, \text{cm}$$
    Explanation: Using Pythagoras’ Theorem to find the length of the tangent.
13. Answer:
    $$L = 2\sqrt{r^2 – d^2} = 2\sqrt{13^2 – 5^2} = 24 \, \text{cm}$$
    Explanation: Use the chord length formula:
    $$L = 2\sqrt{r^2 – d^2}$$ .
14. Answer:
    Segment formula
    Explanation: This requires using both the area of the sector and subtracting the area of the triangle formed.
15. Answer:
    $$r = \frac{31.4}{2\pi} = 5 \, \text{cm}$$
    Explanation: Rearrange the circumference formula to solve for $$r$$ .
16. Answer:
    $$A_{\text{sector}} = \frac{120}{360} \times \pi \times 10^2 = 104.72 \, \text{cm}^2$$
    Explanation: The area of a