Year 11 Maths: Central Angles

What are Central Angles?

A central angle is an angle formed by two lines (or radii) that meet at the centre of a circle. This angle opens up to the arc of the circle, which is the curved part between the two points where the lines touch the circle.

Visualising Central Angles

Imagine a pizza! The point where the pizza is cut (the centre) creates angles between the slices. Each slice has a central angle. The central angle helps us understand how much of the pizza each slice takes up.

Key Rules

1. Degrees in a Circle: A full circle has 360 degrees.
2. Finding the Central Angle: If you know the size of the arc (the curved part of the circle) and the circle’s radius, you can find the central angle using the formula: $$\text{Central Angle (in degrees)} = \frac{\text{Arc Length}}{\text{Radius}} \times \frac{180}{\pi}$$
3. Relationship with Inscribed Angles: The central angle is always twice the size of any inscribed angle that subtends the same arc.

Tips and Tricks

• Always remember: the total of all central angles in a circle adds up to 360 degrees.
• Use a protractor to measure central angles accurately.
• Practice sketching circles and marking central angles to improve your understanding.

Examples

Example 1: Finding a Central Angle

If an arc of a circle is 10 cm long and the radius is 5 cm, what is the central angle?

Using the formula:

Example 2: Relationship with Inscribed Angles

If the central angle is 80 degrees, what is the size of the inscribed angle that subtends the same arc?

Using the rule:

Practice Questions

Easy Level Questions

1. What is a central angle?
2. How many degrees are in a full circle?
3. If a central angle is 90 degrees, how much of the circle does it represent?
4. What is the relationship between a central angle and an inscribed angle?
5. If the central angle is 60 degrees, what is the inscribed angle?
6. How would you measure a central angle?
7. What formula do you use to find the central angle if you have the arc length and radius?
8. What is the total of all central angles in a circle?
9. If the radius of a circle is 4 cm and the arc length is 12.57 cm, what is the central angle?
10. Is a central angle always less than 180 degrees? Why or why not?

Medium Level Questions

11. A central angle measures 120 degrees. What is the corresponding inscribed angle?
12. If the arc length is 15 cm and the radius is 3 cm, what is the central angle?
13. A pizza is cut into 8 equal slices. What is the measure of each central angle?
14. What is the central angle if the arc length is 10π cm and the radius is 10 cm?
15. If the inscribed angle is 30 degrees, what is the central angle?
16. A circle has a radius of 6 cm. If the central angle is 60 degrees, what is the arc length?
17. How do you calculate the central angle given the arc length and radius?
18. If two central angles in a circle are 70 degrees and 110 degrees, what is the third central angle?
19. If a central angle is 150 degrees, what fraction of the circle does it represent?
20. What happens to the size of a central angle if the radius of the circle increases but the arc length stays the same?

Hard Level Questions

21. Given a circle with a radius of 10 cm, find the central angle if the arc length is 25 cm.
22. A central angle is 270 degrees. What is the corresponding inscribed angle?
23. In a circle, the central angles are in the ratio 2:3:5. What are the measures of each central angle?
24. If the radius of a circle is doubled, how does this affect the central angle for a fixed arc length?
25. A central angle of 45 degrees subtends which arc length in a circle of radius 5 cm?
26. If the central angle is 120 degrees, what would be the arc length in a circle of radius 10 cm?
27. How do you find the arc length if you know the central angle in degrees?
28. A circle has three central angles measuring 30 degrees, 60 degrees, and x degrees. What is x?
29. If the arc length is equal to the radius, what will be the measure of the central angle in degrees?
30. A central angle is 180 degrees. Describe what shape it creates on the circle.

Answers and Explanations

Easy Level Answers

1. A central angle is the angle formed at the center of a circle by two radii.
2. 360 degrees are in a full circle.
3. A 90-degree angle represents one-quarter of the circle.
4. The central angle is always twice the inscribed angle that subtends the same arc.
5. If the central angle is 60 degrees, the inscribed angle is 30 degrees.
6. You can measure a central angle using a protractor.
7. The formula is: $$\text{Central Angle} = \frac{\text{Arc Length}}{\text{Radius}} \times \frac{180}{\pi}$$
8. The total of all central angles in a circle is 360 degrees.
9. The central angle is approximately 76.96 degrees.
10. No, a central angle can be greater than 180 degrees, representing more than half the circle.

Medium Level Answers

11. The corresponding inscribed angle is 60 degrees.
12. The central angle is 90 degrees.
13. Each central angle measures 45 degrees.
14. The central angle is 180 degrees.
15. The central angle is 60 degrees.
16. The arc length is 6π cm.
17. You divide the arc length by the radius and multiply by (\frac{180}{\pi}).
18. The third central angle is 180 degrees.
19. It represents (\frac{150}{360} = \frac{5}{12}) of the circle.
20. The size of the central angle will decrease because the angle depends on the radius and arc length.

Hard Level Answers

21. The central angle is 90 degrees.
22. The corresponding inscribed angle is 135 degrees.
23. The measures are 60 degrees, 90 degrees, and 150 degrees.
24. The central angle remains the same for a fixed arc length.
25. The arc length is 3.93 cm (approximately).
26. The arc length is (\frac{20\pi}{3}) cm.
27. The formula for arc length is: $$\text{Arc Length} = \text{Radius} \times \text{Central Angle (in radians)}$$
28. x = 90 degrees.
29. The central angle would be 90 degrees.
30. It creates a semicircle.

Feel free to ask any questions if you need further clarification on central angles!