Areas Related to Circles

Circles appear everywhere around us, from the face of a clock and the blades of a car wiper to the beam of a lighthouse and a slice of pizza. In this chapter you extend the area (pi r^2) and circumference (2 pi r) of a circle to parts of a circle. A sector is the region enclosed by two radii and their arc, and a segment is the region between a chord and its arc. You will distinguish minor and major sectors and segments using the angle theta at the centre. The key results are the length of an arc, (theta/360) x 2 pi r, and the area of a sector, (theta/360) x pi r^2. The area of a minor segment is the area of the sector minus the area of the triangle formed by the two radii and the chord, and major parts are found by subtracting the minor part from the whole circle. You will apply these results to problems such as the area swept by a minute hand or grazed by a tethered horse.

Detailed Worksheet: Areas Related to Circles Section A - Definitions (10 marks) 1. Define a sector of a circle and a segment of a circle. How is a minor sector different from a major sector? (2 marks) 2. Write the formulae for the length of an arc and the area of a sector of angle theta (in degrees) of a circle of radius r. (2 marks) 3. What is the angle of a semicircle and of a quadrant? Write the area of each in terms of r. (2 marks) 4. Explain why the sum of the areas of a minor sector and its corresponding major sector equals the area of the circle. (2 marks) 5. Define the area of a minor segment in words. Write the expression for the area of a minor segment when the angle at the centre is 90 degrees. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the area of a sector of a circle of radius 21 cm with central angle 60 degrees, and the length of the arc of this sector. (Use pi = 22/7) (3 marks) 7. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes. (Use pi = 22/7) (3 marks) 8. A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of (i) the minor segment (ii) the major sector. (Use pi = 3.14) (3 marks) 9. A horse is tied to a peg at one corner of a square grass field of side 15 m by a rope 5 m long. Find (i) the area of the part of the field in which the horse can graze (ii) the increase in grazing area if the rope were 10 m long instead. (Use pi = 3.14) (3 marks) 10. An umbrella has 8 ribs which are equally spaced. Assuming the umbrella to be a flat circle of radius 45 cm, find the area between two consecutive ribs. (Use pi = 22/7) (3 marks) Section C - Diagrams (10 marks) 11. Draw a circle with centre O and radius r. Mark a chord AB that subtends an angle of 90 degrees at O. Shade and label the minor sector OAPB, the minor segment APB and the major segment AQB. Write the formula for the area of each shaded region. (4 marks) 12. Draw a labelled figure of the sector of a circle swept by a minute hand of length 21 cm in 20 minutes. Mark the central angle, find it and calculate the area of the sector. (Use pi = 22/7) (3 marks) 13. Draw a neat figure showing a chord AB of a circle of radius 15 cm that subtends an angle of 60 degrees at the centre O. Explain why triangle OAB is equilateral and find its area. (Use sqrt(3) = 1.73) (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115 degrees. Find the total area cleaned at each sweep of the blades. Explain how the area cleaned would change if the blade length were doubled while the angle stayed the same, and why. (Use pi = 22/7) (5 marks) 15. A chord of a circle of radius 15 cm subtends an angle of 60 degrees at the centre. Find the areas of the corresponding minor and major segments. A student claims the major segment is simply (300/360) x pi r^2. Identify the error in this reasoning. (Use pi = 3.14 and sqrt(3) = 1.73) (5 marks) 16. A brooch is made with silver wire in the form of a circle of diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors. Find (i) the total length of silver wire required (ii) the area of each sector of the brooch. Explain why all 10 sectors have equal areas. (Use pi = 22/7) (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Use pi = 22/7 or 3.14 as stated in each question. Draw all figures neatly with a compass and shade the required regions.