Draft copy. Circles, cylinders and prisms. Circles
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1 12 Circles, cylinders and prisms You are familiar with formulae for area and volume of some plane shapes and solids. In this chapter you will build on what you learnt in Mathematics for Common Entrance Book Two about the special formulae for circles. Circles Can you recall the names of the parts of a circle? The circumference is the name for the line round the outside of the circle. The perimeter of the circle is also called the circumference. The radius is the straight line from the centre of the circle to the circumference. The distance from the centre to the circumference is also called the radius. The diameter is the straight line between two points on the circumference, through the centre. The distance from two points on the circumference, through the centre, is also called the diameter. Diameter = 2 radius Do you remember the formulae for calculating the circumference and area of a circle? Circumference C = πd or C = 2πr Area A = πr 2 The symbol π is the Greek letter pi, the constant that describes the ratio of diameter to circumference. Find the π on your calculator. It should give you You should use the full value of π when using a calculator, if not you can use 3.14 or the fraction 22 7 Circumference Diameter Radius 189
2 12 Circles, cylinders and prisms Example Calculate the circumference of a circle of diameter 14 cm. d = 14 cm C = πd = π 14 = = 44.0 cm (to 3 s.f.) Exercise cm Use the π button on your calculator for this exercise and give rounded answers correct to 3 s.f. 1 Calculate: (i) the circumference (ii) the area of each circle. (a) diameter 8 cm (b) radius 31 cm (c) diameter 1.2 m (d) radius 4.5 m (e) diameter 50 cm (f) radius 12 m 2 My bicycle has wheels of diameter 95 cm. How far will my bicycle move forward in one turn of the wheel? 3 The diameter of this table mat is 14 cm. Work out its area and circumference. 4 I have a glass with a base diameter of 8 cm. Find the area of the base of my glass. 5 The radius of my frisbee is 9 cm. What is the circumference? 6 I bought a large sheet of plywood measuring 1.25 m by 2.5 m. I cut from it the largest circle that I could. What area of plywood is left? As usual, it is a good idea to start with a quick sketch, to make sure that you have the correct dimensions and do not confuse radius and diameter.
3 Fractions of circles Perimeter and area of fractions of circles Some of the shapes you will meet may not be whole circles but parts of circles. A half-circle is a semi-circle. A quarter-circle is a quadrant. To find the perimeter of a semi-circle you have to first find half the circumference and then add on the length of the straight side. Example Calculate: (i) the area (ii) the perimeter of this semi-circle. d = 14 cm r = 7 cm (i) Remember that this is half of a circle, so you must halve the area formula. Area = 1 2 πr2 = 1 2 π 72 = 1 2 π 49 = = 77.0 cm 2 (to 3 s.f.) (ii) Curved length = 1 2 πd = 1 2 π 14 = Here are some more parts of a circle you should know. An arc is a part of the circumference. A sector is a slice formed by two radii and the arc between them. The angle at the centre of a sector, between the two radii, is the sector angle. The straight side is the diameter measuring 14 cm Note you are Perimeter = working with the full = display, not with the rounded answers. = 36.0 cm (to 3 s.f.) 7 cm 50º 191 Fractions of circles
4 12 Circles, cylinders and prisms Exercise 12.2 For question 1 6 use the π button on your calculator and give rounded answers correct to 3 s.f. 1 Calculate the perimeter of this semi-circular carpet cm 3 Calculate the area of this three-quarter circle. 25 cm 2 Calculate the area and perimeter of this quadrant. 14 cm 4 (a) Calculate the distance round the outside of this running track. 40 m 40 m (b) What is the area enclosed by the running track? 5 Find the area of this lily pad cm 6 Look at these shapes. Each is a sector of a circle. Work out the area and perimeter of each sector. (a) 50º 10 cm (b) 225º 35 cm
5 For the next few questions use π = This is the cross-sectional view of the building that won this year s architectural prize. Calculate the area of the cross-section. 8 This is the net of a conical hat for the school play. What length of ribbon do I need to trim the whole outside edge of the net? 9 This is the vertical cross-section through a spinning toy. The top semi-circle has a diameter of 49 mm and the bottom semi-circle a diameter of 98 mm. What is the area of the cross-section? 10 Look at Kim s design. It is made from four semicircles of diameter 14 cm. (a) What is the perimeter of the design? (b) What is the area of the design? (c) Kim draws the design on a piece of card that is 28 cm by 28 cm and then cuts it out. What area of card is left? Calculating the radius and diameter Given the radius or diameter of a circle, you can calculate its perimeter (circumference) or area. It follows that, given the circumference or area, you can calculate the radius or diameter. 14 m 7 m 70 cm 98 mm 49 mm 193 Calculating the radius and diameter
6 12 Circles, cylinders and prisms Example Work out the radius of a circle of area 100 cm 2. Area of circle = πr = πr π = r 2 r 2 = r = = The radius is 5.64 cm (to 3.s.f.). Exercise Calculate the diameter of a circle of circumference 14 cm. 2 Work out the radius of a circle of area 100 cm 2. 3 Calculate the radius of a circle of circumference 12 m. 4 Work out the diameter of a circle of area 250 cm 2. 5 Calculate the radius, in centimetres, of a circle of area 4 m 2. 6 I have a length of wooden trim measuring 2 m. What is the radius of the largest circle that I could trim with it? 7 I have a circular tablecloth of area 5 m 2. What is the radius of the largest table that it could cover? 8 The perimeter of a semi-circle is 20 cm. What is the diameter? More circle problems Some questions need a little more thought. For some of the following problems you may need to use Pythagoras theorem. Be sure to set out your working clearly. Exercise 12.4 Use the π on your calculator for this exercise and give rounded answers correct to 3 s.f. 1 (a) Calculate the diameter of the circle inscribed in a square of side 10 cm. (b) Now work out the length of the diagonal of the square. Always write down the dimensions that you are given first, to make sure that you use the correct formula and substitute the correct value. 10 cm The inscribed circle is inside the square and just touches the sides.
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