1. The area of a park on a map is 500 mm2. If the scale of the map is 1 to 40 000 determine the true area of the park in hectares (1 hectare=104m2).
2. A model of a boiler is made having an overall height of 75 mm corresponding to an overall height of the actual boiler of 6 m. If the area of metal required for the model
Areas of plane figures 173
is 12 500 mm2determine, in square metres, the area of metal required for the actual boiler.
3. The scale of an Ordinance Survey map is 1:2500. A cir- cular sports field has a diameter of 8 cm on the map.
Calculate its area in hectares, giving your answer correct to 3 significant figures. (1 hectare=104m2)
Assignment 10
This assignment covers the material contained in Chap- ters 21 and 22. The marks for each question are shown in brackets at the end of each question.
1. Change the following Cartesian co-ordinates into polar co-ordinates, correct to 2 decimal places, in both degrees and in radians:
(a) (−2.3, 5.4) (b) (7.6,−9.2) (10) 2. Change the following polar co-ordinates into Cartesian
co-ordinates, correct to 3 decimal places:
(a) (6.5, 132◦) (b) (3, 3 rad) (6) 3. A rectangular park measures 150 m by 70 m. A 2 m flower border is constructed round the two longer sides and one short side. A circular fish pond of diameter 15 m is in the centre of the park and the remainder of the park is grass. Calculate, correct to the nearest square metre, the area of (a) the fish pond, (b) the flower borders, (c) the
grass. (10)
4. A swimming pool is 55 m long and 10 m wide. The per- pendicular depth at the deep end is 5 m and at the shallow end is 1.5 m, the slope from one end to the other being uniform. The inside of the pool needs two coats of a protective paint before it is filled with water. Determine how many litres of paint will be needed if 1 litre covers
10 m2. (7)
5. Find the area of an equilateral triangle of side
20.0 cm. (4)
6. A steel template is of the shape shown in Fig. A10.1, the circular area being removed. Determine the area of the template, in square centimetres, correct to 1 decimal
place. (7)
30 mm
45 mm
130 mm
70 mm
70 mm 150 mm
60 mm 50 mm
dia.
30 mm
Fig. A10.1
7. The area of a plot of land on a map is 400 mm2. If the scale of the map is 1 to 50 000, determine the true area of the land in hectares (1 hectare=104m2). (3) 8. Determine the shaded area in Fig. A10.2, correct to the
nearest square centimetre. (3)
20 cm
2 cm
Fig. A10.2
23
The circle
23.1 Introduction
Acircleis a plain figure enclosed by a curved line, every point on which is equidistant from a point within, called thecentre.
23.2 Properties of circles
(i) The distance from the centre to the curve is called the radius,r, of the circle (seeOPin Fig. 23.1).
P R
C B
A O
Q
Fig. 23.1
(ii) The boundary of a circle is called thecircumference,c.
(iii) Any straight line passing through the centre and touching the circumference at each end is called thediameter,d (seeQRin Fig. 23.1). Thusd=2r
(iv) The ratio circumference
diameter =a constant for any circle.
This constant is denoted by the Greek letterπ(pronounced
‘pie’), whereπ=3.14159, correct to 5 decimal places.
Hencec/d=πorc=πdorc=2πr (v) Asemicircleis one half of the whole circle.
(vi) Aquadrantis one quarter of a whole circle.
(vii) Atangentto a circle is a straight line which meets the circle in one point only and does not cut the circle when produced.AC in Fig. 23.1 is a tangent to the circle since it touches the curve at pointBonly. If radiusOBis drawn, then angleABOis a right angle.
(viii) Asectorof a circle is the part of a circle between radii (for example, the portionOXY of Fig. 23.2 is a sector). If a sector is less than a semicircle it is called aminor sector, if greater than a semicircle it is called amajor sector.
S R
T Y O
X
Fig. 23.2
(ix) Achordof a circle is any straight line which divides the circle into two parts and is terminated at each end by the circumference.ST, in Fig. 23.2 is a chord.
(x) Asegmentis the name given to the parts into which a circle is divided by a chord. If the segment is less than a semicircle it is called aminor segment(see shaded area in Fig. 23.2). If the segment is greater than a semicircle it is called amajor segment(see the unshaded area in Fig. 23.2).
(xi) Anarcis a portion of the circumference of a circle. The distanceSRT in Fig. 23.2 is called aminor arcand the distanceSXYTis called amajor arc.
(xii) The angle at the centre of a circle, subtended by an arc, is double the angle at the circumference sub- tended by the same arc. With reference to Fig. 23.3, AngleAOC=2×angleABC.
The circle 175
A P
C O
Q B
Fig. 23.3
(xiii) The angle in a semicircle is a right angle (see angleBQP in Fig. 23.3).
Problem 1. Find the circumference of a circle of radius 12.0 cm.
Circumference,
c=2×π×radius=2πr=2π(12.0)=75.40 cm
Problem 2. If the diameter of a circle is 75 mm, find its circumference.
Circumference,
c=π×diameter=πd=π(75)=235.6 mm
Problem 3. Determine the radius of a circle if its perimeter is 112 cm.
Perimeter=circumference,c=2πr Henceradiusr= c
2π=112
2π =17.83 cm
Problem 4. In Fig. 23.4,ABis a tangent to the circle atB.
If the circle radius is 40 mm andAB=150 mm, calculate the lengthAO.
A
B O
r
Fig. 23.4
A tangent to a circle is at right angles to a radius drawn from the point of contact, i.e.ABO=90◦. Hence, using Pythagoras’
theorem:
AO2=AB2+OB2 from which, AO=
AB2+OB2=
1502+402
=155.2 mm
Now try the following exercise