# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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Solution. The angle required is angle x in figure b:

x = 34Â° Angles on the same arc are equal The coachâ€™s shooting angle is 34Â°, the same as Williamâ€™s.

ANNULUS 35

Example. In figure c, AB is parallel to CD, indicated by the arrows. The straight lines AC and BD intersect at point E. Prove that triangle ABE is an isosceles triangle.

Solution. Write

Angle ACD = angle ABD Angles on the same arc are equal

Angle ACD = angle BAC Alternate angles are equal

Angle ABD = angle BAC Both angles are equal to angle ACD

Therefore, triangle ABE is isosceles, because its base angles are equal.

References: Alternate Angles, Circle Geometry Theorems, Geometry Theorems, Isosceles Triangle, Subtend.

ANNULUS

This is the region between two concentric circles, and is drawn shaded in figure a. Another name for an annulus is a ring. If the radii of the circles are R and r, then the area of the annulus is given as follows:

A = TtR1 â€” 7tr2 Formula for the area of a circle is A = ttR2

A = tt(R2 â€” r2) Factoring, since tt is a common factor

A = tt(R â€” r)(R f r) Factoring, using the difference of two squares

36 ARC

Example. Find the volume of plastic required to make a plastic pipe (figure b) which is 20 cm long and whose inner and outer radii of the circular ends are 3 and 4 centimeters, respectively.

Solution. Write

Volume = area of annulus x length

Volume = tt(R â€” r) (R + r) x length Volume = Tt(4 - 3) (4 + 3) x 20

Volume = 439.8 cm3 (1 dp)

References: Concentric Circles, Factoring, Ms

APPROXIMATION

Reference: Accuracy.

ARC

An arc is a section, or part, of a curve. It can also be a section of a line graph. In the figure the arc AB is part of a circle.

A

When the arc of a circle is less than a semicircle, it is called a minor arc, and when the arc is greater than a semicircle, it is called a major arc. In the figure, the arc AB of the circle is a minor arc because it is less than a semicircle.

Volume of a prism = area of cross section x length

Using formula for area of annulus

Substituting R = 4, r = 3, and length = 20

Using calculator

References: Line Graph, Networks.

ARC LENGTH 37

ARC LENGTH

The length of an arc of a circle can be calculated as in the following example.

Example. Amanda is schooling up her horse George for a show. He is trotting in a circle at the end of a rope held by Amanda, who is at point A in the figure. The length of the rope is 10 meters.

(a) As Amanda turns through an angle of 120Â° find the distance trotted by George along the arc of the circle from G\ to G2.

(b) If George trots a distance of 20 meters, through what angle has the rope turned?

Solution, (a) The circumference C of the whole circle is given by the formula C = 7t x diameter

C = 7t x 20 Substituting 20 meters for the diameter of the circle

C = 2Q7T

The arc length trotted by George is a fraction of C, and this fraction is 120Â°/ 360Â°, where 120Â° is the angle turned through by the rope, and a full turn is 360Â°.

120

Arc length = x 20jr Multiplying the fraction of the full turn by the

circumference

= 20.9 (1 dp)

The distance trotted along the arc of the circle by George is 20.9 meters.

38 AREA

(b) Suppose the angle turned through by the rope is x. The equation is

x

360

x C = 20

360 x 20

x

c

360 x 20

20jr

jc = 114.6Â° (1 dp)

Making x the subject of the formula

Substituting C = 20jr from earlier Using the calculator

The rope turns through an angle of 114.6Â°. For an alternative method using radians refer to the entry Radian.

References: Arc, Circumference, Radian.

AREA

The area of a surface is a measure of the two-dimensional space it occupies. It is measured in square units, which is written as units2. We can find the area of a shape, say a rectangle, by counting the number of squares that its surface occupies. The area of the rectangle in figure a is found by counting the number of square centimeters it occupies. Square centimeters are abbreviated cm2. The area of this rectangle is 12 cm2. Alternatively, the length of the rectangle, which is 4 cm, and the width, which is 3 cm, can be measured and the area calculated using the formula

Area = length x width A = 4 x 3 = 12

The area of this rectangle is 12 cm2.

1 2 3 4

5 6 7 ! 8 I

9 10 11 12

(a)

For some shapes, like circles, it is difficult to find the area accurately by counting squares, because the process would involve piecing together small parts of the circle to make up whole squares, like a jigsaw. The only satisfactory method is to use a formula to find its area.

AREA 39

Example. Calculate the area of the circle in figure b.

(b)

Solution. We need to know the radius of the circle, which is R = 1.5 cm. We write Area = 7t x 1.52 Formula for the area of a circle is Area = tt x R2

Area = 7.07 (to 2 dp) Using a calculator The area of the circle is 7.07 cm 2.

The units commonly used for area are as follows:

â™¦ Square millimeters, mm2.

â™¦ Square centimeters, cm2.

â™¦ Square meters, m2.

â™¦ Hectares, ha.

â™¦ Square kilometers, km2.

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