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

ISBN 0-471-15045-2

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The vertices of a triangle are labeled A, B, and C, and the lengths of the sides of the triangle opposite those vertices are a, b, and c, respectively. The perimeter of the triangle is a T b T c. Half the perimeter of the triangle (semiperimeter) is given

by

a + b + c

Heron’s formula for the area of any triangle is

A = y/s(s — a)(s — b)(s — c)

Example. Jane and her husband, David, are planning to buy a plot of land and build a house on it. The site in which they are particularly interested is triangle-shaped due to a fork in the road (see figure). Jane measures the lengths of the three sides of the triangle section, which are marked on the figure. Use Heron’s formula to find the area of the section.

Solution. Write

a + b + c

35 + 50 + 63

s =------------------------- a = 35, b = 50, c = 63, but the lengths

can be allocated differently

s = 74m

HEXAGON 239

Heron’s formula for the area of any triangle is

A = y/s(s — a)(s — b)(s — c)

We now substitute for s, a, b, and c:

A = /74(74 - 35)(74 - 50)(74 - 63)

A = V761904 A = 873 m2

The area of the triangular section is 873 m2.

References: Area, Triangle, Trigonometry.

HEXAGON

A hexagon is a polygon with six sides. Some hexagons are drawn in figure a; the last one is a regular hexagon. The regular hexagon has six sides of equal length, and its six angles are of equal size, which is 120°. It has six axes of symmetry, and the order of rotational symmetry is six (see figure b).

Evaluating with the calculator To nearest whole number

(b) (c)

The regular hexagon will tessellate, which means it will form a tiling pattern or mosaic. It is this tiling pattern that bees use to build their honeycomb. The regular hexagon is made up of six equilateral triangles (see figure c).

Thousands of years ago the Babylonians knew how to make a regular hexagon in a circle, in the following way:

240 HEXOMINO

Example. Construct a regular hexagon of side 2 cm.

/4

Solution. Using compasses, draw a circle of radius 2 cm. Starting at a point on the circumference, say point A, step off around the circle small arcs with the compasses still of radius 2 cm (see figure d). There will be exactly six steps in getting back to point A again. With a ruler, join up the six points where the arcs cut the circle. There are now six chords equal in length to the radius of the circle. This makes a regular hexagon of side 2 cm.

References: Equilateral Triangle, Polygon, Regular Polygon, Symmetry, Tessellations. HEXAGRAM

If all the sides of a regular hexagon, which is drawn shaded in the figure, are extended, a six-pointed star results which is called a hexagram.

References: Hexagon, Regular Polygon, Star Polygons.

HEXAHEDRON

A hexahedron is a regular polyhedron with six faces, and is another name for a cube. References: Cube, Polyhedron, Regular Polyhedron.

HEXOMINO

When six squares are joined together edge to edge they form ahexomino. The joining of each pair of squares must be along a complete edge. Altogether there are 35 different

HIRE PURCHASE 241

combinations of joining together six squares, and each one is called a hexomino. Some of the hexominoes are nets for a cube. Remember that types like the ones in figure a are regarded as the same hexomino.

(a)

The 35 hexominoes are shown in figure b.

(b)

References: Cube, Pentominoes, Polyominoes, Rectangle, Square.

HIGHEST COMMON FACTOR

Reference: Factor.

HIRE PURCHASE

We use hire purchase (or “buying on time”) when we wish to buy an item from a shop, but decide not to pay for it all at once, but pay for it in stages called installments. Provided you pay some of the money to start with, called the deposit, you will be allowed to take the item with you. For example, suppose you wish to buy a CD player for $700. You will be asked to make a payment straightaway of, say, 10% of the price, which is $70. This is called the deposit for the CD player. Then you will be asked to repay the remaining $630 in stages every month over a period of 2 years, say. But the retailer will add some interest to the $630, because he has to wait 2 years before all

242 HISTOGRAM

his money is paid up. That interest may be 15% of the balance owing. Now, 15% of $630 = $94.50, which is the interest the retailer charges.

So the amount you now owe, and have to pay by installments, is $630 + $94.50 = $724.50. If the payments are monthly, over 2 years there will be 24 payments. Each payment will be $724.50 -f 24 = $30.1875, which can be rounded off to $30.20.

Summary. You pay a deposit of $70 and then $724.80 by 24 monthly repayments of $30.20. The total cost of the CD player is now $794.80

Reference: Percentage.

HISTOGRAM

A histogram is a statistical graph of a frequency distribution, it resembles a bar graph, but there are differences between them. In a histogram there are no spaces between the columns, because it is a graph of continuous data. In theory the area of the column represents the frequency and so the columns do not have to be of equal widths, but in practice the columns of the histogram are usually equal in width, and the frequency is then represented by the height of the column. The following example illustrates how to draw a histogram.

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