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

ISBN 0-471-15045-2

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o

1 1 1 1----------1—*■

-4-2 0

(a)

The symbol for “less than or equal to” is <, and the symbol for “greater than or equal to” is >.

Example 2. What are the numbers less than or equal to —2?

Solution. This solution is exactly the same as the solution for “less than —2,” except that the number —2 is included in the set of solutions (see figure b). There is no “hole” in the line at the point —2 itself.

—i--------1------1-----------1 r

-4 -2 0

(b)

References: Inequality, Interval, Number Line.

GROWTH CURVE 233

GREATER THAN OR EQUAL TO

References: Greater Than, Inequality, Inequations.

GROSS

This is the number 144. Alternatively, one gross can be described as 12 dozen, where one dozen is 12.

Reference: Duodecimal.

GROUPED DATA

Reference: Arithmetic Mean.

GROWTH CURVE

Reference: Exponential Curve.

H

HALF-LIFE

Reference: Exponential Decay.

HECTARE

This is a unit for measuring land areas such as farms, playing fields, school grounds, parks, etc. The abbreviation for hectare is ha. One hectare is equal in size to 10,000 square meters, which is a square measuring 100 meters by 100 meters. Roughly, two average soccer pitches together have the same area as 1 hectare.

50 m

110 m

Example. The figure is a sketch of a park, and the measurements are in meters. Find its area in hectares.

Solution. The easiest way to find the area of the park is to split it into two rectangles A and B, by drawing a dashed line.Write

Area of A = length x width = 150 x 50 = 7500 m2 Area of 15 = 125 x 40

The width of B is 150- 110 = 40m

234

= 5000 m2

HELIX 235

Total area of the park = area of A + area of B = 7500 + 5000 = 12,500 m2

Now write

Area of park in hectares = 12,500 -r 10,000 There are 10,000 m2 in 1 ha Area of park = 1.25 hectares

One hectare is approximately 2.47 acres. This means that to change hectares into acres we multiply the number of hectares by 2.47 To change acres into hectares we divide number of acres by 2.47.

References: Acre, Area, Metric Units.

HEIGHT

This is the vertical distance from the base of an object to its top. Alternatively, height can be described as the altitude of an object. When finding the areas and volumes of some shapes we need to identify their heights. The height H of a cone is shown in figure a. The slant height, or sloping height, is the length 5.

The heights H of two triangles are shown in figure b. The base of each triangle is indicated. Suppose the triangle is twisted and we need to know its height and its base in order to find its area. The height is still the perpendicular distance of the top from the base, as shown in figure c.

References: Altitude, Cone, Triangle, Vertical.

HELIX

A helix is a three-dimensional curve that is formed when a right-angled triangle is wrapped around a cylinder with no overlapping. Suppose the triangle shown in the

236 HEMISPHERE

figure is very long, so that it can be wrapped around the cylinder a few times. The curve formed by the hypotenuse of the triangle is a helix.

This curve is often confused with a spiral, but they are not the same. Examples of the use of screws in the shape of a helix are glue sticks, lipsticks, and swivel chairs.

References: Cylinder, Right Triangle, Spiral.

HEMISPHERE

A hemisphere is half a sphere and is formed when a sphere is divided equally by a single cut. In figure a, the dome of the cathedral is a hemisphere.

The volume of a hemisphere is half the volume of a whole sphere, but extra care must be taken in finding the surface area of a hemisphere, as shown in the following example.

Example. An orange with a diameter of 40 cm is cut in half forming two identical hemispheres. Find the surface area of one of the hemispheres of the orange (see figure b).

Solution. The hemisphere is made up of a curved surface area of peel that is half the surface area of a sphere, plus the area of the circular base where the cut was made.

HEPTOMINO 237

Write

Area of curved surface = | x 4ttR2

Area curved surface of sphere = 4ttR2

= | x 4 x it x 202 Substituting radius = 20 cm

= 2513.3 (to 1 dp) Using it in the calculator

The area of the curved surface is 2513.3 cm2, to 1 dp. Now write

The area of the circular base is 1256.6 cm2.

The total area of the hemisphere is the sum of the curved surface and the circular base. Thus the surface area of one of the hemispheres of the orange = 3769.9 cm2.

References: Diameter, Radius, Sphere, Surface Area.

HENDECAGON

This is an 11-sided polygon, also known as an endecagon.

Reference: Polygon.

HENDECAHEDRON

This is a solid with 11 faces.

Reference: Polyhedron.

HEPTAGON

This is a seven-sided polygon and is also called a septagon.

References: Polygon, Septagon.

HEPTOMINO

Area of circular base = ttR2

= it x 202

Formula for the area of a circle

Substituting radius = 20 cm

1256.6 (to 1 dp)

Reference: Polyominoes.

238 HERON’S FORMULA

HERON’S FORMULA

Heron of Alexandria (also known as Hero) lived about the first century AD and derived a formula for the area of any triangle if the length of each side is known. The notation used to describe this formula is the same as the one used in trigonometry, and is briefly explained here.

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