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Geometry

Geometry is about measuring lines and angles and assessing the relationship between them, so let’s start with some ways of measuring.

• The perimeter of a two-dimensional object is the total length of all its sides. For example, if these sides are straight, it’s a matter of simple addition: A rectangle measuring 4 inches by 5 inches has two sides 4 inches long and two sides 5 inches long, so its perimeter is 4 + 5 + 4 + 5 = 18 inches.

• The area of a four-sided figure is calculated by multiplying the length by the width: In the above example 4 x 5 = 20 square inches (in.2).

• Volume is calculated in the same way, by multiplying the length by the width by the height (or, if you prefer, the area by the height). For instance, a box that is 6 inches high, whose base measures 4 inches by 5 inches, has a volume of 4 x 5 x 6 = 120 cubed inches (in3).

The volume of a pyramid is the area of the base multiplied by the height, divided by 3:

It’s when you get to circles that it all becomes more complicated, because then you have to start dealing with…

☞ PI

Pi (π) is the Greek equivalent of the Roman p and is used in math to represent the ratio of the circumference of a circle to its diameter. Depending on how sophisticated you are as a mathematician, you can say that π = 3.142, 3.14159, or 3.1415926535897932384626433832795, but even then it is not 100 percent exact. Expressed as a fraction, pi is roughly 3, or 22/7.

Before we go on, three more quick definitions:

• The circumference of a circle is its perimeter, the distance around the outside.

• The diameter is the length of a straight line through the middle, from one point on the circumference to another.

• The radius is half the diameter; that is, the distance from the center of the circle to the circumference.

So to calculate the circumference of a circle, you multiply the diameter by π: a circle that is, say, 7 inches in diameter has a circumference of 7 x 22/7 = approximately 22 inches. The formula for this can be expressed as πd, but is usually given as 2(πr).

Area is πr2—that is, π times the radius squared. So a circle of 6 inches radius has an area of 22/7 x (6 x 6) = approximately 113 square inches.

The three-dimensional equivalent to a circle is a sphere, and its volume is calculated by the formula 4/3πr3—that is, four thirds (or one and one third) of the product of π and the radius cubed (multiplied by itself and then by itself again). So a sphere with a radius of 6 inches has a volume of 4/3 x π x(6 x 6 x 6) = approximately 905 cubic inches.

A cone is effectively a pyramid with a circular base, so the pyramid formula applies: A cone with a base 6 inches in diameter and a height of 10 inches has a base area of π x (6 x 6) = approximately 113 square inches, and a volume of:

which equals approximately 377 cubic inches.

☞ TRIANGLES

The area of a triangle is calculated by:

There are three types of triangles, depending on the length of their sides:

• An equilateral triangle has three sides of equal length.

• An isosceles triangle has two sides of equal length.

• A scalene triangle has three sides that are all of different lengths.

The sum total of the angles of a triangle, whatever its shape, is 180°. A right angle is 90°; any angle smaller than 90° is called an acute angle, while anything above 90° but lower than 180° is obtuse. In a right-angled triangle the side opposite the right angle (also always the longest side) is called the hypotenuse, which brings us neatly to…

☞ THE PYTHAGOREAN THEOREM

This theorem states that the square on the hypotenuse is equal to the sum of the squares on the other two sides. The simplest example of this is what is called a 3:4:5 triangle, in which the hypotenuse is 5 inches (or centimeters or miles, it doesn’t matter) and the other two sides are 3 inches and 4 inches.

The square on the side that is 3 inches long is 9 in.2 (3 x 3), the square on the 4-inch side is 16 in.2 (4 x