Chapter 12

Geometry and measurement

Geometry is where the letters stop being placeholders and start standing for things you can measure: a length, a width, the height of a triangle.

That makes it the natural end of the course. Every formula here is an equation, and every one of them is solved with the substitution and the balance rule from the earlier chapters.

Shapes and measurement

Pre-algebra brings in formulas for area, perimeter and missing sides of shapes.

  • Area of a rectangle: A = l × w.
  • Perimeter of a rectangle: P = 2l + 2w.
  • Area of a triangle: A = (1/2)bh.
  • Area is counted in square units, like cm² or m².

A rectangle 8 long and 5 wide

Area: 8 × 5 = 40

Perimeter: 2(8) + 2(5) = 26

Formulas

A few of these are usually memorised. The rest follow from what they describe: profit is what is left after the cost comes off, which is the formula itself stated in words.

Shapes

Area of a rectangle

A = l × w

  • l is the length
  • w is the width

The space inside, counted in square units.

Perimeter of a rectangle

P = 2l + 2w

  • l is the length
  • w is the width

The distance all the way around the outside.

Area of a triangle

A = (1/2)bh

  • b is the base
  • h is the height

The height goes straight up from the base. It is not the slanted side.

Example

A triangle with base 10 and height 6.

(1/2)(10)(6) = 30

Pythagorean theorem

a² + b² = c²

  • a is one shorter side
  • b is the other shorter side
  • c is the hypotenuse, the longest side

It only works in a right triangle, the kind with a square corner.

Example

Sides of 3 and 4.

3² + 4² = c²

9 + 16 = c²

25 = c²

c = √25 = 5

Key words

Much of pre-algebra is vocabulary. A question names what it wants in these terms, and the term decides which arithmetic answers it.

Hypotenuse
The longest side of a right triangle.

Examples

Each one is set out the way a solution is written on paper: the question, the steps in order, then the answer.

Example 1. Finding the hypotenuse

A right triangle has sides of 6 and 8. How long is the hypotenuse?

Solution

  1. Use a² + b² = c²: 6² + 8² = c².
  2. 36 + 64 = c².
  3. 100 = c².
  4. c = √100, so c = 10.

10

The main thing to take away

Pre-algebra is mostly about learning how numbers, variables, equations and formulas work together. The most important skills to build are substitution, order of operations, working with fractions and negative numbers, and solving simple equations. If you feel comfortable with these ideas, you will have a strong foundation for algebra.

Where this leads

The scientific calculator
Squares, square roots and trigonometry, for the triangles whose sides do not come out whole.