Getting around with shapes

Geometry. Text 4.

1. counting things.

Equations:

2+2=4

8:1=8

x+8=11

2+2

Addition. Two added by two equals four/two plus two is four

2-1

Subtraction. Two subtracted by one equals

2x2

Multiplication

2 multiplied by 2 equals 4

4:2

Division

4 divided by 2 equals 2

2. Getting around with measures.

Perimeter

The perimeter of a polygon is the sum of the lengths of all its sides.

Example 1: the perimeter of this rectangle is 7+3+7+3 = 20

Example 2: the perimeter of this regular pentagon is 3+3+3+3+3 = 5×3 = 15

The perimeter of a circle is called the circumference:

It is equal to Pi () times the diameter of the circle. Pi or is a number that is approximately 3.14159.

Area

The area of a figure measures the size of the region enclosed by the figure. This is usually expressed in terms of some square unit. A few examples of the units used are square meters, square centimeters, square inches, or square kilometers.

The area of the triangle is 1/2 × b × h, where b is the base length and h is the height.

The area of a rectangle is the product of its width and length.

Example:

What is the area of a rectangle having a length of 6 and a width of 2.2?

The area is the product of these two side-lengths, which is 6 × 2.2 = 13.2.

The area of a parallelogram is b × h, where b is the length of the base of the parallelogram, and h is the corresponding height. To picture this, consider the parallelogram below:

Area of a Trapezoid

If a and b are the lengths of the two parallel bases of a trapezoid, and h is its height, the area of the trapezoid is

1/2 × h × (a + b).

Getting around with shapes.

Figures that have the same shape are called similar figures. They may be different sizes or turned somewhat.

Example:

The following pairs of figures are similar.

These pairs of figures are not similar.

Congruent Figures

Two figures are congruent if they have the same shape and size.

The pairs below are similar but not congruent.

The pairs below are not similar or congruent.

Transformations

The three main Transformations are:

Translation (slide),

Reflection (flip)

Rotation (turn)

After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths.

If one shape can become another using Turns, Flips and/or Slides, then the two shapes are called Congruent.

Translation

In Geometry, "translation" simply means moving without rotating, resizing or anything else. It slides.

Every point of the shape must move:

- the same distance

- in the same direction.

Slide!

Rotation

When a figure is turned, we call it a rotation of the figure. We can measure this rotation in terms of degrees; a 360 degree turn rotates a figure around once back to its original position.

Turn!

Example:

For the following pairs of figures, the figure on the right is a rotation of the figure on the left.

Reflection

Flip!

If we flip (or mirror) over some line, we say the figure is a reflection over that line.

Examples:

Reflections along a vertical line:

Reflections along a horizontal line:

So, a reflection is a flip over a line which is called the Mirror Line

Every point in the reflection is the same distance from the central line, and the reflection has the same size as the original image.

Resizing

The other important Transformation is Resizing (also called dilation, contraction, compression, enlargement or even expansion). The shape becomes bigger or smaller:

You can use Resizing to make one shape from another, when they are Similar but not Congruent.

Folding

When we talk about folding a plane figure, we mean folding it as if it were a piece of paper in that shape. We might fold this into a solid figure such as a box, or fold the figure flat along itself.

Example:

Folding the figure on the left into a box:

Folding the figure on the left flat along the dotted line:

Symmetric Figure

A figure that can be folded flat along a line so that the two halves match perfectly is a symmetric figure; such a line is called a line of symmetry.

Examples:

The triangle below is a symmetric figure. The dotted line is the line of symmetry.

The square below is a symmetric figure. It has four different lines of symmetry shown below.

The circle below is a symmetric figure. Any line that passes through its center is a line of symmetry!