Practice problems for these concepts can be found at:

Geometry Practice Problems: GED Math

### Polygons

**Plane figures** are two-dimensional objects that reside on a plane. You can think of a plane like a sheet of paper that extends forever in all directions. Special figures are called **polygons**.

A good grasp of geometry requires a knowledge of the basic polygon shapes. Study the following geometry vocabulary.

A

polygonis a closed plane figure made up of line segments.A

triangleis a polygon with three sides.A

quadrilateralis a polygon with four sides.A

pentagonis a polygon with five sides.A

hexagonis a polygon with six sides.A

heptagonor aseptagonis a polygon with seven sides.An

octagonis a polygon with eight sides.

### Quadrilaterals

Four-sided polygons are called **quadrilaterals**, and there are classifications for quadrilaterals.

A quadrilateral with one pair of parallel sides (bases) is called a **trapezoid**. In an isosceles trapezoid, the sides that are not bases are congruent (equal in measure). Because the parallel bases are not the same length in a trapezoid, we call these bases *b*_{1} and *b*_{2}.

A quadrilateral with two pairs of parallel sides is called a **parallelogram**. The two sets of opposite sides that are parallel are equal and congruent in a parallelogram, as shown in the diagram:

Parallelograms are broken down into further subgroups.

- A
**rectangle**is a parallelogram with four right angles. - A
**rhombus**is a parallelogram with four equal and congruent sides. - A
**square**is a parallelogram with both four right angles and four equal and congruent sides. A square is a rhombus, a rectangle, a parallelogram, and a quadrilateral.

### Finding Perimeter and Circumference

The GED provides you with several geometrical formulas. One of these is for the perimeter. **Perimeter** is the distance around a figure. You can calculate the perimeter of a figure by adding up the lengths of all the sides.

ExampleFind the perimeter of a square whose side is 5 cm.

You know that each side of a square is equal. So, if you know that one side of a square is 5 cm, then you know that each side of the square is 5 cm. To calculate the perimeter, add up the lengths of all four sides.

5 + 5 + 5 + 5 = 20

So, the perimeter of a square whose side measures 5 cm is 20 cm.

The perimeter of a circle is called its **circumference**. You can calculate a circle's circumference using either its radius or its diameter.

The **diameter** is a line segment that goes through the center of a circle. The endpoints of the diameter are on the curve of the circle. Any line that begins at the center of a circle and ends on a point on the circle is called a **radius**. A circle's diameter is twice as long as its radius. So, if you know either the radius or the diameter, you can easily find the other.

To calculate circumference, use either of these formulas, where π = 3.14 ( π is a Greek symbol, spelled *pi*, and pronounced "pie"), *d* is the circle's diameter, and *r* is the circle's radius:

*C* = π*d*

*C* = 2π*r*

ExampleFind the circumference of a circle with a diameter of 5 inches.

Because you know the diameter, use the formula that includes the diameter:

C= πd

C= π(5)= (3.14)(5)

= 15.7

The final answer is 15.7 inches.

### Finding Area

**Area** is a measure of the surface of a two-dimensional figure. The following table shows how to calculate the area of different figures.

### Finding Volume

**Volume** is a measure of the amount of space inside a three-dimensional shape. Three-dimensional shapes are sometimes called **solids**.

The formula for calculating the volume of a rectangular solid is:

*V* = *Ah*

where *V* = the volume

*A* = the area of the base

*h* = the height

**Examples**

- Find the volume of a cube that is 3 inches long on each edge.
- Find the volume of a cylinder that has a height of 10 cm and a radius of 5 cm.

Choose the correct formula. The problem tells you that you are measuring the volume of a cube. The formula for the volume of a cube is *V* = *Ah* or *V* = *s*^{3}.

Plug in the known measures and solve:

*V* = 3^{3}

= 3 × 3 × 3

= 27

So the final answer is 27 square inches.

Choose the correct formula. The problem tells you that you are measuring the volume of a cylinder. The formula for the volume of a cylinder is *V* = *Ah* or *V* = π*r*^{2}*h*.

Plug in the known measures and solve:

*V* = π*r*^{2}*h*

= π(5)^{2}(10)

= π(25)(10)

= π(250)

= 785

So the final answer is 785 cubic centimeters.

(Cubic centimeters can also be written as cm3.)

### Surface Area

The surface area of a three-dimensional object measures the area of each of its faces and adds them together. The total surface area of a rectangular solid is double the sum of the areas of the three different faces. For a cube, simply multiply the surface area of one of its sides by 6.

Practice problems for these concepts can be found at: