Geometry Part II

In this geometry worksheet we will cover some of the formulas that you will find on the high school equivalency test, such as perimeter, area, and volume.  These formulas are found on the first page of the high school equivalency math test, so you do not need to memorize the formulas, just know when and how to use them. You can refer to the formula page to see what it looks like.  We’ll go over each formula, step by step, and do several practice exercises.

Formula Page


Polygons
Let’s review a few shapes first.  A closed, flat figure that is made up of three or more sides and angles, is called a polygon.  Polygons are identified by the number of sides they have.  Here are a few of the more common polygons.

Most of the shapes we will be working with will be circles, triangles, and quadrilaterals. A quadrilateral is an enclosed shape with 4 sides and 4 angles.  Below are some of the most commonly talked about quadrilaterals.


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Perimeter
Perimeter is the distance around a shape or figure.  Have you ever heard –"The security guard checked the perimeter of the building"?  That means that he walks around the outside of the building checking for anything unusual. 

To find the perimeter of most shapes, you can just add all of the sides together.  However, we will go over some “shortcut” formulas” that you will see on the high school equivalency formula page.  But just remember, you can always just add the sides instead of using the formula listed.  The way that I remember that perimeter means add is that when I see the word perime+er I think of the “t” as being a big plus sign and that helps me to remember to add all of the sides.

In word problems, you will look for such scenarios as putting trim around a room, building a frame putting a border around a flower garden, running around a track, and fencing a yard.  Look for words like around, enclose, and border.


Perimeter of a Square
The definition of a square is a shape with 4 equal sides and 4 right angles.  Opposite sides are parallel. To find the perimeter of a square, just add all 4 sides or multiply one side by 4.

 The formula for finding the perimeter of a square is: Perimeter = 4 * side

Example:

          P = 4 * s
          P = 4 * 2
          P = 8 inches
(Or you can just add 2+2+2+2)

The perimeter of a square with 2-inch sides is 8 inches.


Another example:
This one's an example of a backward perimeter problem.  You are given the answer and are asked to find one side of a square.  If you know the total distance around a square, how would you find one side? That’s right!  You would divide the total distance by 4.

The perimeter of a square room is 32 feet.  What is the length of each side of the room?
          s = P
                4

          s = 32
                 4
s = 8 feet   Each side of the room is 8 feet long.

Now you try some:
1. What is the perimeter of a square picture that is 11 inches on each side?
2. What is the perimeter of a square garden that measures 9 feet on one side?
3. How long is each side of a square room that has a perimeter of 48 feet?
4. Find the perimeter of a square whose sides are 2/3 inch long.
5. What size square painting will fit inside an antique frame with a perimeter of 56 inches?

Answers: 1. 44 inches; 2. 36 feet; 3. 12 feet; 4. 2 2/3 inches; 5. 14 inches (back to top)

 

Perimeter of a Rectangle
A rectangle is a 4- sided shape with 4 right angles.  Opposite sides are parallel and equal.
The formula for finding the perimeter of a rectangle is: P = 2 * length + 2 * width

Example:

SCAN rectangle p. 42 Math in Daily Living

          P = 2l + 2w
          P = 2(6) + 2(4)
          P = 12 + 8
          P = 20 feet

Let’s try it:
1. How many feet of fencing will Joseph need to fence a rectangular garden that is 200 feet by 150 feet?
2. Find the perimeter of a picture that measures 11 by 12 inches.
3. How much border will Susan need to put around a window that is 12 by 36 inches?
4. Find the perimeter of a yard that is 70 feet long and 35 feet wide.
5. How much braid is needed to go around a tablecloth that is 62 by 80 inches?

Answers:
1. 200 x 2 = 400              150 x 2 = 300               400 + 300 =700 feet
2. 11 x 2 = 22                  12 x 2 = 24                   22 + 24 = 46 inches
3. 12 x 2 = 24                  36 x 2 = 72                  24 + 72 = 96 inches
4. 70 x 2 = 140                35 x 2 = 70                  140 + 70 = 210 feet
5. 62 x 2 = 124                80 x 2 = 160                124 + 160 = 284 inches (back to top)

 

Perimeter of a Triangle
A triangle is a three-sided shape that has three lines that meet and form three angles.
The formula for finding the perimeter of a triangle is:  P = side1 + side2  + side 3

          P = s1  + s2  +  s3
          P = 5 + 4 + 3
          P = 12


Let’s try it:
1. Find the perimeter of a triangle having two sides of 12.3 centimeters and a base of 5.7 centimeters.
2. A triangular piece of land has two sides 196 feet and one side of 100 feet.  How much fencing will be required to enclose it?
3. What is the perimeter of a triangular rose garden that measures 9 feet by 4 feet by 5 feet?
4. What is the perimeter of a triangular flag that is 10 inches on one side, 6 inches on one side, and 8 inches on another side?
5. Find the perimeter of a triangle that measures ½ foot on 2 sides and 5/8 foot on the other side.

Answers:
1.      12.3 + 12.3 + 5.7 = 30. 3 centimeters
2. 196 + 196 + 100 = 492 feet
3. 9 + 4 + 5 = 18 feet
4. 10 + 6 + 8 = 24 inches
5. 1/2 + 1/2 + 5/8 = 4/8 + 4/8 + 5/8 = 13/8 = 1 5/8 feet (back to top)