MATC Alternative Learning Division
Geometry
Introduction
Geometry is the study of shapes. These shapes give a visual picture, which is different from numerals and other abstract ideas in math. We can draw a rectangle to represent a room with certain dimensions. We can then visualize problems.
Such as: “What is the distance around the outside of this room?” and “How much carpet would be needed to cover the floor of this room?” These are questions we will answer after learning more about the three shapes in this unit: rectangles, squares, and triangles.
Rectangles/Squares
Rectangles are figures with four sides like the figures below:
Rectangles have special properties:
1. Any adjoining sides (connected sides) are 90 degrees in measure or create a right angle. Right angles are like the corners on this piece of paper. These sides are perpendicular lines and look like one of the following shapes:
Note: In geometric shapes, a 90-degree angle will usually be marked with a “box”
in the vertex of the angle to define the sides as being perpendicular
(like in the first two angles above).
2. Opposite sides of a rectangle are equal. This means the measure of one side is equal to side that is opposite to it. When all of the sides are equal, it is a special rectangle called a square. See the examples below:
Triangles
Triangles are figures with three sides like the figures below:
If a triangle has a right angle within it, it is called a right triangle. You can tell if an angle measures 90 degrees by the perpendicular notation in the corner of the 90-degree angle. See the example below:
The letters b and h as labels on a triangle have special meaning. The b refers to the base of the triangle, and the h refers to the perpendicular height of the triangle. These sides are important in calculating areas of triangles.
MATC Alternative Learning Division
Geometry
Assigned to: ______
Date Assigned: ______
Subject Area: Geometry – Exercise #1
Topic: Properties of Rectangles and Squares
Fill in the missing measurements on the following rectangles and squares.
Note: It is very important to indicate in your answer what units of measure are being used (inches, feet, and yards…). Your answer is not complete without labeling the units.
MATC Alternative Learning Division
Geometry
Perimeter
Perimeter is the distance around an object. We find the perimeter of rectangles, squares, and triangles by finding the sum of or adding all sides.
See the examples below:
Alternate method: Equations can also be used for finding perimeter.
The perimeter of a rectangle is equal to two times the length plus two times the width.
P = (2´ l) + (2´ w)
Example:
P = (2 ´ 15) + (2 ´ 12)
P = 30 + 24
P = 54 ft.
**See that the answer is the same as if we had added all four sides.
The perimeter of a square is 4 times the length of a side.
P = 4 ´ s
Example:
P = 4 ´ s
P = 4 ´ 10
P = 40 ft.
**See that the answer is the same as if we had added all four sides.
Assigned to: ______
Date Assigned: ______
Subject Area: Geometry – Exercise #2
Topic: Perimeter of Rectangles, Squares, and Triangles
Find the perimeter of the following shapes:
Note: You may add all of the sides or use the equations. Be sure to label your answers with the correct unit of measurement.
MATC Alternative Learning Division
Geometry
Perimeter Applications
Perimeter is used to find the distance around rooms, tablecloths, pieces of land, doors, windows, tables, frames, backyards, gardens, and many other things. Why? We may need to find out how much fencing to buy, how much lace to edge a tablecloth, or how much weather stripping to seal a door. There are many uses for finding perimeter.
Below are examples of perimeter applications:
Example 1:
You want to put lace around the outside of a square tablecloth. One side measures 52 inches. How much lace do you need to buy?
P = adding all sides
P = 52 + 52 + 52 + 52
P = 208 inches
OR
Perimeter of a square = 4 ´ s
P = 4 ´ 52
P = 208 inches
Example 2:
Find the cost of fencing needed for the garden below. The fencing costs $4.00 per foot.
Perimeter = Adding all sides
12.5 + 12.5 + 18 + 18 = 61 feet
OR
Perimeter = (2´ l) + (2´ w)
P = (2 ´ 18) + (2 ´ 12.5)
P = 36 + 25
P = 61 feet
The perimeter or distance around the garden is 61 feet. Each foot of fencing will cost $4.00. To find the total cost, multiply the perimeter (total feet) by the cost per foot.
61 ft. ´ $4.00/ft. = $244.00
Assigned to: ______
Date Assigned: ______
Subject Area: Geometry – Exercise #3
Topic: Perimeter Applications
You may use a calculator on this activity.
1. A rectangular room is 14 feet by 12 feet. A carpenter plans to install molding along the top of each wall. What is the perimeter of the room in feet?
2. How much wire will Joe need to fence a garden that is 300 feet by 160 feet?
3. Find the perimeter of a picture that measures 8 inches by 10 inches.
4. What will it cost to replace molding around the floor of a room that is 12 feet by 10.5 feet if the molding sells for $1.29/ft.?
5. Find the perimeter of a triangle having two sides of 12.3 centimeters and a base of 5.7 centimeters.
6. A triangular piece of property has two sides of 196 feet, 4 inches and a base of 100 feet, two inches. How many feet of fencing will it take to enclose it?
7. If the fence in problem #6 sells for $7.89/ft., how much will this project cost?
8. How many feet of ribbon will be needed to go around a box twice if the box measures
1 1/2 feet on each side?
9. A square has a perimeter of 328 feet. What is the length of each side?
10. A local shop charges $3.15 per foot to place a frame around a square mirror with
24-inch sides. What will the frame cost?
Area
Area is a measure of the surface of a flat figure. Area is measured in square units. Area tells how many square units it takes to cover the space inside the figure.
In the following rectangle, we want to find out how many square units, square feet, are in the shape.
We can visually see there are 12 square units or 12 square feet in the rectangle.
**It is very important to label area measurements in square units to indicate it is measuring a surface and not a distance.
The equation for finding the area of a rectangle is length times width:
Area = Length ´ Width
Example:
Find the area of the rectangle below:
Area = Length ´ Width
A = L ´ W
A = 21 ´ 9
A = 189 square yards
**To find the area of a square,
multiply a side times itself since
all sides are the same.
A = s ´ s
Assigned to: ______
Date Assigned: ______
Subject Area: Geometry – Exercise #4
Topic: Area of Rectangles and Squares
Find the area of the following figures:
Be sure to label your answers with square units.
Area of Triangles
The equation for finding the area of a triangle is easier to understand if we compare it to the equation of a rectangle.
Example:
To find the area of the entire
rectangle, we would use the equation:
A = length ´ width
A = 3 ´ 4
A = 12 square feet
To find the area of a triangle, which in this case
we can see is half of the rectangle above,
we use the equation for area of a triangle, which is:
Area of triangle = 1/2 ´ base ´ height
A = 1/2 ´ b ´ h
For the example triangle above:
A = 1/2 ´ b ´ h
A = 1/2 ´ 4 ´ 3
A = 6 square feet (which is half the area of the rectangle)
Area of Triangles continued
**It is very important in finding the area of a triangle to use the base and the height. The height will have a perpendicular sign (box in the angle) where it meets the base and will be a dashed line if within a triangle. In a right triangle, the right angle of the triangle will define base and height.
See example:
To find the area of this triangle, you would first write down the equation:
A = 1/2 ´ b ´ h
Then determine the base (4) and the perpendicular height (5) from the drawing to use in the equation.
A = 1/2 ´ b ´ h
A = 1/2 ´ 4 ´ 5
A = 10 square feet
Note the other two sides on this triangle are" extra" information. We don't need those for finding area. We would use them for perimeter, however.
Assigned to: ______
Date Assigned: ______
Subject Area: Geometry – Exercise #5
Topic: Area of Triangles
Find the area of the following figures. You may use a calculator on this activity.
Be sure to label your answers in square units.
Area Applications
Area applications are used for finding the area of a surface. Examples of common area applications are total square footage of a house, size of a yard for sod/fertilizer, size of rooms for flooring/paint, size of paintings and many more. These are real-life situations where finding the area of an object can help in practical decisions.
Below are examples of area applications:
Example 1:
Angela's laundry room floor has a square shape. She wants to tile the room. If one side of the room measures 11 feet, what is the area of the floor in square feet (sq. ft.)?
Area = Length ´ Width
A = 11 ´ 11 (all sides are the same)
A = 121 square feet
Example 2:
Mike wants to put new flooring in his living room, which measures 12 feet by 16 feet. Carpeting would cost $3.50/sq. ft. and wood laminate sells for $4.75/sq. ft. How much would each type of flooring cost Mike?
Area = Length ´ Width
A = 12 ´ 16
A = 192 square feet
Carpet option:
192 ´ $3.50 = $672.00
Wood option:
192 ´ $4.75 = $912.00
Assigned to: ______
Date Assigned: ______
Subject Area: Geometry – Exercise #6
Topic: Area Applications
Solve the following area applications. You may use a calculator on this activity.
1. A painting measures 12 inches by 18 inches. How many square inches of glass will it take to cover the surface?
2. Which is larger, a room measuring 9 feet by 11 feet or one measuring 8 feet by 12 feet? How much larger is it?
3. Find the area of a triangle with a base of 4 miles and a height of 7 miles.
4. A triangular-shaped park whose base is 110 feet and whose height is 80 feet is going to be sodded at a cost of $0.89 per square foot. How much is it going to cost?
5. A fertilizer company recommends 6 pounds of fertilizer for every 960 square feet of lawn. How much fertilizer will be required to fertilize an area 80 feet by 120 feet?
6. A new house measures 30 feet by 60 feet. How many square feet are in the house?
7. How many pints of paint are needed to paint a wall 8 feet by 20 feet if a pint will cover 40 square feet?
8. A pyramid is formed by four equal triangles. If each has a base of 186 feet and a height of 103 feet, what is the total surface area of the pyramid?
9. An A-frame house has ends that form triangles, which measure 28 feet at the base and 18 feet at the height. What is the total area of both ends?
10. If a 2-inch square picture is enlarged to 3 times its original size, how many square inches are in the enlarged picture?
Geometry Answer Key
Exercise #1: Properties of Rectangles and Squares
1. a. 15 ft.
b. 12 ft.
2. a. 9.5 in.
b. 63.8 in.
3. a. 25 yd.
b. 25 yd.
4. a. 35.9 cm
b. 4.25 cm
5. a. 56.575 m
b. 89.655 m
Exercise #2: Perimeter of Rectangles, Squares, and Triangles
1. 54 ft.
2. 156 ft.
3. 58 yd.
4. 36.4 yd.
5. 276 ft.
Exercise #3: Perimeter Applications
1. 52 ft.
2. 920 ft.
3. 36 in.
4. $58.05
5. 36 cm
6. 593 ft.
7. $4,678.77
8. 12 ft.
9. 82 ft.
10. $25.20
Exercise #4: Area of Rectangles and Squares
1. 144 sq.in.
2. 400 sq. ft.
3. 2100 sq. m.
Exercise #5: Area of Triangles
1. 40 sq. yd.
2. 144 sq. in.
3. 44.02 sq. ft.
Exercise #6: Area Applications
1. 216 sq. in.
2. 9 by 11= 99 sq.ft., 8 by 12= 96 sq. ft., 9 by 11 bigger by 3 sq. ft.
3. 14 sq. miles
4. $3,916.00
5. 9600 sq. ft. ¸ 960 = 10 ´ 6 = 60 lbs. fertilizer
6. 1800 sq. ft.
7. 160 sq. ft. ¸ 40 = 4 pints
8. 9,579 ´ 4 = 38,316 sq. ft.
9. 504 sq. ft.
10. 36 sq. in.