The perimeter of a polygon is the distance around the outside of the polygon. A polygon is 2dimensional; however, perimeter is 1-dimensional and is measured in linear units. To help us
make this distinction, look at our picture of a rectangular backyard. The yard is 2-dimensional: it
has a length and a width. The amount of fence needed to enclose the backyard (perimeter) is 1dimensional. The perimeter of this yard is the distance around the outside of the yard, indicated
by the red arrow; It is measured in linear units such as feet or meters.
To find the perimeter of a polygon, take the sum of the length of each side. The polygons below are much smaller than a
fenced-in yard. Thus, we use smaller units in our examples, such as centimeters and inches.
Example 1:
Find the perimeter of a triangle with sides measuring 5 centimeters, 9
centimeters and 11 centimeters.
Solution:
P = 5 cm + 9 cm + 11 cm = 25 cm
Example 2:
A rectangle has a length of 8 centimeters and a width of 3 centimeters. Find
the perimeter.
Solution 1:
P = 8 cm + 8cm + 3 cm + 3 cm = 22 cm
Solution 2:
P = 2(8 cm) + 2(3 cm) = 16 cm + 6 cm = 22 cm
In Example 2, the second solution is more commonly used. In fact, in mathematics, we commonly use the following formula for
perimeter of a rectangle:
, where
is the perimeter,
is the length and
is the width.
In the next few examples, we will find the perimeter of other polygons.
Example 3:
Solution:
Example 4:
Solution:
Find the perimeter of a square with each side measuring 2 inches.
= 2 in + 2 in + 2 in + 2 in = 8 in
Find the perimeter of an equilateral triangle with each side measuring 4
centimeters.
= 4 cm + 4 cm + 4 cm = 12 cm
Perimeter, Part II
Unit 1 >
Lesson 1
of 8
A square and an equilateral triangle are both examples of regular polygons. Another method for finding the perimeter of a
regular polygon is to multiply the number of sides by the length of one side. Let's revisit Examples 3 and 4 using this second
method.
Example 3:
Find the perimeter of a square with each side measuring 2 inches.
Solution:
This regular polygon has 4 sides, each with a length of 2 inches. Thus we get:
= 4(2 in) = 8 in
1.
Find the perimeter of a triangle with sides measuring 10 inches, 14
inches and 15 inches.
CLEAR
ANSWER BOX:
=
in
RESULTS
BOX:
Example 4:
Find the perimeter of an equilateral triangle with each side measuring 4 centimeters.
Solution:
This regular polygon has 3 sides, each with a length of 4 centimeters. Thus we get:
= 3(4 cm) = 12 cm
Example 5:
Solution:
Example 6:
Solution:
Find the perimeter of a regular pentagon with each side measuring 3 inches.
= 5(3 in) = 15 in
The perimeter of a regular hexagon is 18 centimeters. How long is one side?
= 18 cm
Let
represent the length of one side. A regular hexagon has 6 sides, so we can divide
the perimeter by 6 to get the length of one side ( ).
= 18 cm ÷ 6
= 3 cm
Summary:
2.
To find the perimeter of a polygon, take the sum of the length of each side. The formula for perimeter of a
rectangle is:
. To find the perimeter of a regular polygon, multiply the number of
sides by the length of one side.
A rectangle has a length of 12 centimeters and a width of 4
centimeters. Find the perimeter.
CLEAR
ANSWER BOX:
=
cm
RESULTS
BOX:
3.
Find the perimeter of a regular hexagon with each side measuring 8
meters.
CLEAR
ANSWER BOX:
=
m
RESULTS
BOX:
4.
The perimeter of a square is 20 feet. How long is each side?
CLEAR
ANSWER BOX:
=
ft
RESULTS
BOX:
5.
The perimeter of a regular pentagon is 100 centimeters. How long is
each side?
CLEAR
ANSWER BOX:
=
cm
RESULTS
BOX:
A square and an equilateral triangle are both examples of regular polygons. Another method for finding the perimeter of a
regular polygon is to multiply the number of sides by the length of one side. Let's revisit Examples 3 and 4 using this second
method.
Example 3:
Find the perimeter of a square with each side measuring 2 inches.
Solution:
This regular polygon has 4 sides, each with a length of 2 inches. Thus
we get:
= 4(2 in) = 8 in
Example 4:
Find the perimeter of an equilateral triangle with each side measuring
4 centimeters.
Solution:
This regular polygon has 3 sides, each with a length of 4 centimeters.
Thus we get:
= 3(4 cm) = 12 cm
Example 5:
Solution:
Example 6:
Find the perimeter of a regular pentagon with each side measuring 3
inches.
= 5(3 in) = 15 in
The perimeter of a regular hexagon is 18 centimeters. How long is one
side?
Solution:
= 18 cm
Let
represent the length of one side. A regular hexagon has 6
sides, so we can divide the perimeter by 6 to get the length of one side
( ).
= 18 cm ÷ 6
= 3 cm
Summary:
To find the perimeter of a polygon, take the sum of the length of each side. The formula
for perimeter of a rectangle is:
. To find the perimeter of a
regular polygon, multiply the number of sides by the length of one side.
Find the perimeter of a triangle with sides measuring 10 inches,
14 inches and 15 inches.
1.
CLEAR
ANSWER BOX:
=
in
RESULTS
BOX:
2.
A rectangle has a length of 12 centimeters and a width of 4
centimeters. Find the perimeter.
CLEAR
ANSWER BOX:
=
cm
RESULTS
BOX:
3.
Find the perimeter of a regular hexagon with each side measuring 8
meters.
CLEAR
ANSWER BOX:
=
m
RESULTS
BOX:
4.
The perimeter of a square is 20 feet. How long is each side?
CLEAR
ANSWER BOX:
RESULTS
BOX:
=
ft
5.
The perimeter of a regular pentagon is 100 centimeters. How long is
each side?
CLEAR
ANSWER BOX:
=
cm
RESULTS
BOX:
The area of a polygon is the number of square units inside the polygon. To understand the
difference between perimeter and area, think of perimeter as the length of fence needed to enclose
the yard, whereas area is the space inside the yard. Perimeter is 1-dimensional and is measured in
linear units such as inches, feet or meters. Area is 2-dimensional: it has a length and a width. Area
is measured in square units such as square inches, square feet or square meters.
To find the area of a rectangle, multiply the length by the width. The formula is:
, where
is the area,
is the length,
is the width, and · means multiply.
A square is a rectangle with 4 equal sides. To find the area of a square, multiply the length of one
side by itself. The formula is:
or
, where A is the area, s is the length of a side, and · means multiply.
Let's look at some examples of finding the area of rectangles.
Example 1:
Find the area of a square with each side measuring 2 inches.
Solution:
= (2 in) · (2 in) = 4 in2
Example 2:
A rectangle has a length of 8 centimeters and a width of 3 centimeters. Find the
area.
Solution:
= (8 cm) · (3 cm) = 24 cm2
In Examples 1 and 2, we found the area given the dimensions of the rectangle. Let's look at some examples in which we are
given the area of the rectangle, and are asked to work backwards to find the missing dimension.
Example 3:
The area of a square is 9 square centimeters. How long is one side?
Solution:
9 cm2 =
·
Since 3 · 3 = 9, we get 3 cm · 3 cm = 9 cm2. So
must equal 3 cm.
= 3 cm
Example 4:
Solution:
The area of a rectangle is 12 square inches and the width is 3 inches. What is the
length?
12 in2 =
· 3 in
Since 4 · 3 = 12, we get (4 in) · (3 in) = 12 in2. So
must equal 4 in.
= 4 in
Summary:
The dimensions of a rectangle are length and width. Given the length and width of a
rectangle, we can find the area. Given the area and one dimension of a rectangle,
we can find the other dimension. The formula for area of a rectangle is:
where
is the length and
is the width.
A square is a rectangle with 4 equal sides The formula for area of a square is:
or
where
is the length of one side.
Exercises
Directions: Read each question below. Click once in an ANSWER BOX and type in your answer; then click ENTER. Your
answers should be given as whole numbers greater than zero. After you click ENTER, a message will appear in the RESULTS
BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.
1.
Find the area of a square with each side measuring 6 inches.
CLEAR
ANSWER BOX:
=
in2
RESULTS
BOX:
2.
A rectangular yard has a length of 12 meters and a width of 4 meters.
Find the area.
CLEAR
ANSWER BOX:
=
m2
RESULTS
BOX:
3.
The area of a square is 25 square centimeters. How long is one side?
CLEAR
ANSWER BOX:
RESULTS
BOX:
=
cm
4.
A rectangular rug has an area of 28 square feet and a length of 7 feet.
What is the width?
CLEAR
ANSWER BOX:
=
ft
RESULTS
BOX:
5.
A piece of paper has an area of 45 square inches and a width of 5
inches. What is the length?
CLEAR
ANSWER BOX:
=
in
RESULTS
BOX:
The area of a polygon is the number of square units inside that polygon. Area is 2-dimensional
like a carpet or an area rug. A triangle is a three-sided polygon. We will look at several types of
triangles in this lesson.
To find the area of a triangle, multiply the base by the height, and then divide by 2. The
division by 2 comes from the fact that a parallelogramcan be divided into 2 triangles. For
example, in the diagram to the left, the area of each triangle is equal to one-half the area of
the parallelogram.
Since the area of a parallelogram is
formula for the area of a triangle is:
, the area of a triangle must be one-half the area of a parallelogram. Thus, the
or
where
is the base,
is the height and · means multiply.
The base and height of a triangle must be perpendicular to each other. In each of the examples below, the base is a side of the
triangle. However, depending on the triangle, the height may or may not be a side of the triangle. For example, in the right
triangle in Example 2, the height is a side of the triangle since it is perpendicular to the base. In the triangles in Examples 1 and
3, the lateral sides are not perpendicular to the base, so a dotted line is drawn to represent the height.
Example 1: Find the area of an acute triangle with a base of 15 inches and a height of 4
inches.
Solution:
=
· (15 in) · (4 in)
=
· (60 in2)
= 30 in2
Example 2: Find the area of a right triangle with a base of 6 centimeters and a height of 9
centimeters.
Solution:
=
· (6 cm) · (9 cm)
=
· (54 cm2)
= 27 cm2
Example 3: Find the area of an obtuse triangle with a base of 5 inches and a height of 8
inches.
Solution:
=
· (5 in) · (8 in)
=
· (40 in2)
= 20 in2
Example 4: The area of a triangular-shaped mat is 18 square feet and the base is 3 feet.
Find the height. (Note: The triangle in the illustration to the right is NOT
drawn to scale.)
Solution:
In this example, we are given the area of a triangle and one dimension, and
we are asked to work backwards to find the other dimension.
18 ft2 =
· (3 ft) ·
Multiplying both sides of the equation by 2, we get:
36 ft2 = (3 ft) ·
Dividing both sides of the equation by 3 ft, we get:
12 ft =
Commuting this equation, we get:
1.
= 12 ft
Find the area of a triangle with a base of 16 feet and a height of 3
feet.
CLEAR
ANSWER BOX:
ft2
=
RESULTS
BOX:
2.
Find the area of a triangle with a base of 4 meters and a height of 14
meters.
CLEAR
ANSWER BOX:
RESULTS
=
m2
BOX:
3.
Find the area of a triangle with a base of 18 inches and a height of 2
inches.
4.
A triangular-shaped piece of paper has an area of 36 square
centimeters and a base of 6 centimeters. Find the height. (Hint: work
backwards)
5.
The area of a triangular-shaped rug is 12 square yards and the height
is 3 yards. Find the base. (Hint: work backwards)