Introduction
The distance formula can be used to find solutions to
many real-world problems. In the previous lesson, the
distance formula was used to find the distance between
two given points. In this lesson, the distance formula will
be applied to perimeter and area problems.
A polygon is a two-dimensional figure formed by three
or more segments. Sometimes we need to calculate the
perimeter or distance around a polygon, as well as find
the area or the amount of space inside the boundary of
a polygon. The distance formula is a valuable tool for
both of these calculations.
1
6.2.2: Calculating Perimeter and Area
Key Concepts
• Situations where you would need to calculate perimeter
include finding the amount of linear feet needed to
fence a yard or a garden, determining the amount of
trim needed for a room, or finding the amount of
concrete needed to edge a statue.
• Perimeter is the sum of the lengths of all the sides of a
polygon.
• The final answer must include the appropriate label
(units, feet, inches, meters, centimeters, etc.).
2
6.2.2: Calculating Perimeter and Area
Key Concepts, continued
• Sometimes the answer is not a whole number. If it is
not, you must simplify the radical and then
approximate the value.
Calculating the Perimeter of a Polygon
1. Calculate the length of each side of the polygon using
the distance formula: (x2 - x1)2 + (y 2 - y1)2 .
2. Add all the lengths of the polygon to find the perimeter.
3. Simplify and approximate the value if necessary.
Be sure to include the appropriate label in your answer.
3
6.2.2: Calculating Perimeter and Area
Key Concepts, continued
• Calculating area is necessary when finding the
amount of carpeting needed for a room in your home,
or to determine how large a garden will be.
• The area of a triangle is found using the formula
1
Area = (base)(height).
2
• The height of a triangle is the perpendicular distance
from the third vertex to the base of the triangle.
4
6.2.2: Calculating Perimeter and Area
Key Concepts, continued
• It may be necessary to determine the equation of the
line that represents the height of the triangle before
calculating the area. For an example of this, see
Example 3 in the Guided Practice.
• Determining the lengths of the base and the height is
necessary if these lengths are not stated in the
problem.
• The final answer must include the appropriate label
(units2, feet2, inches2, meters2, centimeters2, etc.).
5
6.2.2: Calculating Perimeter and Area
Key Concepts, continued
Calculating the Area of a Triangle
1. Find the equation of the line that represents the base of the triangle.
2. Find the equation of the line that represents the height of the triangle.
3. Find the point of intersection of the line representing the height of the
triangle and the line representing the base of the triangle.
4. Calculate the length of the base of the triangle using the distance
formula: (x - x )2 + (y - y )2 .
2
1
2
1
5. Calculate the length of the height of the triangle using the distance
formula: (x - x )2 + (y - y )2 .
2
1
2
1
1
6. Calculate the area using the formula Area = (base)(height) .
2
Be sure to include the appropriate label in your answer.
6
6.2.2: Calculating Perimeter and Area
Key Concepts, continued
• By definition, rectangles have adjacent sides that are
perpendicular.
• The area of a rectangle is found using the formula
Area = (base)(height).
• The lengths of the base and height are found using
the distance formula.
• The final answer must include the appropriate label
(units2, feet2, inches2, meters2, centimeters2, etc.).
6.2.2: Calculating Perimeter and Area
7
Key Concepts, continued
Calculating the Area of a Rectangle
1. Calculate the length of the base of the rectangle using the distance
formula: (x2 - x1)2 + (y 2 - y1)2 .
2. Calculate the length of the height of the rectangle using the distance
formula: (x2 - x1)2 + (y 2 - y1)2 .
1
3. Calculate the area using the formula Area = (base)(height).
2
8
6.2.2: Calculating Perimeter and Area
Common Errors/Misconceptions
• forgetting to simplify radicals
• incorrectly simplifying radicals
• adding x-values and y-values rather than subtracting
them when using the distance formula
• incorrectly finding the height of a triangle
9
6.2.2: Calculating Perimeter and Area
Guided Practice
Example 2
Quadrilateral ABCD has
vertices A (–3, 0), B (2, 4),
C (3, 1), and D (–4, –3).
Calculate the perimeter of
the quadrilateral.
10
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 2, continued
1. Calculate the length of each side of the
quadrilateral using the distance formula.
Calculate the length of AB .
(x2 - x1)2 + (y 2 - y1)2
((2) - (-3))2 + ((4) - (0))2
4). 2
(5) + (4)2
Distance formula
Substitute (–3, 0) and (2,
25 +16
Simplify as needed.
41
AB
The length of
6.2.2: Calculating Perimeter and Area
41
is
11
units.
Guided Practice: Example 2, continued
Calculate the length of BC .
(x2 - x1)2 + (y 2 - y1)2
Distance formula
((3) - (2))2 + ((1) - (4))2
Substitute (2, 4) and (3, 1).
(1)2 + (–3)2
Simplify as needed.
1+ 9
10
The length of BC is 10 units.
12
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 2, continued
Calculate the length of CD .
(x2 - x1)2 + (y 2 - y1)2
((–4) - (3))2 + ((–3) - (1))2
3).
(–7)2 + (–4)2
49 +16
Distance formula
Substitute (3, 1) and (–4, –
Simplify as needed.
CD
65
65
The length of
is
units.
13
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 2, continued
Calculate the length of DA .
(x2 - x1)2 + (y 2 - y1)2
Distance formula
((–3) - (–4))2 + ((0) - (–3))2 Substitute (–4, –3) and (–3,
0).
(1)2 + (3)2
Simplify as needed.
1+ 9
DA
10
The length of
10
is
units.
14
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 2, continued
2. Calculate the perimeter of quadrilateral
ABCD.
Find the sum of the sides of the quadrilateral.
perimeter = AB + BC + CD + DA
= 41+ 10 + 65 + 10
= 2 10 + 41+ 65
» 20.8
The perimeter of quadrilateral ABCD is
2 10 + 41+ 65 » 20.8 units.
6.2.2: Calculating Perimeter and Area
✔
15
Guided Practice: Example 2, continued
16
6.2.2: Calculating Perimeter and Area
Guided Practice
Example 3
Triangle ABC has
vertices A (1, –1),
B (4, 3), and C (5, –3).
Calculate the area of
triangle ABC.
17
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
1. Find the equation of the line that
represents the base of the triangle.
Let AC be the base.
Calculate the slope of the equation that represents
side AC .
18
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
m=
=
3).
=
y 2 - y1
x2 - x1
(-3) - (-1)
(5) - (1)
–2
4
1
=–
2
Slope formula
Substitute (1, –1) and (5, –
Simplify as needed.
–
AC
The slope of the equation the represents side
6.2.2: Calculating Perimeter and Area
is
1
2
19
.
Guided Practice: Example 3, continued
Write the equation of the line that represents side AC .
y – y1 = m(x – x1)
1
y - y1 = - (x - x1)
2
1
y - (-1) = - (x - (1))
2
1
y +1= - (x -1)
2
Point-slope formula
Substitute –
1
2
for m.
Substitute (1, –1) for (x1, y1)
Simplify.
20
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
1
1
y +1= - x +
2
2
1
1
y =- x2
2
Distribute –
1
2
over (x – 1).
Subtract 1 from both sides.
The equation of the line that represents the base of
1
1
the triangle is y = - x - .
2
2
21
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
2. Find the equation of the line that
represents the height of the triangle.
The equation of the line that represents the height is
perpendicular to the base; therefore, the slope of this
line is the opposite reciprocal of the base.
The slope of the line representing the height is 2.
22
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
y – y1 = m(x – x1)
Point-slope form
y – y1 = 2(x – x1)
Substitute 2 for m.
y – (3) = 2(x – (4))
Substitute (4, 3) for (x1, y1).
y – 3 = 2(x – 4)
Simplify.
y – 3 = 2x – 8
Distribute 2 over (x – 4).
y = 2x – 5
Add 3 to both sides.
The equation of the line that represents the height of
the triangle is y = 2x – 5.
23
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
3. Find the point of intersection of the line
representing the height and the line
representing the base of the triangle.
Set the equation of the line representing the base
and the equation of the line representing the height
equal to each other to determine the point of
intersection.
24
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
1
1
- x - = 2x - 5
2other.
2
1
9
- x = 2x 2
2
5
9
- x=2
2
9
x=
5
Set the equations equal to each
1
Add 2 to both sides.
Subtract 2x from both sides.
5
–
Divide both sides by 2 .
9
The point of intersection has an x-value of 5 .
6.2.2: Calculating Perimeter and Area
25
Guided Practice: Example 3, continued
Substitute
9
5
into either equation to find the y-value.
y = 2x – 5
Equation of the line representing
æ9ö
9
height
y = 2ç ÷ - 5
5
è5ø
Substitute for x.
18
y=
–5
5
y=–
7
Simplify.
5
Solve for y.
6.2.2: Calculating Perimeter and Area
26
Guided Practice: Example 3, continued
7
The point of intersection has a y-value of - .
5
æ9 7ö
The point of intersection is ç , - ÷.
è5 5ø
27
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
4. Calculate the length of the base,AC , of the
triangle.
(x2 - x1)2 + (y 2 - y1)2
Distance formula
((5) - (1))2 + ((-3) - (-1))2 Substitute (1, –1) and (5, –
3).
(4)2 + (-2)2
Simplify as needed.
16 + 4
20
AC
2 5
The length of
6.2.2: Calculating Perimeter and Area
2 5
is
28
units.
Guided Practice: Example 3, continued
5. Calculate the length of the height from
point B to the point of intersection.
(x2 - x1)2 + (y 2 - y1)2
Distance formula
ææ 9 ö
ö ææ 7 ö
ö
ççç ÷ - (4)÷÷ + ççç- ÷ - (3)÷÷ Substitute (4, 3) and
èè 5 ø
ø èè 5 ø
ø
2
2
æ 11ö æ 22 ö
ç- ÷ + ç- ÷
è 5ø è 5 ø
2
æ9 7ö
ç ,- ÷ .
è5 5ø
2
Simplify as needed.
29
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
121 484
+
25 25
605
25
605
25
11 5
5
The length of the height is
6.2.2: Calculating Perimeter and Area
11 5
5
units.
30
Guided Practice: Example 3, continued
6. Calculate the area of triangle ABC.
1
Area = (base)(height) Area formula for triangles
2
æ11 5 ö
÷
= 2 5 ç
è 5 ø
2
1
( )
( )(
)
æ 2 5 11 5 ö
1ç
÷÷
= ç
ø
2è
5
Substitute the lengths of the
height and the base of the
triangle.
Simplify as needed.
31
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
æ 22
1ç
= ç
2è
( )( )
5
5
5 ö
÷÷
ø
1 æ 22(5) ö
= ç
÷
2è 5 ø
1
= (22)
2
= 11
The area of triangle ABC
is 11 square units.
✔
32
6.2.2: Calculating Perimeter and Area
Guided Practice: Example 3, continued
33
6.2.2: Calculating Perimeter and Area