Lesson 15Lesson 15Perimeter and AreaPerimeter and Area

Two important measurements you will be Two important measurements you will be expected to find on the Terra Nova are the expected to find on the Terra Nova are the distance around a figure and the area the distance around a figure and the area the figure takes up. This lesson will give you figure takes up. This lesson will give you the formulas you will need to know in order the formulas you will need to know in order to determine those measurements.to determine those measurements.

PerimeterPerimeter

A polygon is a closed figure made up of A polygon is a closed figure made up of straight sides. Perimeter is the name straight sides. Perimeter is the name mathematicians use for the distance mathematicians use for the distance around a polygon. Perimeter is around a polygon. Perimeter is determined by adding up the lengths of all determined by adding up the lengths of all the sides. In some cases, you can use a the sides. In some cases, you can use a formula as a shortcut to adding and get formula as a shortcut to adding and get the perimeter of a figure.the perimeter of a figure.

TriangleTriangle

p = a + b + c (where p = perimeter and a, p = a + b + c (where p = perimeter and a, b, and c are the lengths of the sides).b, and c are the lengths of the sides).

a b

c

SquareSquare

p = 4s (where s = the length of a side).p = 4s (where s = the length of a side).

s

s

RectangleRectangle

p = 2L + 2W (where L = length and W = p = 2L + 2W (where L = length and W = width).width).

W

L

Example 1Example 1

What is the perimeter of a triangle whose What is the perimeter of a triangle whose sides are 5, 12, and 13?sides are 5, 12, and 13?

Strategy: See if there is any shortcut Strategy: See if there is any shortcut formula for the perimeter. If not, add the formula for the perimeter. If not, add the lengths of the sides.lengths of the sides.

There is no shortcut formula so add the There is no shortcut formula so add the lengths of the sides.lengths of the sides.

5 + 12 + 13 = 305 + 12 + 13 = 30

SolutionSolution

The perimeter of the triangle is 30.The perimeter of the triangle is 30.

Example 2Example 2

The Chesterfield Middle School has a The Chesterfield Middle School has a continuous row of shrubs around the front continuous row of shrubs around the front lawn, which is shaped like a rectangle. lawn, which is shaped like a rectangle. The front lawn is 55 feet wide and 24 feet The front lawn is 55 feet wide and 24 feet long. How long is the row of shrubs?long. How long is the row of shrubs?

Strategy: See if there is a shortcut formula Strategy: See if there is a shortcut formula for the perimeter. If not, add the lengths of for the perimeter. If not, add the lengths of the sides.the sides.

The perimeter of a rectangle has a The perimeter of a rectangle has a shortcut formula. It is p = 2L + 2W. shortcut formula. It is p = 2L + 2W.

Substitute 55 for W and 24 for L into the Substitute 55 for W and 24 for L into the formula: p = 2(24) + 2(55) =formula: p = 2(24) + 2(55) =

p = 48 + 110p = 48 + 110

Example 3Example 3

The perimeter of a square end table is 64 The perimeter of a square end table is 64 inches. What is the length of one side of inches. What is the length of one side of the table?the table?

Strategy: See if there is a shortcut formula Strategy: See if there is a shortcut formula for the perimeter you can use.for the perimeter you can use.

The perimeter of a square has the formula The perimeter of a square has the formula p = 4s. Substitute 64 for p. 64 = 4s. p = 4s. Substitute 64 for p. 64 = 4s. Divide both sides by 4, s = 16Divide both sides by 4, s = 16

SolutionSolution

The length of one side of the table is 16 The length of one side of the table is 16 inches.inches.

Circumference of a CircleCircumference of a Circle

See lesson 18, Properties of Circles, for See lesson 18, Properties of Circles, for basic definitions of the parts of circles.basic definitions of the parts of circles.

The distance around a circle is called its The distance around a circle is called its circumference. circumference.

C = 2C = 2r (where C = circumference, r = r (where C = circumference, r = radius, and radius, and 3.14 or 22/7). 3.14 or 22/7).

r

SolutionSolution

The row of shrubs is 158 feet long.The row of shrubs is 158 feet long.

Example 4Example 4

A circular pond in a garden has a radius of A circular pond in a garden has a radius of 7 feet. What is its circumference.7 feet. What is its circumference.

Strategy: Substitute the numbers into the Strategy: Substitute the numbers into the formula for circumference.formula for circumference.

Step 1: Use the formula for the Step 1: Use the formula for the circumference of a circle: C = 2circumference of a circle: C = 2rr

Step 2: Substitute the radius (7 feet) and Step 2: Substitute the radius (7 feet) and (22/7 or 3.14) into the problem. (22/7 or 3.14) into the problem.

C = (2) (22/7) (7)C = (2) (22/7) (7)

SolutionSolution

C = 44 feetC = 44 feet

Area of Triangles, Squares, and Area of Triangles, Squares, and RectanglesRectangles

The following are the formulas for The following are the formulas for the area of polygons that you the area of polygons that you need to know.need to know.

Learn them.Learn them. Keep a formula card in your Keep a formula card in your

notebook.notebook.

TriangleTriangle

A = ½ bh (where A = area, b = base, and h A = ½ bh (where A = area, b = base, and h = height or altitude).= height or altitude).

b

h

Square Square

A = sA = s2 2 (where s = side)(where s = side)

s

s

RectangleRectangle

A = bh (where b = base and h = height).A = bh (where b = base and h = height).

b

h

Example 5Example 5

Desks at the middle school are 24 inches Desks at the middle school are 24 inches by 14 inches. What is the area of the by 14 inches. What is the area of the desk?desk?

Strategy: Substitute the numbers into the Strategy: Substitute the numbers into the area formula. Substitute 24 for L and 14 area formula. Substitute 24 for L and 14 for W and multiply. 24 x 14 = 336 in.for W and multiply. 24 x 14 = 336 in.22

SolutionSolution

The area of each desk is 336 square The area of each desk is 336 square inches.inches.

Area is always measured in square units. Area is always measured in square units.

Example 6Example 6

Find the area of the triangle shown here.Find the area of the triangle shown here.

Strategy: Substitute the base and height Strategy: Substitute the base and height into the area formula for triangles.into the area formula for triangles.

12 cm

8 cm

A = ½ bhA = ½ bh

base = 12; height = 8base = 12; height = 8

A = ½ x 12 x 8 =A = ½ x 12 x 8 =

A = 48A = 48

SolutionSolution

The area of the circle is 48 sq. cm.The area of the circle is 48 sq. cm.

Area of a CircleArea of a Circle

You should know the formula for the area You should know the formula for the area of a circle: A = of a circle: A = rr22 (where A = area, r = (where A = area, r = radius, and radius, and = 3.14 or 22/7). = 3.14 or 22/7).

r

Example 7Example 7

A circular pond has a radius of 7 feet. A circular pond has a radius of 7 feet. What is the area?What is the area?

Strategy: Substitute the numbers into the Strategy: Substitute the numbers into the formula for area of a circle.formula for area of a circle.

Step 1: Use the formula A = Step 1: Use the formula A = rr22.. Step 2: Substitute the radius (7 feet) and Step 2: Substitute the radius (7 feet) and

(22/7) into the formula. (22/7) into the formula.

A = A = rr22

A = (22/7)(7A = (22/7)(722))

A = A = 22 x 7 x 722 x 7 x 7

77

A = 22 x 7A = 22 x 7

A = 54A = 54

SolutionSolution

The area of the circle is 154 square feet.The area of the circle is 154 square feet.

C = (2 ) (22/7) (7)C = (2 ) (22/7) (7)