Lesson 15 Perimeter and Area

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

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

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

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

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

Example 1  What is the perimeter of a triangle whose sides are 5, 12, and 13?  Strategy: See if there is any shortcut formula for the perimeter. If not, add the lengths of the sides.  There is no shortcut formula so add the lengths of the sides.  = 30

Solution  The perimeter of the triangle is 30.

Example 2  The Chesterfield Middle School has a continuous row of shrubs around the front lawn, which is shaped like a rectangle. The front lawn is 55 feet wide and 24 feet long. How long is the row of shrubs?

 Strategy: See if there is a shortcut formula for the perimeter. If not, add the lengths of the sides.  The perimeter of a rectangle has a shortcut formula. It is p = 2L + 2W.  Substitute 55 for W and 24 for L into the formula: p = 2(24) + 2(55) = p = p =

Example 3  The perimeter of a square end table is 64 inches. What is the length of one side of the table?  Strategy: See if there is a shortcut formula for the perimeter you can use.  The perimeter of a square has the formula p = 4s. Substitute 64 for p. 64 = 4s. Divide both sides by 4, s = 16

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

Circumference of a Circle  See lesson 18, Properties of Circles, for basic definitions of the parts of circles.  The distance around a circle is called its circumference.  C = 2  r (where C = circumference, r = radius, and  3.14 or 22/7). r

Solution  The row of shrubs is 158 feet long.

Example 4  A circular pond in a garden has a radius of 7 feet. What is its circumference.  Strategy: Substitute the numbers into the formula for circumference.  Step 1: Use the formula for the circumference of a circle: C = 2  r  Step 2: Substitute the radius (7 feet) and  (22/7 or 3.14) into the problem. C = (2) (22/7) (7) C = (2) (22/7) (7)

Solution  C = 44 feet

Area of Triangles, Squares, and Rectangles  The following are the formulas for the area of polygons that you need to know.  Learn them.  Keep a formula card in your notebook.

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

Square  A = s 2 (where s = side) s s

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

Example 5  Desks at the middle school are 24 inches by 14 inches. What is the area of the desk?  Strategy: Substitute the numbers into the area formula. Substitute 24 for L and 14 for W and multiply. 24 x 14 = 336 in. 2

Solution  The area of each desk is 336 square inches.  Area is always measured in square units.

Example 6  Find the area of the triangle shown here.  Strategy: Substitute the base and height into the area formula for triangles. 12 cm 8 cm

A = ½ bh base = 12; height = 8 A = ½ x 12 x 8 = A = 48

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

Area of a Circle  You should know the formula for the area of a circle: A =  r 2 (where A = area, r = radius, and  = 3.14 or 22/7). r

Example 7  A circular pond has a radius of 7 feet. What is the area?  Strategy: Substitute the numbers into the formula for area of a circle.  Step 1: Use the formula A =  r 2.  Step 2: Substitute the radius (7 feet) and  (22/7) into the formula.

A =  r 2 A = (22/7)(7 2 ) A = 22 x 7 x 7 7 A = 22 x 7 A = 54

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

 C = (2 ) (22/7) (7)