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Measurement and Geometry: Perimeter and Circumference of Geometric Figures

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses perimeter and circumference of geometric figures. By the end of the module students should know what a polygon is, know what perimeter is and how to find it, know what the circumference, diameter, and radius of a circle is and how to find each one, know the meaning of the symbol π π and its approximating value and know what a formula is and four versions of the circumference formula of a circle.

Section Overview

  • Polygons
  • Perimeter
  • Circumference/Diameter/Radius
  • The Number π π
  • Formulas

Polygons

We can make use of conversion skills with denominate numbers to make measure­ments of geometric figures such as rectangles, triangles, and circles. To make these measurements we need to be familiar with several definitions.

Polygon

A polygon is a closed plane (flat) figure whose sides are line segments (portions of straight lines).

Polygons

Four shapes, each completely closed, with various numbers of straight line segments as sides. graphics2.png graphics3.png graphics4.png

Not polygons

Four shapes. One three-sided open box. One oval. One oval-shaped object with one flat side, and one nondescript blob.graphics6.pnggraphics7.pnggraphics8.png

Perimeter

Perimeter

The perimeter of a polygon is the distance around the polygon.

To find the perimeter of a polygon, we simply add up the lengths of all the sides.

Sample Set A

Find the perimeter of each polygon.

Example 1

A rectangle with short sides of length 2 cm and long sides of length 5 cm.

Perimeter = 2 cm + 5 cm + 2 cm + 5 cm = 14 cm Perimeter = 2 cm + 5 cm + 2 cm + 5 cm = 14 cm

Example 2

A polygon with sides of the following lengths: 9.2cm, 31mm, 4.2mm, 4.3mm, 1.52cm, and 5.4mm.

Perimeter = 3.1   mm 4.2   mm 4.3   mm 1.52 mm 5.4   mm + 9.2   mm ̲ 27.72 mm Perimeter = 3.1   mm 4.2   mm 4.3   mm 1.52 mm 5.4   mm + 9.2   mm ̲ 27.72 mm

Example 3

A polygon with eight sides. It is not an octagon, but can be visualized as one large rectangle with two  smaller rectangles connected to it.

Our first observation is that three of the dimensions are missing. However, we can determine the missing measurements using the following process. Let A, B, and C represent the missing measurements. Visualize

A polygon with eight sides. It is not an octagon, but can be visualized as one large rectangle with two  smaller rectangles connected to it.  The height and width are measured and labeled with variables, A, B, and C.

A = 12m-2m = 10m A = 12m-2m = 10m size 12{A="12m - 2m"="10m"} {}
B = 9m + 1m - 2m = 8m B = 9m + 1m - 2m = 8m size 12{B="9m"+"1m-2m"="8m"} {}
C = 12m - 1m = 11m C = 12m - 1m = 11m size 12{C="12m-1m"="11m"} {}

Perimeter = 8 m 10 m 2 m 2 m 9 m 11 m 1 m +   1 m ̲ 44 m Perimeter = 8 m 10 m 2 m 2 m 9 m 11 m 1 m +   1 m ̲ 44 m

Practice Set A

Find the perimeter of each polygon.

Exercise 2

Exercise 3

Circumference/Diameter/Radius

Circumference

The circumference of a circle is the distance around the circle.

Diameter

A diameter of a circle is any line segment that passes through the center of the circle and has its endpoints on the circle.

Radius

A radius of a circle is any line segment having as its endpoints the center of the circle and a point on the circle.
The radius is one half the diameter.

A circle with a line directly through the middle, ending at the edges of the shape. The entire length of the line is labeled diameter, and the length of the portion of the line from the center of the circle to the edge of the circle is labeled radius.

The Number π π

The symbol π π, read "pi," represents the nonterminating, nonrepeating decimal number 3.14159 … . This number has been computed to millions of decimal places without the appearance of a repeating block of digits.

For computational purposes, π π is often approximated as 3.14. We will write π3.14π3.14 size 12{π approx 3 "." "14"} {} to denote that π π is approximately equal to 3.14. The symbol "≈" means "approximately equal to."

Formulas

To find the circumference of a circle, we need only know its diameter or radius. We then use a formula for computing the circumference of the circle.

Formula

A formula is a rule or method for performing a task. In mathematics, a formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but possibly unknown, quantities.

If CC size 12{C} {}, dd size 12{d} {}, and rr size 12{r} {} represent, respectively, the circumference, diameter, and radius of a circle, then the following two formulas give us directions for computing the circum­ference of the circle.

Circumference Formulas

  1. C=π dC=π d size 12{C=πd} {} or C3.14dC3.14d size 12{C approx left (3 "." "14" right )d} {}
  2. C=2 πrC=2 πr size 12{C=2πr} {} or C23.14rC23.14r size 12{C approx 2 left (3 "." "14" right )r} {}

Sample Set B

Example 4

Find the exact circumference of the circle.

A circle with a dashed line from one edge to the other, labeled d = 7 in.

Use the formula C=π dC=π d size 12{C=πd} {}.

C = π 7 in . C = π 7 in . size 12{C=π cdot 7 ital "in" "." } {}

By commutativity of multiplication,

C = 7 in . π C = 7 in . π size 12{C=7 ital "in" "." cdot π} {}

C=7 πin.C=7 πin. size 12{C=7π` ital "in" "." } {}, exactly

This result is exact since π π has not been approximated.

Example 5

Find the approximate circumference of the circle.

A circle with a dashed line from one edge to the other, labeled d = 6.2 mm.

Use the formula C=π dC=π d size 12{C=πd} {}.

C 3 . 14 6 . 2 C 3 . 14 6 . 2 size 12{C approx left (3 "." "14" right ) left (6 "." 2 right )} {}

C 19 . 648   mm C 19 . 648   mm size 12{C approx "19" "." "648"" mm"} {}

This result is approximate since π π has been approximated by 3.14.

Example 6

Find the approximate circumference of a circle with radius 18 inches.

Since we're given that the radius, rr size 12{r} {}, is 18 in., we'll use the formula C=2 πrC=2 πr size 12{C=2πr} {}.

C 2 3 . 14 18   in . C 2 3 . 14 18   in . size 12{C approx left (2 right ) left (3 "." "14" right ) left ("18"" in" "." right )} {}

C 113 . 04   in . C 113 . 04   in . size 12{C approx "113" "." "04"" in" "." } {}

Example 7

Find the approximate perimeter of the figure.

A cane-shaped object of an even thickness, with one straight portion and one portion shaped in a half-circle. The thickness is 2.0cm, the length of the straight portion is 5.1cm, and the radius of the semicircle portion is 6.2cm.

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is 6.2 cm - 2.0 cm = 4.2 cm6.2 cm - 2.0 cm = 4.2 cm size 12{6 "." "2 cm - 2" "." "0 cm "=" 4" "." "2 cm"} {}.

The width of the bottom part of the rectangle is 2.0 cm.

Perimeter = 2.0 cm 5.1 cm 2.0 cm 5.1 cm (0.5 ) ( 2 ) ( 3.14 ) ( 6.2 cm ) Circumference of outer semicircle.     + ( 0.5 ) ( 2 ) ( 3.14 ) ( 4.2 cm ) ̲ Circumference of inner semicircle 6.2 cm- 2.0 cm = 4.2 cm The 0.5 appears because we want the  perimeter of only half a circle. Perimeter = 2.0 cm 5.1 cm 2.0 cm 5.1 cm (0.5 ) ( 2 ) ( 3.14 ) ( 6.2 cm ) Circumference of outer semicircle.     + ( 0.5 ) ( 2 ) ( 3.14 ) ( 4.2 cm ) ̲ Circumference of inner semicircle 6.2 cm- 2.0 cm = 4.2 cm The 0.5 appears because we want the  perimeter of only half a circle.

Perimeter 2.0     cm 5.1     cm 2.0     cm 5.1     cm 19.468 cm + 13.188 cm ̲ 48.856 cm Perimeter 2.0     cm 5.1     cm 2.0     cm 5.1     cm 19.468 cm + 13.188 cm ̲ 48.856 cm

Practice Set B

Exercise 4

Find the exact circumference of the circle.
A circle with a line through the middle, ending at the edges of the circle. The line is labeled, d = 9.1in.

Solution

9.1 π π in.

Exercise 5

Find the approximate circumference of the circle.
A circle with a line through the middle, ending at the edges of the circle. The line is labeled, d = 1.8in.

Solution

5.652 mm

Exercise 6

Find the approximate circumference of the circle with radius 20.1 m.

Solution

126.228 m

Exercise 7

Find the approximate outside perimeter of
A shape best visualized as a hollow half-circle. The thickness is 1.8mm, and the diameter of the widest portion of the half-circle is 16.2mm.

Solution

41.634 mm

Exercises

Find each perimeter or approxi­mate circumference. Use π=3.14π=3.14 size 12{π=3 "." "14"} {}.

Exercise 8

Exercise 9

A triangle with sides of length 8mm, 9.3mm, and 3.8mm.

Exercise 10

Exercise 11

A four-sided polygon with sides of length 0.04ft, 0.07ft, 0.04ft, and 0.095ft.

Exercise 12

Exercise 13

A circle of radius 6m.

Exercise 14

Exercise 15

A circle of diameter 8mm.

Exercise 16

Exercise 17

A half-circle of diameter 1.1mm.

Exercise 18

Exercise 19

A quarter-circle of radius 5 in.

Exercise 20

Exercise 21

A shape best visualized as a rectangle connected to a half-circle on top. The rectangle's height is 4.1in, and the rectangle's width is 7.8in.

Exercise 22

Exercise 23

A shape best described as a rectangle with two half-circle slices taken out of the top and bottom. The rectangle's height is 18m, and the radius of the circles is 6m.

Exercise 24

Exercise 25

A tube in the shape of a half-circle with straight ends. The ends have a width of 2mm, and the inner side of the circular tube has a radius of 7mm.

Exercise 26

Exercise 27

A complex shape.

Exercises for Review

Exercise 28

((Reference)) Find the value of 281310916281310916 size 12{2 { {8} over {"13"} } cdot sqrt {"10" { {9} over {"16"} } } } {}.

Solution

8.5 or 172172 size 12{ { {"17"} over {2} } } {} or 812812 size 12{8 { {1} over {2} } } {}

Exercise 29

((Reference)) Find the value of 815+710+2160815+710+2160 size 12{ { {8} over {"15"} } + { {7} over {"10"} } + { {"21"} over {"60"} } } {}.

Exercise 30

((Reference)) Convert 7878 size 12{ { {7} over {8} } } {} to a decimal.

Solution

0.875

Exercise 31

((Reference)) What is the name given to a quantity that is used as a comparison to determine the measure of another quantity?

Exercise 32

((Reference)) Add 42 min 26 sec to 53 min 40 sec and simplify the result.

Solution

1 hour 36 minutes 6 seconds

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