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Chapter 9—Basic Geometry

9-2 Finding Perimiters and Circumferences

 When you complete the work for this section, you should be able to do the following: Explain the meaning of perimeter. Describe how to find the perimeter of any triangle or quadrilateral. Given the length of all sides, calculate the perimeter of a triangle, square, rectangle, parallelogram, and trapezoid. Explain the meaning of circumference. Given the diameter or radius of a circle, calculate its circumference.

 Definitions The perimeter of a plane figure is the distance around its outside borders. The distance around a circle is not called perimeter. Instead, it is called the circumference.
• To find the perimeter of triangles, squares, rectangles, parallelograms, and trapezoids, simply add up the lengths of the sides.
• You need to use a special formula, C = 2pr, to find the circumference of a circle. This is described later in this lesson.

Perimeter of a Triangle

The perimeter, or total distance around, a triangle is equal to the sum of the lengths of the three sides.

Equation

 The equation for the perimeter of any triangle is: P = a + b + c Where: a, b, and c are the lengths of the three sides P is the perimeter

Example

Examples and Exercises

 Use these interactive examples and exercises to strengthen your understanding and build your skills: Note: Use this figure only for reference. The proportions do not necessarily match those cited in the problem.

Quadrilateral is a technical term for any 4-sided figure, including squares, rectangles, parallelograms, and trapezoids.

The perimeter of any quadrilateral is simply the distance around the outside of the figure. The perimeter is equal to the sum of the lengths of the four sides.

• For the rectangle: P = a + b + c + d
• For the square: P = a + b + c + d
• For the parallelogram: P = a + b + c + d
• For the trapezoid: P = a + b + c + d

The formlas for the rectangle, square, and parallelogram are seldom shown this way, however. The reason is that these figures tend to have pairs of sides that have equal length. The basic equations, in other words, can be simplified.

For a rectangle, sides a and c have the same length. Likewise, sides b and d have the same length. One pair of sides is called the length of the rectangle, and the other pair of equal sides is called the width.  So the permieter of a rectangle is more commonly shown in terms of  the length and width of its sides.

Equations

 The equation for the perimeter of any rectangle is: P = 2(l + w) Where: l is the length of one pair of parallel sides w is the length of the second pair of parallel sides P is the perimeter

Example

 Problem For a certain rectangle, one set of parallel sides is 8 cm long and the second set is 12 cm long. What is the perimeter of this rectangle? Procedure Cite the appropriate equation P = 2(l + w) Assign the given values P = 2( 8 + 12 ) Solve the equation and simplify P = 2( 20 ) P = 40 Solution The perimeter of this rectangle is 40 cm.

Examples and Exercises

 Use these interactive examples and exercises to strengthen your understanding and build your skills: Note: Use this figure only for reference. The proportions do not necessarily match those cited in the problem.

As far as the perimeter is concerned, a parallelogram is identical to a rectangle.

Equation

 The equation for the perimeter of any parallelogram is: P = 2(a + b) Where: a is the length of one pair of parallel sides b is the length of the second pair of parallel sides P is the perimeter

Example

 Problem The parallel sides of a certain parallelogram measure 16 ft and 24 ft.  What is the perimeter? Procedure Cite the appropriate equation P = 2(a + b) Assign the given values P = 2( 16 + 24) Solve the equation and simplify P = 2( 40 ) P = 80 Solution The perimeter of this parallelogram is 80 ft.

Examples and Exercises

 Use these interactive examples and exercises to strengthen your understanding and build your skills: Note: Use this figure only for reference. The proportions do not necessarily match those cited in the problem.

All four sides of a square have the same length. You can find the perimeter of any plane figure by simply adding up the lengths of the sides: P = s + s + s + s.  But since the four sides of a square have the same length, it is even simpler to find the perimter by multiplying the length of a side by 4: P = 4s.

Equation

 The equation for the perimeter of any square is: P = 4s Where: s is the length of the sides P is the perimeter

Example

 Problem A certain square measures 6 inches on each side. What is the perimeter of this square? Procedure Cite the appropriate equation P = 4s Assign the given values P = 4 · 6 Solve the equation P = 24 Solution The distance around this particular square is 24 inches.

Examples and Exercises

 Use these interactive examples and exercises to strengthen your understanding and build your skills:

Perimeter of a Trapezoid

None of the sides of a trapezoid are necessarily equal, so the basic equation for the perimeter cannot be simplified. It has the be the basic relationship P = a + b + c + d.

Equation

 The equation for the perimeter of any trapezoid is: P = a + b + c + d Where: a, b, c, d are the lengths of the four sides P is the perimeter

Example

The Problem

The sides of a trapezoid measure 4 in, 6 in, 8 in, and 2 in. What is the perimeter of this trapezoid?

The Solution

 Cite the appropriate equation P = a + b + c + d Assign the given values P = 4 +  6 + 8 + 2 Solve the equation and simplify P =  20

So the perimeter of this trapezoid is 20 inches.

Examples and Exercises

 Use these interactive examples and exercises to strengthen your understanding and build your skills: Note: Use this figure only for reference. The proportions do not necessarily match those cited in the problem.

Circumference of a Circle

Finding the distance around a circle—the circumference—isn't quite as straightforward as finding the perimeter of straight-line plane figures such as triangles and quadralaterals.

Equation

 The equation for the circumference of any circle is: C = 2pr or C = pd Where: p = approximately 3.14 or 22/7 r is the radius of the circle d is the diameter of the circle C is the circumference

Example

The Problem

What is the circumference of a circle that has a radius of 15 cm?

The Solution

 Cite the appropriate equation C = 2pr Assign the given values C = 2 · 3.14 · 15 Solve the equation and simplify C = 94.2

The circumference is 94.2 cm.

Example

The Problem

What is the circumference of a circle that has a diameter of 100 ft?

The Solution

 Cite the appropriate equation C = pd Assign the given values C = 3.14 · 100 Solve the equation and simplify C = 314

The circumference is equal to 314 ft.

Examples and Exercises

 Use these interactive examples and exercises to strengthen your understanding and build your skills:

Summary of Equations for the Perimeter or Circumference of Plane Figures

 Triangle P = a + b + c Square P = 4s Rectangle P = 2(l + w) Parallelogram P = 2(a + b) Trapezoid P = a + b + c + d Circle C = 2pr C = pd

Examples & Exercises

These interactive examples and exercises give you a chance to test your understanding of finding the distance around all of these plane figures.