Geometry Part 1BPerimeter

ByJulia Arnold, Dick Gill and Marcia Tharp

for Elementary Algebra Math 03 online

Perimeter is the distance around an object.

If the object is a rectangle, then it has 4 sides and opposite sides are equal in length and parallel to each other.The formula for its perimeter is P = 2L + 2W where L is length and W is width of the rectangle.

If the object is a circle, we call the perimeter the circumference and ancient mathematicians found its formula to be .dC

In many perimeter problems involving rectangles, you maybe asked to find the dimensions. The dimensions of a rectangleare the width by length. For example, picture frames are categorized by their dimensions, such as 8 by 10, or 11 by 14.The sum of the dimensions represents half of the perimeter.

A picture frame that is an 8 by 10 would have a perimeter of2(8 + 10) or 36 inches. What would be the perimeter of an11 by 14 picture frame?

The dimensions are only half the perimeter.

Answer: An 11 by 14 inch picture frame would have a perimeter of 2(11+14) = 50 inches.

Example 1: Suppose we have a rectangle with a perimeter of 26 inches.We are told that the length is 10 more than twice the width.Can we find the dimensions of the rectangle?

Define the variables:Let x = width, then 10 + 2x = length

The sum of width and length is half the perimeter, thus...

x + 10 + 2x = 26/23x + 10 = 13

3x = 3 x = 1 inch This is the width.

2x + 10 = 2(1) + 10 = 12 which is the length.The dimensions are 1 by 12.

Example 2. The perimeter of a rectangular lot is 440 meters. If the length is 150 m. find the width.

Let x = width

2L + 2W = P 2(150) + 2x = 440 300 + 2x = 440300 + 2x - 300 = 440 - 300 2x = 140 x = 70 m, the width

or L + W = P/2150 + x = 220x = 220 - 150= 70 m, the width

Example 3. The length of a rectangle is 10 more than twice the width. Find the length and the width if the perimeter is 56 inches.

Go ahead and guess now. Remember that 2L + 2W = P or2(L+W) = P or L + W = P/2

Let x = the width2x + 10 = the length

2L + 2W = P 2(2x + 10) + 2x = 56 4x + 20 + 2x = 56 6x + 20 - 20 = 56 - 20 6x = 36 x = 6 inches, the width 2x + 10 = 2(6) + 10 = 22 inches, the length

Or 2x + 10 + x = 56/23x + 10 = 283x = 18x = 6 inches, the width2(6)+10=22 inches, the length

Example 4. The perimeter of a triangle is 85 cm. If the middle side is 5 cm. larger than the smallest side and the large side is twice thesmallest side, find each side.

2L + 2W = P only works for rectangles. Perimeter is the distancearound the outside of the figure. For a triangle then, the perimeteris the sum of the three sides.

Let x = the smallest sidex + 5 = the middle side 2x = the largest side

x + (x + 5) + 2x = 85 4x + 5 - 5 = 85 - 5 4x = 80 x = 20 cm, the shortest side x + 5 = 25 cm, the middle side and 2x = 40 cm

How would you find the perimeter of this architect’s home plan?

10’

This shape consists of a rectangle with twoOverlapping circles. Can we find the perimeterIf we know the dimensions of the rectangle?

20’

Example 5:

This shows the outside of the figure whichIs the perimeter that we must find. The perimeterIs actually two lengths of the rectangle and twoSemi-circles.

We need the formula for the circumference of a circle which is C = 2 pi( r) Where r is the radius of the circle.An alternate formula is C = pi d where d is the diameter of the circle. pi is the number which is approximated by 3.141592654

20’

10’

Since the two semicircles are the same size the two parts make one complete circle. Thus the perimeter is

P = 20 + 20 + (10) = 40 + 10Which is approximately 71.4’20’

10’

Is the Greek symbol for Pi

Other four sided figures which have opposite sides equal are: parallelograms which have opposite sides parallel as well.

A rhombus which has four equal sides with opposite sides parallel

What is the difference between rectangles, squares, parallelograms and rhombuses?

A square is a rectangle which has 4 equal sides and every angle is a right angle.

SquareRectangle

A rectangle is a parallelogram because opposite sides are equal in both and opposite sides are parallel in both.

The difference between the two is the rectangle has four right angles.

A parallelogram has no right angles.

A square is a rhombus, because all four sides are equal and opposite sides are parallel

A rhombus is also a parallelogram because opposite sides are equal and parallel.A square

hasfour right angles.

A rhombus has no right angles.

If a figure is called a quadrilateral, then that means ithas four sides, but they could all be different measurements and they do not have to have opposite sides parallel.

Your Turn

Work out the problems on the nextslides to see if you understand thisconcept.

1. The perimeter of a rectangular lot is 440 meters. If the length is 150 m. find the width.Complete Solution

2. The perimeter of a triangle is 85 cm. If the middle side is 5 cm. larger than the smallest side and the large side is twice the smallest side, find each side. Complete Solution

3. A rectangular lot has a perimeter of 80 feet and length of 25 feet.What is the width of the lot?

4. Mr. Green wants to make a rectangular garden plot. He wants the length to be twice the width plus 3 feet. If the perimeter of the lot is 66 feet, what are the dimensions of the lot?

Complete Solution

Complete Solution

5. Mr. Fixit wants to fence in a rectangular lot where one side will be the wall of his barn. He has 60 feet of fencing available. If the length is 4 feet less than twice the width. What are the dimensions of the lot?

Complete Solution

6. A quadrilateral has sides measuring 16.5 in. , 17.3 in. , 21.8 in., 29.2 in.What is its perimeter?

7. An Equilateral triangle has a side measuring 1.28 cm. What is its perimeter?

8. An Isosceles triangle has its equal side measuring 15.4 in. and its 3rd side measuring 23.9 in. What is its perimeter?

9. A square measures 1.63 m. What is its perimeter?

10. A rhombus measures 8.06 ft. on a side. What is its perimeter?

11. A parallelogram measures 47.2 on one side and 36.8 on another. What is its perimeter?

12. A rectangle has dimensions 57.9 by 39.8 in. What is its perimeter?

Complete SolutionsWhen complete go to next slide.

13. Find the circumference of the circle whose radius is 14.3 cm.Round to the nearest tenth.14. Find the circumference of the circle whose diameter is 8.4 in.Round to the nearest tenth.

Find the perimeters of the indicated figures:15.

12 ft.

6 ft.7 ft.

11 ft.

18 ft.

16.60.8 ft.

46.0 ft.

17.

28 cm.

12 cm.

14 cm.

16 cm

12 cmComplete Solutions

18. 15.3 cm.

19.6 cm

Complete Solutions

Find the perimeter of this figure.

1. The perimeter of a rectangular lot is 440 meters. If the length is 150 m. find the width.

2. The perimeter of a triangle is 85 cm. If the middle side is 5 cm. larger than the smallest side and the large side is twice the smallest side, find each side.

Length + width = 1/2 perimeter150 + x = 440/2150 + x = 220x = 70The width is 70

X = smallest sidex + 5 = middle side2x = largest sidex + x + 5 + 2x = 854x + 5 = 854x = 80x = 20

X = 20x + 5 = 252x = 40check20 + 25 + 40 = 85

Return to Problems

3. A rectangular lot has a perimeter of 80 f eet and length of 25 f eet.What is the width of the lot?

Let x = widthx + 25 = 80/2x + 25 = 40x = 15 The width of the lot is 15 feet.

Let x = width, length = 2x + 3x + 2x + 3 = 66/23x + 3 = 333x = 30x = 10 2x + 3 = 23 The dimensions are 10 x 23 feet.

Return to Problems

4. Mr. Green wants to make a rectangular garden plot. He wants the length to be twice the width plus 3 feet. If the perimeter of the lot is 66 feet, what are the dimensions of the lot?

5. Mr. Fixit wants to f ence in a rectangular lot where one side will bethe wall of his barn and represent the lenth. He has 60 f eet off encing available. I f the length is 4 f eet less than twice the width.What are the dimensions of the lot?

Barn Wall

Three sides fencing

Let x = width2x – 4 = length

2x-4

x x

Add 3 sides: x + x + 2x – 4 = 60 4x – 4 = 60 4x = 64 x = 16 2x – 4 = 28

The dimensions are16 by 28 feet.

Return to Problems

.

6. A quadrilateral has sides measuring 16.5 in. , 17.3 in. , 21.8 in., 29.2 in.What is its perimeter? 84.8 inches

7. An Equilateral triangle has a side measuring 1.28 cm. What is its perimeter? 1.28*3= 3.84 cm

8. An Isosceles triangle has its equal side measuring 15.4 in. and its 3rd side measuring 23.9 in. What is its perimeter? 15.4*2+23.9 = 54.7 in.

9. A square measures 1.63 m. What is its perimeter? 1.63*4 = 6.52 m

10. A rhombus measures 8.06 ft. on a side. What is its perimeter? 8.06*4 = 32.24 ft.

11. A parallelogram measures 47.2 on one side and 36.8 on another. What is its perimeter? 47.2*2+36.8*2 = 168

12. A rectangle has dimensions 57.9 by 39.8 in. What is its perimeter?2(57.9+39.8)= 195.4

Return to Problems

13. Find the circumference of the circle whose radius is 14.3 cm.Round to the nearest tenth. Ans. 2 (14.3) = 89.814. Find the circumference of the circle whose diameter is 8.4 in.Round to the nearest tenth. Ans (8.4) =26.4

Find the perimeters of the indicated figures:15.

12 ft.

6 ft.7 ft.

11 ft.

18 ft.

16.60.8 ft.

46.0 ft.

17.

28 cm.

12 cm.

14 cm.

16 cm

12 cm

6+7+11+18+12= 54 ft.

60.8*2+46+46*pi/2=239.9 ft.

3/4 of a rectangleplus 1/2 of a circle

28+14+16=58

58 + 12+28+12+14+12+16+12=164cm

Return to Next ProblemsReturn to same questions

18. 15.3 cm.

19.6 cm

This shape is a combination of a rectangle and two semi-circles.The two semi-circles = one circle.Top of Rectangle + bottom of rectangle + circumference of circle = perimeter15.3 + 15.3 + 19.6 = 30.6 + 19.6 = 92.2 cm

Back to beginning of questions End show

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