Transcript Geometry - TCC: Tidewater Community College

Geometry Part 1B
Perimeter
By
Julia 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 C  d .
In many perimeter problems involving rectangles, you may
be asked to find the dimensions. The dimensions of a rectangle
are 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.
The dimensions are only half the perimeter.
A picture frame that is an 8 by 10 would have a perimeter of
2(8 + 10) or 36 inches. What would be the perimeter of an
11 by 14 picture frame?
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 = 26/2
3x + 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 = 440
300 + 2x - 300 = 440 - 300
2x = 140
x = 70 m, the width
or L + W = P/2
150 + x = 220
x = 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 or
2(L+W) = P or L + W = P/2
Let x = the width
2x + 10 = the length
Or 2x + 10 + x = 56/2
3x + 10 = 28
3x = 18
x = 6 inches, the width
2(6)+10=22 inches, 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
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 the
smallest side, find each side.
2L + 2W = P only works for rectangles. Perimeter is the distance
around the outside of the figure. For a triangle then, the perimeter
is the sum of the three sides.
Let x = the smallest side
x + 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?
Example 5:
20’
10’
This shape consists of a rectangle with two
Overlapping circles. Can we find the perimeter
If we know the dimensions of the rectangle?
This shows the outside of the figure which
Is the perimeter that we must find. The perimeter
Is actually two lengths of the rectangle and two
Semi-circles.
20’
10’
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
Since the two semicircles are the same size the two
parts make one complete circle. Thus the perimeter is
P = 20 + 20 + (10) = 40 + 10
Which 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.
Rectangle
Square
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 square
has
four right
angles.
A
rhombus
has no
right
angles.
A rhombus is also a
parallelogram because
opposite sides are equal and
parallel.
If a figure is called a quadrilateral, then that means it
has 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 next
slides to see if you understand this
concept.
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?
Complete Solution
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
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?
When complete go to next slide.
Complete Solutions
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.
15.
Find the perimeters of the indicated figures:
6 ft.
12 ft.
17.
7 ft.
28 cm.
11 ft.
12 cm.
18 ft.
16.
60.8 ft.
14 cm.
46.0 ft.
16 cm
Complete Solutions
12 cm
Find the perimeter of this figure.
18.
15.3 cm.
19.6 cm
Complete Solutions
1. The perimeter of a rectangular lot is 440 meters. If the length is 150 m.
find the width.
Length + width = 1/2 perimeter
150 + x = 440/2
150 + x = 220
x = 70
The width is 70
Return to Problems
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.
X = smallest side
x + 5 = middle side
2x = largest side
x + x + 5 + 2x = 85
4x + 5 = 85
4x = 80
x = 20
X = 20
x + 5 = 25
2x = 40
check
20 + 25 + 40 = 85
3.
A rectangular lot has a perimeter of 80 feet and length of 25 feet.
What is the width of the lot?
Let x = width
x + 25 = 80/2
x + 25 = 40
x = 15 The width of the lot is 15 feet.
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?
Return to Problems
Let x = width, length = 2x + 3
x + 2x + 3 = 66/2
3x + 3 = 33
3x = 30
x = 10 2x + 3 = 23 The dimensions are 10 x 23 feet.
5.
Mr. Fixit wants to fence in a rectangular lot where one side will be
the wall of his barn and represent the lenth. 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?
Barn Wall
x
x
2x-4
Three sides fencing
Let x = width
2x – 4 = length
Add 3 sides: x + x + 2x – 4 = 60
4x – 4 = 60
4x = 64
x = 16
2x – 4 = 28
The dimensions are
16 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) = 89.8
14. Find the circumference of the circle whose diameter is 8.4 in.
Round to the nearest tenth. Ans
(8.4) =26.4
15.
Find the perimeters of the indicated figures:
6 ft.
12 ft.
7 ft.
6+7+11+18+12= 54 ft.
11 ft.
18 ft.
16.
46.0 ft.
60.8 ft.
17.
28 cm.
28+14+16= 58 + 12+
28+12+14+
58
12+16+12=
164cm
12 cm.
14 cm.
3/4 of a rectangle
plus 1/2 of a circle
16 cm
60.8*2+46+46*pi/2=239.9 ft.
Return to same questions
Return to Next Problems
12 cm
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 = perimeter
15.3 + 15.3 + 19.6
= 30.6 + 19.6 = 92.2 cm
Back to beginning of questions
End show
Go to Geometry Part II
Similar Triangle