Perimeter Area and
Circumference
The perimeter of a polygon is the distance around the outside of the polygon. It’s
equal to the sum of the lengths of the sides of the polygon.
The area of a figure can be thought of as the space in the plane that the figure
takes up.
The distance around the outside of a circle is called the circumference, rather
than the perimeter
Unit Square
A unit square is a square whose sides are each one unit in length.
The unit can be inches, meters, miles, centimeters or any other
length measurement. We say the area of the unit square equals
one square unit (1 sq. in., or 1 sq. m. etc.)
Area = 1 sq. unit
Rectangle
A rectangle is a quadrilateral with 4 right angles.
Rectangle
l
w w
l
To find the perimeter of a rectangle we need to know the length of its sides, l and w.
Because opposite sides are equal we get the formula
Perimeter = 2l + 2w
Rectangle
width
length
To find the area of a rectangle see how many unit squares will fit in it.
The number of unit squares that will fit in the rectangle equals the area
of the rectangle. Notice that the total number of unit squares that will fit
in the rectangle equals the number of squares across times the number of
squares down. This leads us to Area = length x width
Rectangle
w = 6 ft.
l = 23 ft.
What is the perimeter of this rectangle?
Rectangle
w = 6 ft.
l = 23 ft.
Perimeter = 2l + 2w = 2(23) + 2(6) = 46 + 12 = 58 ft.
Rectangle
w = 6 ft.
l = 23 ft.
What is the area of this rectangle?
Rectangle
w = 6 ft.
l = 23 ft.
Area = 23 x 6 = 138 sq. ft.
Square
A square is a special type of rectangle with length = width.
If we call the length of a side s then
Perimeter = 4s
Area = s2
Square
5.5 yds
What is the perimeter of this square?
Square
5.5 yds
Perimeter = 4s = 4(5.5) = 22.0 yds.
Square
5.5 yds
What is the area of this square?
Square
5.5 yds
2
Area = s = 5.52 = 30.25 sq. yds.
PARALLELOGRAM
A parallelogram is a quadrilateral with opposite
sides parallel. Opposite sides are also equal.
PARALLELOGRAM
s
b
The perimeter of a parallelogram is equal to the sum of the
lengths of its sides. Because opposite sides are equal we get
Perimeter = 2b + 2s
PARALLELOGRAM
To find the area of a parallelogram drop a line from the upper
corner to the line below, forming a right triangle.
PARALLELOGRAM
Move the
created triangle
to the other side
of the
parallelogram...
...creating a rectangle
whose area
is the same as
the original
parallelogram.
The area of the height
rectangle is
equal to the
base of the
parallelogram times
its height. Thus the base
area of the parallelogram
itself is equal to its base times Area = (base) x (height)
its height
PARALLELOGRAM
height
base
Area = (base)(height)
PARALLELOGRAM
150 in.
175 in.
230 in.
What is the perimeter of this parallelogram?
PARALLELOGRAM
150 in.
175 in.
230 in.
Perimeter = 2b + 2s = 2(230) + 2(175) = 460 + 350 = 810 in.
PARALLELOGRAM
150 in.
175 in.
230 in.
What is the area of this parallelogram?
PARALLELOGRAM
150 in.
175 in.
230 in.
Area = 230 x 150 = 34,500 sq. in.
PARALLELOGRAM
12.5 yds. 10.5 yds.
7 yds.
What is the perimeter of this parallelogram?
PARALLELOGRAM
12.5 yds. 10.5 yds.
7 yds.
Perimeter = 2b + 2s = 2(7) + 2(12.5) = 14 + 25 = 39 yds.
PARALLELOGRAM
12.5 yds. 10.5 yds
7 yds.
What is the area of this parallelogram?
PARALLELOGRAM
12.5 yds. 10.5 yds
7 yds.
Area = 7 x 10.5 = 73.5 sq. yds.
TRIANGLE
A triangle is a three sided figure.
TRIANGLE
s2 s1
s3
The perimeter of a triangle equals the sum of the lengths of its sides.
Perimeter =s1 + s2 + s3
TRIANGLE
To determine the area
of a triangle begin by
drawing a second triangle
the exact size and shape
of the first but rotated
180 degrees
(upside down)...
TRIANGLE
…line the two triangles
up to make a parallelogram
which will be twice
the area of either triangle.
For a parallelogram
we already know height
Area = (base) x (height).
From this we can
conclude that for
a triangle its area
will be 1/2(base)(height) base
TRIANGLE
height
base
Area = 1/2 (base)(height)
TRIANGLE
9.5 ft.
7 ft.
5 ft
11.5 ft.
What is the perimeter of this triangle?
TRIANGLE
9.5 ft.
7 ft.
5 ft
11.5 ft.
Perimeter = s1 + s2 + s3 = 11.5 + 7 + 9.5 = 28.0 ft.
TRIANGLE
9.5 ft.
7 ft.
5 ft
11.5 ft.
What is the area of this triangle?
TRIANGLE
5 ft
11.5 ft.
Area = (1/2)(base)(height)= (1/2)(11.5)(5) = 28.75 sq. ft.
TRIANGLE
43 cm. 45 cm. 58 cm.
39 cm.
What is the perimeter of this triangle?
TRIANGLE
43 cm. 45 cm. 58 cm.
39 cm.
Perimeter = s1 + s2 + s3 = 39 + 45 + 58 = 142 cm.
TRIANGLE
43 cm. 45 cm. 58 cm.
39 cm.
What is the area of this triangle?
TRIANGLE
43 cm.
45 cm. 58 cm.
39 cm.
Area = (1/2)(base)(height) = (1/2)(39)(43) = 838.5 sq. cm.
TRAPEZOID
b
B
A trapezoid is a quadrilateral with 2 sides parallel. The two
parallel sides are called the bases and are labeled B and b.
TRAPEZOID
b
h h
B
To find the area start by drawing a diagonal line making two triangles.
The area of the lower triangle is (1/2)(B)(h).
The area of the upper triangle is (1/2)(b)(h).
The height of both triangles is h, the distance between the two bases.
TRAPEZOID
b
h h
B
The area of the trapezoid is the sum of the areas of the two triangles.
Area = (1/2)(B)(h) + (1/2)(b)(h)
Factoring (1/2)(h) out of each term we get
Area = (1/2)(h)(B + b)
Trapezoid
20 in.
21 in. 18 in. 21 in.
50 in.
What is the perimeter of this trapezoid?
Trapezoid
20 in.
21 in. 18 in. 21 in.
50 in.
Perimeter = B + b + s1 + s2 = 50 + 20 + 21 + 21 = 112 in.
Trapezoid
20 in.
18 in.
50 in.
Area = (1/2)(h)(B + b) = (1/2)(18)(50 + 20) = 630 sq. in.
Trapezoid
33 m.
9 m.
16 m.
What is the area of this trapezoid?
Trapezoid
33 m.
9 m.
16 m.
Area = (1/2)(h)(B + b) = (1/2)(9)(33 + 16) =220.5 sq. m.
Circle
A circle is the set of points which are all the same distance from a given point called the center.
The diameter (d) is the length of a segment drawn from one side of the circle to the other going
through the center.
The radius (r) is the length of a segment drawn from the center to a point on the circle. The
radius equals 1/2 the diameter.
The circumference (C) is the distance around the outside of the circle.
For any circle, regardless of its size, the ratio C/d is the constant (approx. 3.14159). This
leads us to the formula Circumference = d
= 2r
r
d
The area of a circle is Area = r2
Circle
5 ft.
What is the circumference and area of this circle?
Circle
5 ft.
Circumference = d = (5) = 3.14(5) = 15.70 ft.
r = (1/2)d
Area = r2 = (2.5)2 = 6.25 = 3.14(6.25) = 19.635 sq ft.
Circle
2.2 m.
What is the circumference and area of this circle?
Circle
2.2 m.
d = 2r
Circumference = d = 4.4 = 3.14(4.4) = 13.82 m.
Area = r2 = (2.2)2 = 4.84 = 3.14(4.84) = 15.20 sq. m.