# Area and Perimeter of Polygons

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Finding Area, Perimeter, and Circumference of circle, square, and other polygons. Area refers to the space occupied by a shape or an object or a surface. Perimeter refers to the measure of the length of the outline or boundary of a shape, an object, or a surface.
1. FINDING AREA, PERIMETER, AND CIRCUMFERENCE
Area, perimeter, and circumference are all measures of two-dimensional shapes. These
are things you can think of as flat: a football field, a piece of paper, or a pizza. You're
probably not interested in how high they are, but you might want to know their:
● Perimeter or Circumference. This is the total length of a shape's outline. If you
built a fence around its edge, how long would that fence be? If you walked
around the edges of this area, how far would you have gone? The length of a
straight-sided shape's outline is called its perimeter, and the length of a circle's
outline is called its circumference.
● Area. This is the total amount of space inside a shape's outline. If you wanted to
paint a wall or irrigate a circular field, how much space would you have to cover?
1. The perimeter of any triangle is the sum of its sides: a + b + c
perimeter = a + b + c
c 5 P=a+b+c
a 3 P=3+5+4
P = 12
b 4
2. The area of any triangle is half its base times its height.
area = 1/2 bh
A = 1/2 bh
h 3 A = 1/2 * 3 * 4
A = 1/2 * 12
b 4 A=6
It doesn't matter which of the triangle's short legs is the "base" and which is the
"height": you get the same solution either way.
A = 1/2 bh
h 4 A = 1/2 * 4 * 3
A=2*3
A=6
b 3
2. 1. A square is a kind of rectangle, and the perimeter of any rectangle is the sum of its
four sides. Since all sides of a square are the same,
perimeter = 4s
P = 4s
s 3 P=4*3
P = 12
s 3
2. The area of a square is equal to any one of its sides times any other: s * s . Since
that's the same as s squared,
area = s2
A = s2
s 3 A = 32
A=3*3
s 3 A=9
1. The perimeter of a rectangle is the sum of its four sides. Since a rectangle has two
equal short sides (width, w ) and two equal long sides (length, l ),
perimeter = 2l + 2w
P = 2l + 2w
w 3 P = (2 * 7) + (2 * 3)
P = 14 + 6
l 7 P = 20
2. The area of a rectangle is equal to its length times its width.
area = l * w
A=l*w
w 3 A=3*7
A = 21
l 7
3. Like squares and rectangles, parallelograms are quadrilaterals: they have four sides
and four interior angles. In a parallelogram those angles are not right angles, but
the opposite sides must still be parallel to each other.
1. The perimeter of a parallelogram is the sum of its four sides. Since a parallelogram
has two equal short sides (width, w ) and two equal long sides (length, l ),
perimeter = 2l + 2w
P = 2l + 2w
w 4 P = (2*5) + (2 * 4)
P = 10 +8
l 5 P = 18
2. The area of a parallelogram is equal to its base (another name for length) times
its height. Its height is not the same as its width: height is measured by a vertical line
perpendicular (at right angles to) the base.
area = b * h
A=b*h
h 3 A=3*5
A = 15
b 5
A trapezoid is also a quadrilateral: it has four sides, but only two are parallel.
1. The perimeter of a trapezoid is the sum of its four sides.
perimeter = a + b + c + d
b 2
P=a+b+c+d
a c 4 3 P=2+3+4+5
P = 14
d 5
2. To find the area of a trapezoid, we use its two bases and its height:
area = 1/2 (b1 + b2) (h) A = 1/2 (b 1 + b 2 ) * (h)
b1 2 A = 1/2 (2 + 5) * 3
A = 1/2 * 7 * 3
h 3 A = 1/2 * 21
A = 10.5
b2 5
4. To find a circle's circumference or area, you first need to know either its
radius: r , the distance from its center to any point on its outer edge, or its
diameter: d , the length of a straight line through the circle's center that touches
any two points on the outer edge.
A circle's radius is always exactly half its diameter.
r d
1. The circumference of any circle equals two times its radius multiplied by pi
(π, approximately 3.14). We can also say it equals pi times its diameter.
circumference = 2 π r OR πd
C=2 π*3
r 3 C=6* π
C ≈ 18.84
Because 3.14 is only an approximate value for pi, we replace the "equals" sign (=)
with the "approximately equals" sign ( ≈). For accuracy, some teachers prefer
to use the symbol: the circumference of this circle is 6 π .
2. To find the area of a circle, square its radius and multiply the result by pi .
2
area = πr
A = πr 2
r 3 A = 32 * π
A = (3 * 3) * π
A=9* π
A = 9π or ≈ 28.26
Created Fall 2012
Pocatello ISU Math Center Idaho Falls
REND 327 Student Success Center CHE 220
208-282-3662 www.isu.edu/success/math 208-282-7925