How to Find the Perimeter of Different Geometrical Shapes?

Amita Ray Feb 8, 2019
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Having trouble in understanding how to find perimeters of different geometrical shapes? ScienceStruck comes to your rescue by making geometry easier than ever!

Fun Fact

The perimeter or circumference of the Earth is 24,901 miles, i.e., almost 40,075 kilometers!
In mathematics, geometry deals with the shape, size, relative positions, the three dimensional orientation of figures in space. It deals with three basic measurements of shapes:area, volume, and perimeter.
Area is the measure of the extent of a two-dimensional figure or shape; surface area can be described as the measurement of the extent of the surface of an object. It is the measure of the three-dimensional space enclosed by an object.
Perimeter can simply be described as the length of the path that surrounds a two-dimensional shape. In other words, it is the distance around the shape. Let us now take a look on how to find perimeter of different geometrical shapes.

SQUARE

A square is a quadrilateral that has all four sides and all four angles equal (all are 90°).
Example: To find the perimeter of a square with sides 5 cm, we use the formula shown in the figure.
P = a + a + a + a
P = 5 + 5 + 5 + 5
P = 20 cm
The same formula can be used to calculate the perimeter of a rhombus.

RECTANGLE

A rectangle is a quadrilateral that has all four angles equal (all are 90°). The opposite sides of a rectangle are equal in length (whereas the adjacent sides aren't).
Example: To find the perimeter of a rectangle, we use the formula shown in the figure.
l = 15 cm
b = 25 cm
P = 2 (15 + 25)
P = 2 (40)
P = 80 cm
You can use the same formula to find the perimeter of a parallelogram.

CIRCLE

A circle can be described as a set of points that are at equal distance from the center. The perimeter of a circle is known as its circumference, denoted by C.
Example: To find the circumference, we use the formula shown in the figure.
r = 7 cm or d = 14 cm (d = r + r)
C = 2πr or πd
C = 2 X 3.14 X 7 or 3.14 X 14
C = 43.96 cm

SEMICIRCLE

A semicircle, simply put, is half a circle, its perimeter will thus be half of that of a circle.
Example: To find the perimeter of a semicircle, we use the formula shown in the figure.
r = 7 cm or d = 14 cm (d = r + r)
P = πr or πd/2
P = 2 X 3.14 X 7 or 3.14 X 14/2
P = 21.98 cm

SECTOR

A sector can be described as a part of a circle.
Example: To find the perimeter of a sector, we use the formula shown in the figure.
ϴ = 60°
r = 7 cm
P = 60/360 X 2 X 3.14 X 7
P = 7.33 cm

TRIANGLE

A triangle is a polygon that has three sides and three vertices. Let us take three cases into consideration in order to determine its perimeter.
1. When all the three sides are known.
To find the perimeter of a triangle, we use the formula shown in the figure.
a = 14 cm
b = 16 cm
c = 15 cm
P = 14 + 16 + 15
P = 45 cm
2. For a right-angled triangle when its hypotenuse is unknown.
To find the perimeter of a right-angled triangle, we use the formula shown in the figure.
b = 3 cm
h = 4 cm
P = b + h + b2 + h 2
P = 3 + 4 + 32 + 4 2
P = 3 + 4 + 5
P = 12 cm

If any other side is unknown, you can use the Pythagorean formula to find the side first and then calculate the perimeter.
3. For any other triangle when only two sides and the angle they make are known.
We first need to find the length of the side using the law of cosines,
When a, b, and c are the length of sides of a triangle, and A, B, and C are opposite angles of sides a, b, and c, respectively; we can find the length of the unknown side (let say, c) with the formula:

c2 = a 2 + b 2 - 2 a.b cos(C)
Example:
a = 4 cm
b = 2 cm
c2 = 4 2 + 2 2 - 2 4.2 cos(45)
c2 = 16 + 4 - 2 (0.876)
c2 = 20 - 1.752
c2 = 18.284
c = 4.272 cm

P = a + b + c
P = 4 + 2 + 4.272
P = 10.272 cm

TRAPEZOID

A trapezoid is a quadrilateral with at least one pair of parallel lines. The parallel lines are called the base of the trapezoid, and the other sides are known as the legs of the trapezoid. The distance between the parallel lines is known as the height of the trapezoid.
Let us consider three different scenarios to find the perimeter.
1. When all sides are known.
To find the perimeter of a trapezoid, we use the formula shown in the figure.
a = 4 cm
b = 16 cm
c = 5 cm
d = 8 cm
P = 4 + 16 + 5 + 8
P = 33 cm
2. When its sides (legs) are unknown.
To find the perimeter of a trapezoid, we use the formula shown in the figure.
b = 16 cm
h = 3 cm
d = 8 cm

P = b + d + h ((1/Sin(C)) + (1/ Sin(A)))
P = 16 + 8 + 3 ((1/Sin(53)) + (1/Sin(45)))
P = 16 + 8 + 33.3
P = 57.3 cm
3. When one of the base and height are unknown.
Imagine if we were to cut the trapezoid from two sides in such a way that length of the bases are equal, and when we join the cut portion, we get a triangle like the one shown in the figure.
When A and C are equal; all the three angles are of 60°. This triangle is an equilateral triangle, and thus, when the length of the side is added to the base, we get the length of the larger base. When the angles are unequal; the sum of the angles subtracted by 180°.
The area of this triangle can be calculated by the formula

A = ½ X a X c X sin (B)

To find the perimeter of a trapezoid,
a = 4 cm
c = 6 cm
d = 11 cm
A = 53°
C = 65°
B = 78°

Area = ½ X 4 X 6 X sin 78
Area = 6.12 cm2
Base of triangle= (Area)/(½ x a x Sin(C))

Base = (6.12)/(½ x 4 x Sin(65))
Base = (6.12)/(2 x 0.826)
Base = 3.70 cm
Base of the trapezoid = 11 + 3.70 = 14.70 cm

Now that we have the sides and base of the trapezoid, we can find the perimeter.
P = 14.7 + 4 + 6 + 11
P = 35.7 cm

POLYGON

Any closed figure where the line segments do not intersect each other gives rise to a polygon. The sum of the internal angle of a polygon is always 360°, and they are named according to the number of sides they possess.
1. A regular polygon has all its sides equal, so when the number of sides and the length of each side is known the perimeter of the polygon can be calculated using the formula shown in the figure.
Example: If a hexagon has sides of length 5 cm, its perimeter can be calculated as

s= 5 cm
n= 6
P= n X s
P= 5 X 6 = 30 cm
2. When the length of the sides of a polygon is not known, then its perimeter can be calculated using the formula given here.
s = 2 X a X tan (180/n)
Here, a is the apothem.
Apothem is a line segment from the center of the polygon to the midpoint of a side.

s = 2 X r X tan (180/n)
r is the radius.
The distance from the center of a regular polygon to any vertex.
Example: For a hexagon of apothem 4 cm, its side can be calculated as.
s = 2 x 4 x tan (180/6)
s = 8 x tan (30)
s = 8 x 0.58
s = 4.62 cm

P = 6 x 4.62 = 27.71 cm

For a hexagon of radius 4 cm, its side can be calculated as.
s = 2 x 4 x sin (180/6)
s = 8 x sin (30)
s = 8 x 0.5
s = 4.00 cm

P = 6 x 4.00 = 24 cm
3. For an irregular polygon when all its sides are unequal, we can calculate its perimeter by simply adding the length of all of its sides.
Example: An irregular polygon of six sides
S1 = 8 cm
S2 = 6 cm
S3 = 4 cm
S4 = 7cm
S5 = 5 cm
S6 = 4 cm

P = S1 + S2 + S3 + S4 + S5 + S6
P = 8 + 6 + 4 + 7 + 5 + 4
P = 36 cm
We know that geometry can be a bit daunting at first (believe us, we know) but keep practicing and you will surely keep getting better with every attempt.
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