how to find the perimeter of different geometrical shapes
How to Find the Perimeter of Different Geometrical Shapes
Having trouble in understanding how to find perimeters of different geometrical shapes? Buzzle 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!INDEX
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 a certain point (known as 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.
a. 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
b. 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.
c. 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)
For 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.
a. 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
b. 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 = 16 + 8 + 33.3
P = 57.3 cm
| P = b + d + h | ( | 1 | + | 1 | ) |
| Sin(C) | Sin(A) |
| P = 16 + 8 + 3 | ( | 1 | + | 1 | ) |
| Sin(53) | Sin(45) |
| 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 |
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.
a. 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 shown below.
n = 6 (hexagon has six sides)
s = 5 cm
P = 6 x 5
P = 30 cm
c. 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