What is the Perimeter?
We were introduced to the word in geometry. The perimeter of a closed shape is the total length of the boundary of the shape. The shape may consist of straight line segments or curves. The figures made up of line segments are called polygons. By adding the lengths of all the line segments of a polygon we can calculate its perimeter.
Perimeter is a one dimensional quantity and it is expressed in feet, yards, centimeters, meters or miles.
For example, if you want to fence a piece of agricultural land with barbed wire, the length of wire required for fencing is calculated by finding the perimeter of the land.
Perimeter of Different Figures
- Perimeter of a Triangle:
The triangle is a closed geometric figure with only three sides. The perimeter of a triangle is the sum of the length of all three sides.
Perimeter of a triangle: P = a + b + c, where a, b, and c are the side lengths of the triangle.
- Perimeter of a Rectangle:
A rectangle is a closed geometrical shape with four sides. This shape has opposite sides that are congruent and parallel, with each interior angle being a right angle. The perimeter of a rectangle is twice the sum of its length and width.
Perimeter of a rectangle: P = 2(l + w), where l and w are the length and width of the rectangle.
- Perimeter of a Parallelogram:
A parallelogram is a closed geometrical shape with four sides. The opposite sides are congruent and parallel in a parallelogram. The perimeter of a parallelogram is double the sum of the lengths of a pair of adjacent sides.
Perimeter of a parallelogram: P = 2(a + b), where a and b are side lengths of the parallelogram.
- Perimeter of a Square and a Rhombus:
Both a square and a rhombus are closed geometrical shapes with four sides. In a square each interior angle is a right angle but not in a rhombus. The perimeter of both these quadrilaterals is four times the side length.
Perimeter of a rhombus: P = 4s, where s is the side length of rhombus.
Perimeter of a square: P = 4s, where s is the side length of the square.
- Perimeter of a Kite:
The kite is a closed geometrical shape with four sides in which both pairs of adjacent sides are equal in length. The perimeter of a kite is double the sum of the length of a pair of unequal sides.
Perimeter of kite: P = 2(a + b), where a and b are the unequal side lengths of a kite.
- Perimeter of a Trapezoid
A trapezoid is a closed geometrical shape with four sides, like squares, parallelograms and rectangles. This shape has exactly one pair of opposite sides that are parallel to each other. The perimeter of a trapezoid is the sum of the length of all its sides.
Perimeter of a trapezoid: P = a + b + c + d , where a and b are the side lengths of the parallel sides, and c and d are the lengths of the non parallel sides.
- Perimeter (or Circumference) of a Circle:
In geometry, a circle is a closed curve that can be obtained by tracing a line that is at a fixed distance from a fixed point. A circle does not have edges or vertices like polygons. The perimeter of a circle is called circumference, and is double of pi times radius.
Circumference of a circle: C = 2\(\pi\)r, where r is the radius of a circle.
- Perimeter of a Regular Polygon:
A regular polygon is a closed geometric shape in which all the sides are equal. The perimeter of a regular polygon with ‘n’ sides is ‘n’ times the length of one side.
Perimeter of above regular polygon: P = ns, where n is the number of sides and ‘s’ is the side length of the regular polygon.
- Perimeter of an Irregular Polygon:
An irregular polygon is a closed geometric shape made up of line segments. In an irregular polygon, the measurement of at least one side is different from the measure of the other sides. The perimeter of an irregular polygon is the sum of all its sides.
Perimeter of above irregular polygon: P = a + b + c + d + e + f, where a, b, c, d, e, and f are the side lengths of the irregular polygon.
Solved Examples
Example 1: Find the perimeter of the given trapezoid.
Solution:
AD = 14 cm, CD = 10 cm, AB = 6 cm, BC = 15 cm
Perimeter of the trapezoid:
P = a + b + c + d Formula of perimeter of trapezoid
= 6 + 10 + 14 + 15 Substitute values
= 45 cm Add
So, the perimeter of the trapezoid is 45 centimeters.
Example 2: Find the missing length in the shape in the image. THe perimeter of this shape is 90 cm.
Solution:
Let the missing length in the shape, EF = x cm
P = AB + BC + CD + DE + EF + FG + GH + HQ Formula for the perimeter of irregular polygons
90 = 23 + 18 + 3 + 4 + x + 8 + 20 + 11 Substitute the values
90 = x + 87 Add
3 = x Subtract each side by 87
So, the length of EF is 3 centimeters.
Example 3: Annie transformed a rectangular shaped wire of length 88 centimeters, into the shape of a circle. What is the radius of this circle? (use \(\pi=\frac{22}{7}\))
Solution:
C = 2\(\pi\)r Formula for the circumference of a circle
88 = 2 x \(\frac{22}{7}\) x r Substitute the values
616 = 44 x r Multiply each side by 7
14 = r Divide each side by 44
So, the radius of the circle is 14 centimeters.
Example 4: John has a small piece of garden in which he decides to grow a few flowering plants. This garden is in the shape of an equilateral triangle. The length of the garden’s boundary is 33 meters. Find the length of the side of the triangle.
Solution:
Let x meter be the side length of the equilateral triangle.
P = x + x + x Perimeter formula of triangle
33 = 3x Substitute
11 = x Divide each side by 3
So, the length of the garden in the shape of an equilateral triangle is 11 meters.
The perimeter is the length covered by the boundary of a closed shape.
The units of perimeter are centimeters, millimeters, meters, kilometers, inches, yards, miles, feet and other related units of length.
The perimeter of a shape is the length covered by the shape’s boundary. However, when we speak of the area, it is the region covered by the boundary of the figure.
The perimeter of ‘n’ sided regular polygons is ‘n’ times the side length of the polygon.