What is Perimeter?
The perimeter is the total distance around the edge of a two-dimensional shape. It’s a fundamental concept in geometry that helps us understand and measure the boundaries of shapes. The perimeter is expressed in linear units, such as meters (m), centimeters (cm), or inches (in). Perimeter applies to flat shapes like squares, rectangles, triangles, circles, and more. It does not apply to three-dimensional objects. Each shape has a specific formula for calculating its perimeter. Let’s understand how each formula is derived with step-by-step explanations.
Perimeter Formulas for Common 2D Shapes
Perimeter of a Square
The perimeter of a square is the total distance around the outside of the square. Since a square has four equal sides, its perimeter is four times the length of a side.
The formula for calculating the perimeter of a square of side a units is:
Example:
Calculate the perimeter of a square with side length 8 m.
Side length = a = 8 m
Perimeter = 4a = 4×8 = 32 meters = 32 m
Perimeter of a Rectangle
A rectangle has two pairs of equal sides. We can find the perimeter by adding the lengths of all sides together.
The perimeter of a rectangle can be calculated using the following formula:
Perimeter of a Rectangle = 2x(Length+Breadth) = 2(l+b)
Length is the longer side of the rectangle. Breadth (Width) is the shorter side of the rectangle.
Example:
Calculate the area of a rectangle whose length is 8 cm and breadth is 5 cm.
Perimeter = 2x(l+b) = 2x(8+5) = 2×13 = 26 cm
Perimeter of a Parallelogram
A parallelogram is a four-sided polygon (quadrilateral) where opposite sides are parallel and equal in length.
The perimeter of a parallelogram can be calculated using the formula:
- a is the length of one pair of opposite sides.
- b is the length of the other pair of opposite sides.
Example:
If a parallelogram sides of length 6 cm and 4 cm, its perimeter is:
centimeters.
Perimeter of a Triangle
A triangle is a polygon with three sides and three angles. The perimeter of a triangle is the total distance around the triangle, calculated by adding the lengths of all its sides.
The perimeter of a triangle can be calculated using the formula:
Where:
- a is the length of one side.
- b is the length of the second side.
- c is the length of the third side.
Types of Triangles
- Equilateral Triangle: All three sides are equal.
- Isosceles Triangle: Two sides are equal.
- Scalene Triangle: All sides are of different lengths.
The formula for perimeter remains the same regardless of the type of triangle.
Example:
The lengths of the sides of a triangle are 5 cm, 7 cm and 10 cm. Find its perimeter.
Perimeter = 5+7+10 = 22 cm
Perimeter of a Rhombus
A rhombus is a four-sided polygon (quadrilateral) where all sides are equal in length. The perimeter of a rhombus can be calculated using the formula:
Perimeter of a Rhombus = 4xa
a is the length of a side.
Example:
If each side of the rhombus measures 6 cm, the perimeter would be:
Perimeter = 4×6 = 24 cm
Perimeter of a Trapezium (Trapezoid)
A trapezium is a quadrilateral with at least one pair of parallel sides. The perimeter of a trapezium can be calculated using the formula:
Perimeter of a Trapezium or Trapezoid = a + b + c + d
- a and b are the lengths of the two parallel sides.
- c and d are the lengths of the other two sides.
Example:
If the parallel sides of a trapezium are 8 cm and 5 cm, and the other two sides measure 4 cm and 6 cm,
P = 8+5+4+6 = 23 cm
Perimeter (Circumference) of a Circle
The perimeter of a circle is commonly referred to as its circumference. The circumference is the total distance around the circle.
- If you were to take a piece of string and wrap it around a circular object, the length of that string would represent the circumference.
- The length of that string is seen as times the diameter of the circle it forms, or 2 times the radius.
The formula for calculating the circumference of a circle is:
Circumference of a Circle = 2𝜋r = 𝜋d
(pi) is a mathematical constant approximately equal to 3.14159 or $\frac{22}{7}$.
is the radius of the circle, which is the distance from the center of the circle to any point on its circumference.
d is the diameter of the circle, which is twice the radius.
Example:
Calculate the circumference of a circle with a radius of 14 cm.
A = 2𝜋r = 2x$\frac{22}{7}$x14 = 88 cm
Perimeter Formula Chart
The below table lists the formulas to find the perimeter of common 2D shapes.
Understanding the “why” behind each perimeter formula is crucial for truly grasping the concept of perimeter and applying it effectively in real-world scenarios. In our daily lives, we encounter various physical boundaries, such as fencing a yard, enclosing a garden, or outlining a room. Knowing the reasoning behind specific formulas enables us to apply them accurately to solve practical problems, like determining how much fencing material is needed or how to plan the layout of a space. This understanding not only helps in tackling mathematical challenges but also empowers students to utilize their knowledge in meaningful ways. Building this foundation is vital for cultivating a deeper appreciation of mathematics and its significance in our everyday activities.
FAQs on Perimeter Formulas
- Yes, for irregular shapes, you need the length of each side. However, for regular polygons, knowing one side length and the number of sides is enough.
- If the side length of a square doubles, the perimeter also doubles because perimeter depends linearly on side length. For example, if the side length changes from 5 to 10, the perimeter goes from 4×5=20 to 4×10=40.
- The circumference of a circle is directly proportional to the diameter. If the diameter doubles, the circumference also doubles because: Circumference=π×Diameter
- No, the perimeter cannot be zero because it represents the total length around a shape. All closed shapes must have a positive perimeter.
- Yes, shapes can have the same perimeter but different areas. For instance, a long, narrow rectangle and a square might both have a perimeter of 20 units, but their areas would be different.
- Circles have a circumference instead of a perimeter. The term “perimeter” usually refers to polygons, while “circumference” specifically describes the distance around a circle.