Area and Perimeter Formulas (original) (raw)
Last Updated : 23 Jul, 2025
**Area and Perimeter Formulas are the fundamental formulas in Mensuration that help us calculate the area and perimeter for any geometric shape in mathematics. Area and Perimeter Formulas have equal importance in real life for various calculations like finding the area of land or finding the perimeter of land for boundaries and many more.
In this article, we will explore all the different formulas to find the Area and Perimeter of various geometric shapes such as Triangles, Squares, Rectangles, Rhombus, Parallelogram, Circle, Semicircle, and Ellipse. We will also cover the solved examples of the area and perimeter formula. Let's start our learning on the topic of Area and Perimeter Formula.
Table of Content
- What is Area and Perimeter?
- Area and Perimeter Formulas for All Shapes
- Area and Perimeter Formulas Table
- Difference Between Area and Perimeter
- Examples on Area and Perimeter Formula
What is Area and Perimeter?
Area and Perimeter are fundamental measurements in geometry that define the major characteristics of any geometric shape such as polygons and circles. Let's discuss both of these concepts in detail:
Area Definition
Area is the region enclosed by a closed shape and it depends upon the dimensions and properties of any shape under consideration. The different two-dimensional shapes have different areas.
Area is typically expressed in square units, such as square meters (m2), square feet (ft2), or square centimetres (cm2), depending on the system of measurement.
Perimeter Definition
Perimeter is nothing but the measurement of the boundary of any two-dimensional shape i.e., if all sides are straight lines we can find the perimeter just by adding all the side lengths. In other words, the perimeter is the distance around the outer boundary of a two-dimensional shape.
Similarly, the perimeter of any shape depends on the dimensions and properties of any geometric shape under consideration. Perimeter is usually measured in linear units, such as meters (m), feet (ft), or centimetres (cm).
Area and Perimeter Formula for All Shapes
This article discuss various shapes and their area and perimeter, these shapes are:
Let's discuss area and perimeter formula for all these shapes in detail.
Triangle: Area and Perimeter Formula
The polygon with three sides is called as triangle.
Area of Right Angle Triangle
The area of the right-angled triangle with base b and height h is given by:
**Area of Right Angle Triangle = (1/2) × b × h
Where,
- **b is the Base of Triangle, and
- **h is the Height of Triangle.
Area of Equilateral Triangle
The area of the equilateral triangle with side s is given by:
**Area of Equilateral Triangle = √3a 2 /4
Where **a is the side of Equilateral Triangle.
Area of Isosceles Triangle
The area of the equilateral triangle with side s is given by:
**Area of Isosceles Triangle, A = (1/2) × a 2 sin θ
Where, a is side of the triangle
Read more about **Types of Triangles.
Area of Triangle: Heron's Formula
The area of a triangle with sides a, b and c is given by Heron's formula. The Heron's formula is given below:
**Area of triangle = √[s(s - a)(s - b)(s - c)]
Where,
- **s = (a + b + c) / 2, is the semiperimter of triangle, and
- **a, **b and **c are sides of triangle.
Perimeter of Triangle
The perimeter of a triangle with sides a, b and c is given by:
**Perimeter of triangle = a + b + c
Where a, b, c are sides of triangle.
Square: Area and Perimeter Formula
Square is a 2-D closed quadrilateral with all sides equal.
Area of Square
The area of the square with side a is given by:
**Area of Square = a × a = a 2
Where a is side of square.
Perimeter of Square
The perimeter of square of side a is given by:
**Perimeter of Square = 4a
Where a is side of square.
Rectangle: Area and Perimeter Formula
Rectangle is a 2-D closed quadrilateral with opposite sides equal.
Area of Rectangle
The area of the rectangle with length l and breadth b is given by:
**Area of rectangle = l × b
Where,
- **l is length of rectangle, and
- **b is breadth of rectangle.
Perimeter of Rectangle
The perimeter of the rectangle with length l and breadth b is given by:
**Perimeter of rectangle = 2(l + b)
Where,
- **l is length of rectangle, and
- **b is breadth of rectangle.
Parallelogram: Area and Perimeter Formula
The quadrilateral with two pairs of parallel sides and opposite angles is called a parallelogram.
Area of Parallelogram
The area of the parallelogram with height h and base b is given by:
**Area of parallelogram = b × h
Where,
- **b is base of parallelogram, and
- **h is height of parallelogram.
Perimeter of Parallelogram
The perimeter of the parallelogram with length l and breadth b is given by:
**Perimeter of parallelogram = 2(l + b)
Where,
- **l is length of parallelogram, and
- **b is breadth of parallelogram
Rhombus: Area and Perimeter Formula
The parallelogram with all equal sides is called rhombus.
Area of Rhombus
The area of the rhombus with diagonals d1 and d2 is given by:
**Area of square = (d 1 × d 2 )/2
Where, **d 1and **d 2 are length of diagonals of rhombus.
Perimeter of Rhombus
The perimeter of rhombus of side a is given by:
**Perimeter of Rhombus = 4a
Where, a is side of rhombus.
Polygon: Area and Perimeter Formula
A 2-D closed figure with at least three straight lines and angles is called polygon.
Area of Polygon
The area of the polygon is given by the half of product of apothem and perimeter.
**Area of Polygon = (1 /2) × Perimeter × Apothem
Where **apothem is the perpedicular length of side from center of polygon.
Perimeter of Polygon
The perimeter of the polygon is given by the sum of all sides of the polygon.
**Perimeter of Polygon = Sum of all Sides
Circle: Area and Perimeter Formula
Circle is the 2-D shape which has a center and drawn using equal distance from the center called the radius of the circle.
Area of Circle
The area of the circle with radius r is given by:
**Area of circle = πr 2
Where **r is radius of circle.
Perimeter of Circle
The perimeter of the circle with radius r is given by:
**Perimeter of circle = 2πr
Where **r is radius of circle.
Semicircle: Area and Perimeter Formula
The half of the circle is called a semicircle.
Area of Semicircle
The area of the semicircle with radius r is given by:
**Area of semicircle = (πr 2 )/2
Where **r is radius of semicircle.
Perimeter of Semicircle
The perimeter of the semicircle with radius r is given by:
**Area of semicircle = πr + 2r
Where **r is radius of semicircle
Ellipse: Area and Perimeter Formula
An ellipse is set of all the points from a plane whose distance from two fixed points is constant.
Area of Ellipse
The area of ellipse with the semi-major axis and semi-minor axis a and b respectively is given by:
**Area of ellipse = πab
Where,
- **a is the semi-major axis, and
- **b is semi-minor axis of ellipse.
Perimeter of Ellipse
The perimeter of ellipse with the semi-major axis and semi-minor axis a and b respectively is given by:
**Perimeter of ellipse = 2π√[(a 2 + b 2 ) / 2]
Where,
- **a is the semi-major axis, and
- **b is semi-minor axis of ellipse.
**Also, Check
Area and Perimeter Formulas Table
The table added below shows various Area and Perimeter Formulas of various figure,
| Table for Area and Perimeter Formulas | |||
|---|---|---|---|
| Shape | Area | Perimeter | Variables description |
| Triangle | A = 1/2(b × h) | P = a + b + c | b = base, h = heighta,b and c are sides of triangle |
| Rectangle | A = l × b | P = 2(l+b) | l = length, b = breadth |
| Square | A = s × s | P = 4 × s | s = side |
| Circle | A = πr2 | P = 2πr | r = radius, π = 22/7 or 3.14 |
| Ellipse | A = π×b | P = π(a+b) | a = semi major axisb = semi minor axis |
| Parellelogram | A = b × h | P = 2(a+b) | b = base, h = heighta and b are the opposite sides |
| Rhombus | A = 1/2 (d1 × d2) | P = 4 × a | d1, d2 = diagonalsa = side of rhombus |
| Trapezium | A = 1/2 × (a+b) × h | P = Sum of all Sides | a,b = length of parallel sides, h = height |
Difference Between Area and Perimeter
Differences between Area and Perimeter are listed in the table below,
| Area vs Perimeter | |
|---|---|
| Area | Perimeter |
| Area is a measure of a region's size on a surface. The region is a closed 2D figure. | Perimeter is a measure of the length of boundary of any closed 2D shape. |
| Area is measured in square units, i.e. m2, cm2, mm2, etc. | Perimeter is expressed in units, i.e. m, cm, mm, etc. |
| Example: Top of a Table | Example: Boundary of a Table |
**Read More,
Examples on Area and Perimeter Formula
**Example 1: Find the area of the circle with the radius 4 cm.
**Solution:
Area of circle = πr2 where r is radius of circle
⇒ Area of circle = π(4)2
⇒ Area of circle = 16π cm2
**Example 2: Find the perimeter of square with side 7 cm.
**Solution:
Perimeter of square = 4a where a is side of square
⇒ Perimeter of square = 7 × 4
⇒ Perimeter of square = 28 cm
**Example 3: Find the area of the rectangle with length and breadth 9cm and 3 cm respectively.
**Solution:
Area of rectangle = l × b where l and b are length and breadth of rectangle
⇒ Area of rectangle = 9 × 3
⇒ Area of rectangle = 27 cm2
**Example 4: Find the area of the right-angled triangle with base and height 5cm and 8 cm respectively.
**Solution:
Area of right-angled triangle = (1 / 2) × b × h
⇒ Area of right-angled triangle = (1 / 2) × 5 × 8
⇒ Area of right-angled triangle = 20 cm2
**Example 5: Find the perimeter of the triangle with sides 4 cm, 6cm and 10 cm.
**Solution:
Perimeter of triangle = a + b + c where a, b and c are sides of triangle
⇒ Perimeter of triangle = 4 + 6 + 10
⇒ Perimeter of triangle = 20 cm
**Example 6: Find the area of the semicircle with radius 2cm.
**Solution:
Area of semicircle = (πr2)/2
⇒ Area of semicircle = (π22)/2
⇒ Area of semicircle = 2π cm2
**Example 7: Find the area of the square field whose side is 450 m.
**Solution:
Area of square = a × a
⇒ Area of square = 450 × 450
⇒ Area of square = 202500 m2
Practice Problems on Area and Perimeter Formulas
**Problem 1: Find the perimeter of the rectangle with length and breadth 6 cm and 3 cm.
**Problem 2: Find the perimeter of the circle whose diameter is 8 cm.
**Problem 3: Find the area of a scalene triangle with sides 10 cm, 14 cm and 16 cm respectively.
**Problem 4: Find the area of the equilateral triangle with a side 9 cm.
**Problem 5: Find the perimeter of a semicircle with a diameter of 6 cm.
**Problem 6: Find the area of a rectangle with a length of 10 cm and a breadth 7 cm.
**Problem 7: Find the perimeter of a square whose side length is 5 cm.
**Problem 8: Calculate the area of a trapezoid with bases of lengths 8 cm and 12 cm and a height of 5 cm.
**Problem 9: Determine the area of a circle with a radius of 4 cm.
**Problem 10: Find the perimeter of a triangle with sides measuring 7 cm, 9 cm and 12 cm.
Conclusion
Understanding the formulas for calculating area and perimeter is essential in geometry as they allow us to quantify the space within shapes and the distance around them. Mastery of these concepts is crucial for the solving various practical problems in fields such as the architecture, engineering and everyday life. By applying these formulas correctly one can enhance their spatial awareness and problem-solving skills.