Area and Perimeter - Definition, Formulas and Examples

Area and Perimeter

Edited By Team Careers360 | Updated on Jul 02, 2025 05:11 PM IST

Area and perimeter are the two important properties of 2D shapes. When we talk of the concept of Perimeter, it means the total distance of the boundary of any shape whereas the area describes the region within it. It is commonly used in daily life that it has become an important part of our activities. In this article we will learn about area and perimeter of all shapes such as rectangle, square, triangle, rhombus, difference between area and perimeter, etc.

This Story also Contains

Area and Perimeter Formulas for all Shapes
Area and Perimeter of 2D shapes
Difference Between Area and Perimeter
Solved Examples on Area and Perimeter

Area and Perimeter Formulas for all Shapes

Before looking into the area and perimeter formulas for all shapes, let us look what is area and perimeter.

What is Area?

Area is the region within the shape of an object or in other terms, the space covered by the figure or any two-dimensional geometric shape, in an xy plane. This physical quantity depends on the dimensions and properties of the shape under consideration.

What is Perimeter?

The term Perimeter is defined as the total distance or length of boundary around a shape in a 2d or xy plane. It is calculated by adding the lengths of the sides of the shape. For example if we take a round of a circular park and at the end calculate the total distance covered, we get the perimeter.

Now let us look into the area and perimeter of 2D shapes.

Area and Perimeter of 2D shapes

We should note that if two objects have a similar shape then it is never compulsory that their area when calculated gives out same results. The condition to be met is that their dimension must also be equal.

For example, there are two rectangle boxes, with length as $P1$ and $P2$ and breadth as $Q1$ and $Q2$. So the areas of both the rectangular boxes, say $A1$ and $A2$ will be equal only if $P1=P2$ and $Q1=Q2$.

Area and Perimeter of Rectangle

A rectangle is a four sided shape in which opposite sides are equal and all angles measure 90 degrees.

Formula of area and perimeter of rectangle

Perimeter of a Rectangle $=2(p+q)$
Area of Rectangle $=p \times q$

Where $p,q$ are the length and breadth respectively.

Area and Perimeter of Square

A square is a shape with all four sides equal with all angles measuring 90 degrees.

Area and Perimeter of Square Formula

Area and perimeter of square is given as:

- Perimeter of a Square $=4 x$
- Area of a Square $=x^2$

Where $x$ is the side of square.

Area and Perimeter of Triangle

A triangle is a three-sided shape where the sum of all angles of the triangle is $180^{\circ}$.

Triangle area and perimeter

The area and perimeter of triangle formula is

- Perimeter of a triangle $=p+q+r$, where $p, q$ and $r$ are the three different sides of the triangle.
- Area of a triangle $=\frac{1}{2} \times b \times h$; where $b$ is the base and $h$ is the height of the triangle.

Area and Perimeter of Circle

A circle is a round geometric shape with no vertex.

Area and perimeter of circle is given as:

- Circumference(Perimeter) of Circle $=2 \pi r$
- Area of Circle $=\pi r^2$

where $ r$ is radius of circle

Area and Perimeter of Rhombus

A rhombus is a four sided shape with all equal sides.

Formulas to Calculate Area of Rhombus
Using Diagonals	$\frac{1}{2}\times d_1 \times d_2$
Using Base and Height	$A = b \times h$
Using Trigonometry	$A=b^2 \times \sin (a)$

Difference Between Area and Perimeter

Following table lists the difference between area and perimeter.

Area	Perimeter
Area is the region covered by a shape	Perimeter is total distance covered by the boundary of a shape
Area is measured in square units (m², cm², in², etc.)	Perimeter is measured in units (m, cm, in, feet, etc.)
Example: Area of rectangular ground is equal to product of its length and breadth.	Example: Perimeter of a rectangular ground is equal to sum of all its four boundaries, i.e, 2(length + breadth).

Solved Examples on Area and Perimeter

Example 1: Given the radius of a circle is 20 cm . Find area and perimeter of circle.
Solution: Given, radius $=20 \mid \mathrm{cm}$
Hence, Area $=\pi \times r^2$

$
A= \frac{22}{7} \times 20 \times 20
$

Area of circle $=1257.14$ sq.cm.
Circumference, $C=2 \pi r$
Perimeter of circle $=2 \times \frac{22}{7} \times 20=125.7 \mathrm{~cm}$
So, Area and Perimeter of circle $=1257.14$ sq. $\mathrm{cm}, 125.7 \mathrm{~cm}$ respectively.

Example 2: The length of the side of a square is 5 cm . Calculate area and perimeter of square.

Solution: Given, length of the side, $a=5 \mathrm{~cm}$
Area of square $=a^2=5^2=25$ sq.cm
Perimeter of square $=4 a=4 \times 5=20$ sq.cm.
So, Area and Perimeter of square $=25 \mathrm{sq} . \mathrm{cm}, 20 \mathrm{~cm}$ respectively.

Example 3: The length of rectangular field is 12 m and width is 6 m . Calculate the area and perimeter of rectangle.

Solution: Given, Length = 12m
Width $=6 \mathrm{~m}$
Therefore, Area of rectangle $=$ length $\times$ width $=12 \times 6=72$ sq.m.
Perimeter of rectangle $=2(1+b)=2 \times 18=36 \mathrm{~m}$.
So, Area and Perimeter of rectangle $=72 \mathrm{sq} . \mathrm{cm}, 36 \mathrm{~cm}$ respectively.

Example 4: What is the area of triangle with base 6 cm and height 10 cm ?
Solution: Area of triangle $= \frac{1}{2} \times b \times h= \frac{1}{2} \times 6 \times 10=30 \mathrm{sq} \mathrm{cm}$.

Example 5: What is the perimeter of triangle of sides $3 \mathrm{~cm}, 4 \mathrm{~cm}, 5 \mathrm{~cm}$ ?
Solution: Perimeter of triangle $=$ Sum of all sides $=3 \mathrm{~cm}+4 \mathrm{~cm}+5 \mathrm{~cm}=12 \mathrm{~cm}$.

List of Topics Related to Area and Perimeter

Area of Circle	Area of Isosceles Triangle
Area of Rectangle	Area of Sphere
Area	Area of Quadrilateral
Area of Parallelogram	Area of Square
Area of Equilateral Triangle	cm to inches converter

Frequently Asked Questions (FAQs)

1. What is difference between area and perimeter?

The area is the region covered by shape or the space occupied by it in $x y$ plane. Perimeter is nothing but the total boundary length covered by the shape. Unit of area is square unit and for perimeter it is same as the unit itself.

2. What is the formula for perimeter?

Perimeter is defined as the length of the boundary of the shape (i.e) Perimeter = Sum of length all the sides.

3. What is the area and perimeter of circle?

Area of circle $=\pi r^ 2$
Perimeter of circle $=2 \pi$.

4. Give an example of area and perimeter.

If a square has side length of 4 cm then,
Area of square $=$ side $^2=4 \mathrm{~cm}^2=16 \mathrm{sq} \mathrm{cm}$
Perimeter of square $=$ sum of all sides $=4+4+4+4=64 \mathrm{~cm}$.

5. What is the formula for area and perimeter of rectangle?

Area of rectangle $=$ Length $\times$ Breadth
Perimeter of rectangle $=2(1+b)$

6. What is the relationship between the circumference and diameter of a circle?

The circumference of a circle is always π (pi) times its diameter. This relationship is expressed in the formula C = πd, where C is the circumference and d is the diameter. This constant ratio between circumference and diameter for all circles is what defines π and makes it such an important constant in mathematics.

7. Why is the formula for the area of a circle not simply πd, where d is the diameter?

The formula for the area of a circle is not πd because this would significantly underestimate the actual area. The correct formula, πr² or π(d/2)², accounts for the fact that area is a two-dimensional measure. Using πd would only capture a one-dimensional aspect of the circle (similar to its circumference) and wouldn't accurately represent the way area scales with the circle's size. The squared term in πr² is crucial for correctly representing the two-dimensional nature of area.

8. Why is the formula for the area of a circle (A = πr²) different from the formula for its circumference (C = 2πr)?

The formulas are different because they measure different aspects of the circle. The circumference (C = 2πr) is a one-dimensional measurement of length, so it's directly proportional to the radius. The area (A = πr²) is a two-dimensional measurement of surface, so it's proportional to the square of the radius. The squared term in the area formula reflects the fact that area grows much faster than circumference as the radius increases.

9. Why can't we use a simple formula like 2πr for the area of a circle, similar to its circumference formula?

We can't use a formula like 2πr for the area because area is a two-dimensional measure, while circumference is one-dimensional. The area grows much faster than the circumference as the radius increases. The formula πr² correctly captures this quadratic growth of area with respect to radius. If we used 2πr for area, it would severely underestimate the actual area for any circle larger than a unit circle (r = 1).

10. How does the concept of dimensional analysis help understand the formulas for circle area and circumference?

Dimensional analysis helps verify the correctness of formulas by ensuring that the units on both sides match. For circumference (C = 2πr), both sides have units of length. For area (A = πr²), both sides have units of length squared. This analysis confirms that r² is necessary in the area formula to match the square units of area, while a single r is sufficient for the linear units of circumference.

11. How does the area of a circle compare to the area of an inscribed square?

The area of a circle is always larger than the area of its inscribed square (a square that fits perfectly inside the circle, touching it at four points). The ratio of the circle's area to the inscribed square's area is π/2, or approximately 1.57. This means the circle's area is about 57% larger than the area of its inscribed square.

12. Can a circle and a rectangle have the same perimeter but different areas?

Yes, a circle and a rectangle can have the same perimeter but different areas. This is because the shape that encloses the maximum area for a given perimeter is a circle. So, a circle will always have a larger area than any rectangle with the same perimeter. This principle is known as the isoperimetric inequality.

13. How does the area of a circle change if its radius is halved?

If the radius of a circle is halved, its area is reduced to one-fourth of the original area. This is because the area is proportional to the square of the radius (A = πr²). When r is replaced by r/2, the new area becomes π(r/2)² = πr²/4, which is 1/4 of the original area πr².

14. How does the concept of pi (π) relate to the area and perimeter of a circle?

Pi (π) is crucial in calculating both the area and perimeter of a circle. For the perimeter (circumference), π represents how many times longer the circumference is compared to the diameter (C = πd or 2πr). For the area, π is part of the proportionality constant that relates the square of the radius to the area (A = πr²). In both cases, π captures the fundamental circular nature of these measurements.

15. How does the area of a circle compare to the area of a square with sides equal to the circle's diameter?

The area of a circle is always smaller than the area of a square with sides equal to the circle's diameter. The circle's area is π/4 (approximately 0.7854) times the area of the square. This relationship demonstrates why π is slightly greater than 3, as the circle fits inside the square with some space left over.

16. Can two circles with different radii have the same perimeter?

No, two circles with different radii cannot have the same perimeter (circumference). The circumference of a circle is directly proportional to its radius, given by the formula C = 2πr, where C is the circumference and r is the radius. Therefore, a larger radius always results in a larger circumference.

17. How does the area of a semicircle relate to the area of a full circle?

The area of a semicircle is exactly half the area of a full circle with the same radius. If A is the area of a full circle, then the area of a semicircle is A/2. This relationship holds true because a semicircle is formed by cutting a circle exactly in half along its diameter.

18. Why does squaring the radius have such a significant effect on a circle's area?

Squaring the radius has a significant effect on a circle's area because area is a two-dimensional measure. When you increase the radius, you're increasing the circle's size in two directions simultaneously (width and height). This results in a quadratic growth in area, which is why the formula for circle area (A = πr²) includes the radius squared.

19. What is the difference between area and perimeter of a circle?

The area of a circle is the amount of space inside the circle, measured in square units. It represents how much surface the circle covers. The perimeter of a circle, also called circumference, is the distance around the circle, measured in linear units. It represents the length of the circular boundary.

20. How does changing the radius affect the area of a circle?

The area of a circle is directly proportional to the square of its radius. This means that if you double the radius, the area increases by a factor of four. If you triple the radius, the area increases by a factor of nine, and so on. This relationship is expressed in the formula A = πr², where A is the area and r is the radius.

21. Why is π (pi) used in circle formulas?

π (pi) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It appears in formulas related to circles because it captures this fundamental relationship. π is approximately equal to 3.14159, but it's an irrational number, meaning its decimal representation never ends or repeats.

22. What is the relationship between the radius and diameter of a circle?

The diameter of a circle is always twice the length of its radius. In other words, diameter = 2 × radius, or d = 2r. Conversely, the radius is half the diameter: radius = diameter ÷ 2, or r = d/2. This relationship is fundamental to understanding circle geometry and is used in many circle-related calculations.

23. How does the concept of pi (π) emerge from the relationship between a circle's circumference and its diameter?

Pi (π) emerges as the constant ratio between any circle's circumference and its diameter. If you divide the circumference of any circle by its diameter, you always get π (approximately 3.14159...). This ratio is the same for all circles, regardless of their size. This consistency across all circles is what makes π a fundamental constant in mathematics and gives it its significance in circle geometry.

24. What happens to the ratio of a circle's area to its circumference as the circle gets larger?

As a circle gets larger, the ratio of its area to its circumference increases. This is because the area grows quadratically (proportional to r²) while the circumference grows linearly (proportional to r). The exact relationship is given by A/C = r/2, where A is the area, C is the circumference, and r is the radius. So, as r increases, this ratio increases linearly with it.

25. How does the area of a circular ring (annulus) relate to the difference in areas of two circles?

The area of a circular ring (annulus) is equal to the difference in areas between the larger circle (outer boundary) and the smaller circle (inner boundary). If R is the radius of the larger circle and r is the radius of the smaller circle, the area of the annulus is given by π(R² - r²). This formula comes directly from subtracting the area of the smaller circle from the area of the larger circle.

26. Why is the area of a circle not simply the product of its circumference and radius?

The area of a circle is not simply the product of its circumference and radius because this would overestimate the area. The actual area is πr², while the product of circumference and radius is 2πr² (since C = 2πr). The correct area is exactly half of this product. This relationship reflects the fact that the circle's area doesn't extend all the way to its circumference at every point, but rather tapers off towards the edge.

27. How does the concept of limits from calculus relate to finding the area of a circle?

The concept of limits in calculus is crucial for deriving the formula for a circle's area. One approach is to consider the circle as the limit of a regular polygon with an increasing number of sides. As the number of sides approaches infinity, the polygon's area approaches πr². This limit process provides a rigorous foundation for the circle area formula and demonstrates the deep connection between geometry and calculus.

28. What is the relationship between the perimeters of a circle and its inscribed regular hexagon?

The perimeter of a circle is always larger than the perimeter of its inscribed regular hexagon. The ratio of the circle's circumference to the hexagon's perimeter is π/3, or approximately 1.047. This means the circle's circumference is about 4.7% longer than the perimeter of the inscribed hexagon. This relationship is often used in approximations and in understanding the nature of π.

29. Why is the ratio of a circle's area to the square of its radius always equal to π?

The ratio of a circle's area to the square of its radius is always π because this is the definition of π in terms of circle geometry. The formula A = πr² can be rearranged to π = A/r². This constant ratio holds for all circles, regardless of their size, and it's what makes π a fundamental constant in mathematics. It captures the essential relationship between a circle's size (represented by its radius) and its area.

30. How does the concept of dimensional consistency help verify the correctness of circle formulas?

Dimensional consistency requires that all terms in an equation have the same units. For the circumference formula (C = 2πr), both sides have units of length. For the area formula (A = πr²), both sides have units of length squared. This consistency verifies that these formulas are dimensionally correct, which is a necessary (though not sufficient) condition for their validity. It helps catch errors and ensures that our mathematical descriptions align with physical reality.

31. What is the relationship between the areas of a circle, its inscribed square, and its circumscribed square?

The area of a circle is always between the areas of its inscribed and circumscribed squares. If r is the radius of the circle, the area relationships are:

32. How does the area of a circular segment relate to the area of the corresponding sector and triangle?

A circular segment is the region of a circle bounded by a chord and an arc. Its area can be calculated by subtracting the area of a triangle from the area of the corresponding sector. If θ is the central angle in radians, and r is the radius, the areas are:

33. What is the relationship between the areas of two circles if one has twice the radius of the other?

If one circle has twice the radius of another, its area will be four times larger. This is because the area is proportional to the square of the radius (A = πr²). When the radius is doubled, the area becomes π(2r)² = 4πr², which is four times the original area πr². This quadratic relationship between radius and area is a fundamental concept in circle geometry.

34. How does the area of a sector relate to the central angle and the area of the full circle?

The area of a sector is directly proportional to its central angle and the area of the full circle. If θ is the central angle in radians, the area of the sector is given by (θ/2π) × πr² = (θ/2)r². This formula represents the fraction of the full circle's area that the sector occupies, which is the same as the fraction of 360° that the central angle represents.

35. How does the area of a circle change if its circumference is doubled?

If the circumference of a circle is doubled, its area increases by a factor of four. This is because doubling the circumference means doubling the radius (since C = 2πr), and the area is proportional to the square of the radius (A = πr²). When r becomes 2r, the new area is π(2r)² = 4πr², which is four times the original area.

36. What is the significance of the number 2 in the formula for circle circumference (C = 2πr)?

The number 2 in the circumference formula (C = 2πr) represents the fact that the diameter is twice the radius. The formula can be written as C = πd, where d is the diameter. Since d = 2r, we get C = π(2r) = 2πr. The 2 ensures that we're using the radius in the formula while maintaining the fundamental relationship between circumference and diameter (C = πd).

37. How does the area of a circular sector change as its central angle increases from 0° to 360°?

As the central angle of a circular sector increases from 0° to 360°, its area increases linearly from 0 to the full area of the circle. The area of a sector is given by A = (θ/360°) × πr², where θ is the central angle in degrees. This formula shows that the sector's area is directly proportional to its central angle, with the full circle area (πr²) corresponding to 360°.

38. How does the area of a circle compare to the area of an equilateral triangle with side length equal to the circle's diameter?

The area of a circle is larger than the area of an equilateral triangle with side length equal to the circle's diameter. If d is the diameter of the circle (and thus the side length of the triangle), the areas are:

39. How does the concept of integration from calculus relate to finding the area of a circle?

Integration in calculus provides a rigorous method for deriving the area of a circle. By considering the circle as composed of infinitesimally thin concentric rings, we can set up an integral from 0 to r (the radius) of the circumference function 2πx dx. This integral, ∫(0 to r) 2πx dx, evaluates to πr², confirming the standard area formula. This approach demonstrates how calculus can be used to solve geometric problems and provides insight into why the area formula has its particular form.

40. What is the relationship between the area of a circle and the area of a regular hexagon inscribed within it?

The area of a circle is always larger than the area of its inscribed regular hexagon. If r is the radius of the circle, the areas are: