How to Calculate Perimeter and Area for Common Shapes
For a two-dimensional figure, perimeter refers to the boundary or path around a shape. On the other hand, the area of a two-dimensional figure is the space occupied within the surface of a shape. There are various types of shapes, but the common ones are square, rectangle, triangle, circle, etc. In this content, you will be able to know the perimeter and area of basic shapes.
Let’s start!
1. Rectangle
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A rectangle is a shape whose opposite sides are equal, and all the angles are right angles (90 degrees).
Perimeter of rectangle = \[2 ( a + b )\]
Area of rectangle = \[ a \times b \]
2. Square
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A square is a shape whose all four sides are equal, and all the angles are 90 degrees.
Perimeter of square = \[ 4 \times a \]
Area of square = \[ a^{2} \]
3. Circle
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A circle refers to a round shape that contains no edges or corners.
Perimeter of circle = \[2 \pi r\] (r = radius)
Area of circle = \[ \pi r^{2} \]
Note: Here the value of pi is either \[\frac{22}{7} \] or 3.14. You can use any one of them if not mentioned in the question.
4. Triangle
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A triangle is a shape with three angles and three straight lines. Triangles can be classified into three kinds, such as:
Equilateral Triangle
Perimeter of equilateral triangle = 3 a
Area of equilateral triangle = \[ \frac{1}{4} \times \sqrt{3} \times a^{2} \]
Isosceles Triangle
Perimeter of isosceles triangle = 2s + b
Area of isosceles triangle = \[\frac{1}{2} \times\] b \[\times\] hb
Scalene Triangle
Perimeter of scalene triangle = a + b + c
Area of scalene triangle = \[\frac{1}{2} \times b \times h \]
5. Parallelogram
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This shape is a quadrilateral whose opposite sides are parallel.
Perimeter of parallelogram = \[2 ( a + b ) \]
Area of parallelogram = \[b \times h\]
6. Rhombus
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It is a parallelogram whose sides are equal.
Area of rhombus = \[a \times h \]
Perimeter of Rhombus = \[4 \times a \]
7. Trapezoid
This shape is a quadrilateral which has a minimum of 1 pair of parallel sides.
Perimeter of trapezoid = \[a_1 + a_2 + b_1 + b_2 \]
Area of trapezoid = \[(\frac{( a1 + a2 )}{2}) \times h \]
8. Regular N-Gon
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A regular polygon refers to a polygon whose number of sides and angles are the same.
Area of regular n-gon = \[\frac{1}{2} \times ( a \times n \times s) \]
Perimeter of Regular n-gon = \[n \times s \]
Here are some illustrative examples you can go through to understand the solving procedure.
Ex 1. A rectangular field has length 12 m and breadth 10 m. What will be the area as well as the perimeter of that field?
Solution. Length of the rectangular field = 12 m
Breadth of the rectangular field = 10 m
Therefore, area of the field =\[ l \times b = 12 \times 10 = 120 m^2\]
And perimeter = \[2 ( l + b ) = 2 ( 10 + 12 ) = 44 m\]
Ex 2. Find the perimeter of circles whose radius are (i) 14cm (ii) 10m and (iii) 4km.
Solution.
According to the formula \[ 2 \pi r = 2 \times 3.14 \times 14 cm = 87.92 cm \]
\[ 2 \pi r = 2 \times 3.14 \times 10 m = 62.8 m \]
\[ 2 \pi r = 2 \times 3.14 \times 4 km = 25.12 km \]
Ex 3. If a rhombus has base and height 10 cm and 7 cm respectively, calculate its area.
Solution. With regards to the question base = 6 cm
Height = 8 cm
Therefore, the area of rhombus = \[b \times h\]
= \[10 \times 7 cm^{2} \]
= \[70 cm^{2} \]
This material is mainly for students who belong to standard VII, so here only the basic formulas are provided. There are some other methods also to solve perimeter and area specifically for shapes like rhombus, triangles, etc. which you will learn in higher classes.
If you want to refer to other solved examples of area and perimeter numerical, download the Vedantu app today.
FAQs on Perimeter and Area Explained with Formulas
1. What is the fundamental difference between perimeter and area?
The fundamental difference lies in what they measure. Perimeter is the total length of the boundary of a closed two-dimensional shape; it's a one-dimensional measure of distance. Think of it as the length of a fence needed to surround a garden. In contrast, area is the total space or region enclosed within that boundary; it's a two-dimensional measure of surface. This would be the amount of grass needed to cover the entire garden.
2. How are the perimeter and area of common shapes like squares and rectangles calculated?
The formulas depend on the properties of the shape:
For a Square: Since all four sides are equal (let's call the side 's'), the perimeter is P = 4 × s. The area is calculated as A = s × s or s².
For a Rectangle: With a length ('l') and width ('w'), the perimeter is the sum of all four sides, given by the formula P = 2 × (l + w). The area is found by multiplying the length and width: A = l × w.
3. Why are perimeter and area measured in different units?
Perimeter and area are measured in different units because they represent different physical quantities. Perimeter measures length (a one-dimensional property), so it is expressed in linear units like centimetres (cm), metres (m), or inches. Area, however, measures the surface covered, which is a two-dimensional space. It is expressed in square units (like cm², m², or square inches) because it quantifies how many unit squares can fit inside the shape.
4. What are some real-life examples where we use the concepts of perimeter and area?
Both concepts are used frequently in daily life.
Perimeter examples: Fencing a backyard, putting a decorative border around a room, measuring the length of a running track, or framing a picture.
Area examples: Calculating how much paint is needed to cover a wall, buying carpet for a room, determining the size of a plot of land, or finding out how much seed is needed for a lawn.
5. How is the area of a triangle calculated if it's not a right-angled triangle?
The formula for the area of any triangle, regardless of its type, is Area = ½ × base × height. The 'base' can be any side of the triangle. The 'height' (or altitude) is the perpendicular distance from the base to the opposite vertex. It is a common misconception that the height must be one of the sides; this is only true for right-angled triangles.
6. Can two different rectangles have the same perimeter but different areas? Explain how.
Yes, this is a key concept. Two rectangles can have the same perimeter but enclose different areas. For example, consider a perimeter of 20 cm. A rectangle with a length of 8 cm and a width of 2 cm has a perimeter of 2(8+2) = 20 cm and an area of 8 × 2 = 16 cm². Another rectangle with a length of 6 cm and a width of 4 cm also has a perimeter of 2(6+4) = 20 cm, but its area is 6 × 4 = 24 cm². This shows that the shape of the rectangle affects its area, even if the perimeter is constant.
7. How is the distance around a circle (circumference) related to its perimeter?
The circumference is the special term used for the perimeter of a circle. It is calculated using the formula C = 2 × π × r, where 'r' is the radius of the circle and 'π' (pi) is a special mathematical constant approximately equal to 3.14159. So, circumference is simply the perimeter of a circular shape.
8. If you double the side of a square, what happens to its perimeter and area?
If you double the side of a square, the perimeter and area do not increase by the same factor.
The perimeter will double. The original perimeter is 4s. The new perimeter will be 4 × (2s) = 8s, which is twice the original.
The area will become four times larger. The original area is s². The new area will be (2s)² = 4s², which is four times the original. This demonstrates that area changes with the square of the change in side length.
9. How do you find the area of a parallelogram?
The area of a parallelogram is calculated by multiplying its base by its height: Area = base × height. It's important to note that the 'height' is the perpendicular distance between the base and the opposite side, not the length of the slanted side. You can visualise this by cutting a triangle from one side and moving it to the other to form a rectangle with the same base and height, which is why the formula is similar to that of a rectangle.
