CBSE Notes Class 7 Maths Chapter 9 Perimeter and Area

Chapter 9 of Class 7 Maths introduces the concepts of Perimeter and Area. Mensuration is the process or art of measuring, applied to anything that can be measured. In geometry, perimeter refers to the total boundary length of a closed figure, while area is the amount of surface covered by a shape. A planar region, which has two dimensions (length and breadth), is measured in terms of its area. These concepts are essential for solving practical problems like determining the length of fencing required or the amount of material needed for flooring.

1.0Triangle

1. Perimeter of a Triangle

Perimeter = Sum of Sides = a + b +c 

2. Area of a Triangle:

$\text{ Area }=\frac{1}{2}\times \text{ base }\times \text{ height }$

3. Area of an Equilateral Triangle:

$\text{ Area }=\frac{\sqrt{3}}{4}{a}^2,$where (a) is the side length.

NOTE: ALSO WRITE h in equilateral Triangle. 

4. Median/Altitude/Angle Bisector: For an equilateral triangle, the median (or altitude) is: $\text{ Median }=\frac{\sqrt{3}}{2}a$
5. Circumcircle and Incircle: If (R) is the radius of the circumcircle and (r) is the radius of the incircle: R = 2r
6. Perimeter of an Equilateral Triangle:$\text{ Perimeter }=3a$
7. Area of an Isosceles Triangle:$\text{ Area }=\frac{b\sqrt{4a^2-b^2}}{4}$

8. Height of an Isosceles Triangle:$\text{ Height }=\sqrt{a^2-\left(\frac{b^2}{4}\right)}$

2.0Quadrilateral

Area of Any Quadrilateral: The area is the sum of the areas of two triangles formed by a diagonal.

1. If diagonals are inside the quadrilateral

$\text{ Area }=\frac{1}{2}\times \text{ diagonal }\times (\text{ Sum of the two perpendiculars })$

$A=\frac{1}{2}\times d\times \left(P_1+P_2\right)$ 

2. If the diagonal falls outside the figure, then

$\text{ Area of the quadrilateral }=\frac{1}{2}\times \text{ diagonal }\times (\text{ difference of the perpendiculars })$

$\text{ Area }=\frac{1}{2}\times d\times \left(P_1-P_2\right)$

3.0Square

A square is a quadrilateral where:

1. All sides are equal: AB = BC = CD = DA = a 
2. Diagonals are equal: AC = BD = d 
3. Diagonals bisect each other at 90°, with $a^2=\frac{d^2}{2}$

Perimeter of a Square:

$\text{ Perimeter }=4\times \text{ side }=4a$

Area of a Square:

• $\text{ Area }={\text{ side }}^2=a^2$
• $\text{ Area }=\frac{\text{ diagonal }}{2}=\frac{d^2}{2}$

Diagonal of a square $=\sqrt{2}(sides)$

4.0Rectangle

A rectangle is a quadrilateral whose, 

1. Opposite sides equal and parallel: $AB=CD=\ell$ (length), AD = BC = b (breadth)
2. Equal diagonals that bisect each other: AC = BD = d 
3. All angles $\angle A=\angle B=\angle C=\angle D=90^{\circ }$

Diagonal relation:

$d^2=l^2+b^2$

Area of a Rectangle:

$\text{ Area }=\text{ length }\times \text{ breadth }=l\times b$

Perimeter of a Rectangle:

$\text{ Perimeter }=2\text{ (length }+\text{ breadth })=2(l+b)$

5.0Parallelogram

A parallelogram is a quadrilateral with:

1. Opposite sides equal and parallel: AB = CD = b, AD = BC = a 
2. Diagonal AC = d 
3. Height h between bases

4. Perimeter of a Parallelogram:

$\text{ Perimeter }=2(\text{ sum of adjacent sides })=2(a+b)$

5. Area of a Parallelogram:

$\text{ Area }=\text{ base }\times \text{ height }=b\times h$

$\text{ Area }={\mathrm{ah}}_1={\mathrm{bh}}_2$

6.0Trapezium

1. Area of a Trapezium:

$\text{ Area }=\frac{1}{2}(\text{ sum of parallel sides })\times (\text{ perpendicular distance between them })$

$\text{ Area }=\frac{1}{2}(a+b)h$

7.0Circle

A circle consists of all points in a plane equidistant from a fixed point called the center (O).

The distance from the center to any point on the circle is the radius (r). 

Any line passing through the center and ending at both sides of the circumference is called the diameter (AC').

8.0Circumference and Area of the Circle

The circumference (C) is the perimeter of a circle, and the area (A) is the space it encloses.

• Radius: OB = r 
• Diameter: AC' = d = 2r 

Formulas:

• Circumference: $C=2\pi r=\pi d$
• Area: $A=\pi r^2$
• Correlation: $A=\frac{C^2}{4\pi }$

9.0Semicircle

Area of semicircle $=\frac{\pi r^2}{2}$

Perimeter of semicircle (including the straight edge) = πr + 2r

10.0Pathway 

1. Pathway Outside the Plot: 

For a rectangular plot ABCD with length $\ell$ and breadth b , and a pathway of width W  outside the plot, the area of the pathway is:  

$A_0=2W(\ell +b+2W)$

2. Pathway Inside the Plot:  

If the pathway is inside the plot, the area is:  

$A_1=2W(\ell +b-2W)$

3. Parallel Paths:   

For the same rectangular plot with parallel paths SU and TV, each of width W, the area of the two parallel paths is:  

$A=W(\ell +b-W)$

11.0Circular Pathway

1. Pathway Outside the Circle:

For a circle ABC with radius r and a pathway of width W outside it, the area of the pathway is:  

$\text{ Area }=\pi W(2r+W)$

2. Pathway Inside the Circle:

For a pathway of width W inside the circle, the area is:  

$\text{ Area }=\pi W(2r-W)$

Circular Ring:

If R and r are the radii of the outer and inner circles respectively, with W = R - r (width of the ring), the area of the ring is:  

$\text{ Area }=\pi \left(R^2-r^2\right)=\pi (R+r)W$

12.0Sample Questions on Perimeter and Area

1. How do you find the perimeter of a rectangle?  

Add the lengths of all sides or use $P=2(\ell +b)$.

2. How do you find the area of a circle?  

Use $A=\pi r^2$, where r is the radius.