Perimeter is the length of the boundary of any kind of shape. The area is a measure of the surface area covered by any two-dimensional shape. For three-dimensional shapes, we have the flat surface area and curved surface area.

Square: It's a polygon with four equal sides. The perimeter of a square is the length of the boundary. Since a square is made up of 4 equal sides, the perimeter of a square is equal to four times the length of the side.

The area covered by the square is calculated by the square of the side. If the length of a square is denoted by L, then,

The perimeter of the square= 4L

Area of the Square= L²

NCERT Question

Q.  If the perimeter of the area is 320m. Find the area of a square park.

Answer:
Given:
The perimeter of the square park= 320m

Let the side of the square park = Lm

Then, 4 \( \times \) L = 320

L= 320/4 = 80m 

Now, the area of the square park = L\( \times \)L = 80m \( \times \) 80m = 6400m²

Rectangle: A rectangle is also a polygon with 4 sides but with 2 pairs of equal and opposite sides. The shorter sides of a rectangle are called breadth and the longer sides are called length. 

The perimeter of a rectangle is the length of the boundary. Since a rectangle is made up of 4 sides in which 2 each are of equal length, the perimeter is equal to two times the sum of the length of the longer and shorter side.

The area of a rectangle is equal to the product of the length of the longer and the shorter side that is equal to the area of the region occupied by the rectangle.

If breadth = B

Length = L

Then, the perimeter of the rectangle = 2(L+B)

Area of the rectangle= L \( \times \) B

NCERT Question:

Q. The breadth and the length of a rectangular piece of land are 500m and 300m, respectively. Find

1.  Its area 

2. The cost of the land, if the cost of 1m2 is ₹ 10,000.

Answer:-

Given: 

Length of the land = 500m

The breadth of the land = 300m

1. Area of the rectangular land = Length×Breadth = 500m \( \times \) 300m = 150000m2

2. Now, cost per unit square metre of the land = rs. 10,000

Then, the cost of land = area of the land \( \times \) rs. 10,000

= 15,0000 \( \times \) 10,000 = Rs. 15,00000000

If we cut a rectangle along the diagonal then we get a triangle with side lengths equal to the breadth, length and diagonal of the rectangle. Also note that the length of the diagonal of the rectangle can be calculated by using the Pythagorean theorem, as per which, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of both the other sides since the angle between them is 90 degrees.

Since the diagonal divides the rectangle into two equal parts, the triangles obtained by the division occupy a region half the area of the rectangle.

The perimeter of the triangle obtained by such division is equal to the sum of the shorter, longer sides of the rectangle and the diagonal.

A parallelogram is a mathematical figure that has 2 pairs of equal, opposite and parallel sides. If you divide a parallelogram through a diagonal, it will give 2 triangles. So you can consider a parallelogram to be equal to 2 triangles of the same area.

Therefore, the area of a parallelogram is equal to the area of 2 triangles.  

Area of parallelogram = 2 \( \times \) 1/2x height \( \times \) base 

                                    = height \( \times \) base

Base: any side of the parallelogram 

Height: perpendicular drawn from the opposite vertice to the base of the parallelogram.

The figure shows a parallelogram ABCD. 

Base: CD or AB both are of equal length.

Height: AE

NCERT Question

Q. Find the area of the parallelogram whose height and base are of length 4cm and 7cm respectively.

Answer:-

Given: 

Height of parallelogram = 4 cm

The base of parallelogram = 7 cm

Then,

Area of parallelogram = base \( \times \) height

= 7 \( \times \) 4

= 28 cm²

A circle is a round figure formed by joining different points which are at an equal distance from the central point. The central point is called the centre of the circle and the distance from the centre to any of these points is called the radius of the circle.

The line passing through the centre from one point to the other point of the circle is called the diameter of the circle. The perimeter of the circle is also called the circumference of the circle which is equal to 2πr. Where r = radius of the circle, π = 22/7 or 3.14.

The area of the circle is given by π²r.

NCERT Question class 7 Perimeter and Area

Q. A circle of radius 3cm is cut out from a circular sheet of radius 4cm. Find the area of the remaining sheet. (Take π = 3.14)

Answer:-

Given: 

The radius of the circular sheet (R)  = 4 cm

The radius of the circle (r)= 3 cm

Then,

The area of the remaining sheet = area of the circular sheet - area of the circle.

= π(R² – r²)

= 3.14 (4² – 3²)

= 3.14 (16 – 9)

= 3.14 × 7

= 21.98 cm²

So, the area of the remaining sheet is 21.98 cm².

Many units of length are interconvertible such as kilometres, metres, centimetres, millimetres.

10 millimetres = 1 centimetre

100 centimètres = 1 mètre

1000 metres = 1 kilometre

1 cm² = 100 mm²

1 m² = 10000 cm²

1 hectare = 10000 m²

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