Perimeter and Area Lesson: Definitions, Formulas, Calculations & Practice Questions

Lesson Overview

• What Is Perimeter and Area?
• Difference Between Area and Perimeter
• How to Find Area and Perimeter​ For Different Shapes
• Practice Problems for Perimeters and Areas

Perimeter and area are concepts in mathematics that help us measure and describe shapes. They provide valuable information about the size and shape of an object, allowing us to compare and analyze different figures. 

Perimeter and area have practical applications in fields like architecture, engineering, and design.

What Is Perimeter and Area?

The perimeter of a two-dimensional shape is the total length of its boundary. It's calculated by adding up the lengths of all the sides of the shape.

For example:

A rectangle has sides of 5 cm and 3 cm.

Perimeter = 5 cm + 3 cm + 5 cm + 3 cm = 16 cm

Area is the amount of space a two-dimensional shape occupies. It's measured in square units (like square centimeters or square meters).

For example:

A square has sides of 4 meters. Area = 4 meters * 4 meters = 16 m²

Difference Between Area and Perimeter

Feature	Area	Perimeter
Definition	The space a shape occupies	The total length of the shape's boundary
Measurement	Square units (e.g., cm², m²)	Units of length (e.g., cm, m)
Calculation	Often involves multiplying sides (e.g., length x width)	Involves adding up all the side lengths
Analogy	The amount of paint needed to cover a surface	The length of fence needed to enclose an area
Changes with shape	Two shapes can have the same perimeter but different areas	Two shapes can have the same area but different perimeters

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How to Find Area and Perimeter​ For Different Shapes

Finding the area and perimeter of different shapes depends on their unique properties and formulas. Each shape has specific formulas to calculate its area and perimeter. These formulas use measurements like side lengths, radius, height, and width.

Applying the Formulas

1. Determine the type of shape you're working with (e.g., square, circle, triangle).

2. Measure the relevant dimensions of the shape (e.g., side lengths for a square, radius for a circle).

3. Substitute the measurements into the appropriate formula for the shape.

4. Perform the calculations to find the area and perimeter.

Table of Formulas

Shape	Area	Perimeter	Terms
Square	A = a²	P = 4a	a = length of a side
Rectangle	A = l × w	P = 2(l + w)	l = length, w = width
Triangle	A = ½ × b × h	S = a + b + c	b = base, h = height, a, b, and c are the sides of the triangle
Circle	A = π × r²	Circumference = 2πr	r = radius of the circle
Parallelogram	A = b × h	P = 2(a + b)	a = side, b = base, h = vertical height
Trapezoid	A = ½h(b₁ + b₂)	P = a + b₁ + c + b₂	h = height, b₁ and b₂ are the lengths of the parallel sides, a and c are the lengths of the non-parallel sides

Remember:

• π is a mathematical constant approximately equal to 3.14159.
• The distance from the center of a circle to its edge.
• The perpendicular distance from the base of a shape to its opposite side or vertex.

Practice Problems for Perimeters and Areas

Q1. A square garden has a side length of 12 meters. Find its area and perimeter.

Solution:

Area- 

The formula for the area of a square is Area = side * side.

We know the side length is 12 meters.

Substituting, we get Area = 12 m * 12 m = 144 m².

Perimeter- 

The formula for the perimeter of a square is Perimeter = 4 * side.

Again, the side length is 12 meters.

Substituting, we get Perimeter = 4 * 12 m = 48 m.

Solution:

Area- 

The formula for the area of a rectangle is Area = length * width.

We have a length of 20 meters and a width of 10 meters.

Substituting, we get Area = 20 m * 10 m = 200 m².

Perimeter-

The formula for the perimeter of a rectangle is Perimeter = 2 * (length + width).

Substituting the length and width, we get Perimeter = 2 * (20 m + 10 m) = 2 * 30 m = 60 m.

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Q3.  A triangular sail has a base of 8 meters and a height of 6 meters. Find its area and perimeter (assume it's an isosceles triangle with two sides of length 10 meters).

Solution:

Area-

The formula for the area of a triangle is Area = (1/2) * base * height.

The base is 8 meters and the height is 6 meters.

Substituting, we get Area = (1/2) * 8 m * 6 m = 24 m².

Perimeter-

The formula for the perimeter of a triangle is Perimeter = side1 + side2 + side3.

Since it's an isosceles triangle, two sides are 10 meters each, and the base is 8 meters.

Substituting, we get Perimeter = 10 m + 10 m + 8 m = 28 m.

Q4. A circular pizza has a radius of 15 centimeters. Find its area and circumference (perimeter).

Solution:

Area- 

The formula for the area of a circle is Area = π * radius².

The radius is 15 centimeters.

Substituting, we get Area = π * 15 cm * 15 cm ≈ 706.86 square centimeters (using π ≈ 3.14159).

Circumference- The formula for the circumference of a circle is Circumference = 2 * π * radius.

Substituting the radius, we get Circumference = 2 * π * 15 cm ≈ 94.25 centimeters.

Q5.  A parallelogram-shaped parking lot has a base of 30 meters, a height of 15 meters, and a slanted side length of 20 meters. Find its area and perimeter.

Solution:

Area-

The formula for the area of a parallelogram is Area = base * height.

The base is 30 meters and the height is 15 meters.

Substituting, we get Area = 30 m * 15 m = 450 m².

Perimeter-

The formula for the perimeter of a parallelogram is Perimeter = 2 * (base + side).

We use the base (30 m) and one of the slanted sides (20 m).

Substituting, we get Perimeter = 2 * (30 m + 20 m) = 2 * 50 m = 100 m.

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Q6. A trapezoid-shaped table has bases of 120 cm and 80 cm and a height of 70 cm. The two non-parallel sides are each 75 cm long. Find its area and perimeter.

Solution:

Area-

The formula for the area of a trapezoid is Area = (1/2) * (base1 + base2) * height.

We have base1 = 120 cm, base2 = 80 cm, and height = 70 cm.

Substituting, we get Area = (1/2) * (120 cm + 80 cm) * 70 cm = (1/2) * 200 cm * 70 cm = 7000 cm².

Perimeter-

The formula for the perimeter of a trapezoid is Perimeter = side1 + side2 + base1 + base2.Substituting all the side lengths, we get Perimeter = 75 cm + 75 cm + 120 cm + 80 cm = 350 cm.