Perimeter and Area: Class 7 Maths NCERT Guide

Welcome to the world of shapes and spaces! In this chapter on Perimeter and Area, we'll explore two very important measurements that we use in our daily lives, often without even realizing it. Imagine you want to put a fence around your garden – you'd need to know its perimeter. Now, if you want to buy carpet for your room, you'd need to know its area. Perimeter is the distance around a closed figure, while area is the amount of surface it covers.

This chapter builds on what you've learned in Class 6 and introduces exciting new concepts for shapes like parallelograms and circles. By the end of this guide, you will master the formulas for calculating the perimeter and area of squares, rectangles, triangles, parallelograms, and circles. You'll also learn how to apply these concepts to solve real-world problems. Let's start measuring our world!

What are Perimeter and Area?

Let's break down these two key ideas with a simple example. Think of a rectangular park.

Perimeter is the total length of the path or boundary that encloses the park. If you were to walk exactly one lap around the edge of the park, the total distance you cover is its perimeter. It is a one-dimensional measurement, so its units are simple length units like metres (m), centimetres (cm), or kilometres (km).

  • Formula for Rectangle: Perimeter = 2 × (length + breadth)
  • Formula for Square: Perimeter = 4 × side

Area is the total space enclosed within that boundary. It's the entire grassy field inside the park where you can play. Area tells you how much surface the shape covers. Since it measures a two-dimensional space (length and width), its units are 'square units' like square metres (m²), square centimetres (cm²), etc. This is like counting how many 1m by 1m squares can fit inside the park.

  • Formula for Rectangle: Area = length × breadth
  • Formula for Square: Area = side × side = side²

Area of Parallelograms and Triangles

  • Understanding Area of a Parallelogram: A parallelogram looks like a slanted rectangle. To find its area, you don't multiply the adjacent sides. Instead, the formula is: Area = base × height. The 'height' is the perpendicular distance from the base to the opposite side. Example: A parallelogram has a base of 10 cm and a height of 5 cm. What is its area? Formula: Area = base × height Calculation: Area = 10 cm × 5 cm = 50 cm² * Answer: The area is 50 square centimetres.
  • Understanding Area of a Triangle: A triangle can be seen as half of a rectangle or a parallelogram. This gives us a clue for its area formula: Area = (1/2) × base × height. Just like with a parallelogram, the 'height' must be the perpendicular line from the base to the opposite vertex. Example: Find the area of a triangle with a base of 12 cm and a corresponding height of 8 cm. Formula: Area = (1/2) × base × height Calculation: Area = (1/2) × 12 cm × 8 cm = 6 cm × 8 cm = 48 cm² * Answer: The area of the triangle is 48 square centimetres.

How to Calculate Circumference and Area of a Circle

  1. Step 1: Understand the Key Parts of a Circle — First, identify the radius (r), which is the distance from the center of the circle to any point on its edge. The diameter (d) is the distance across the circle passing through the center (d = 2r). Also, we use a special number called pi (π), which is approximately 22/7 or 3.14.
  2. Step 2: Calculate the Circumference (Perimeter) — The perimeter of a circle is called its circumference. The formula is Circumference (C) = 2πr. For a circle with a radius of 7 cm, C = 2 × (22/7) × 7 cm = 44 cm.
  3. Step 3: Calculate the Area — The area of a circle is the space it covers. The formula is Area (A) = πr². Don't mix this up with the circumference formula! For a circle with a radius of 7 cm, A = (22/7) × (7 cm)² = (22/7) × 49 cm² = 22 × 7 cm² = 154 cm².
  4. Step 4: Check Your Units — Always remember, circumference is a length, so its unit is cm or m. Area is a surface, so its unit is cm² or m². This is a very common place to make mistakes!

Important Tips for Your Exam

Pay close attention to these points to avoid common errors:

  1. Units, Units, Units! Always write the correct units. Perimeter is in cm or m. Area is in cm² or m². Volume (which you'll learn later) is in cm³ or m³. Writing the wrong unit can lose you marks.
  2. Circle Formulas: Don't confuse Circumference (2πr) and Area (πr²). A good way to remember is that 'Area' sounds like 'Square', and the area formula has r-squared (r²).
  3. Perpendicular Height: For triangles and parallelograms, the 'height' is always the perpendicular height, not the length of the slanted side. The question will often give you both lengths to test if you know which one to use.

Practice Questions with Solutions

  • Q: The length and breadth of a rectangular field are 50 m and 30 m respectively. Find its perimeter and the cost of turfing the field at a rate of ₹20 per square metre. A: Step 1: Find the perimeter of the field. Perimeter = 2 × (length + breadth) Perimeter = 2 × (50 m + 30 m) = 2 × 80 m = 160 m. Step 2: Find the area of the field. Area = length × breadth Area = 50 m × 30 m = 1500 m². Step 3: Calculate the cost of turfing. Cost = Area × Rate per square metre Cost = 1500 m² × ₹20/m² = ₹30,000. Final answer: The perimeter is 160 m, and the cost of turfing is ₹30,000.
  • Q: A wire is bent into the shape of a square with a side of 11 cm. If the same wire is rebent to form a circle, what will be the area of the circle? (Use π = 22/7) A: Step 1: Find the total length of the wire from the square's perimeter. Length of wire = Perimeter of square = 4 × side Length of wire = 4 × 11 cm = 44 cm. Step 2: This length is now the circumference of the circle. Use it to find the radius. Circumference = 2πr 44 cm = 2 × (22/7) × r 44 = (44/7) × r r = (44 × 7) / 44 r = 7 cm. Step 3: Calculate the area of the circle using the radius found. Area = πr² Area = (22/7) × (7 cm)² = (22/7) × 49 cm² Area = 22 × 7 cm² = 154 cm². Final answer: The area of the circle is 154 cm².
  • Q: Find the area of a parallelogram with a base of 20 cm and a corresponding height of 9 cm. A: Step 1: Identify the given values. Base (b) = 20 cm Height (h) = 9 cm Step 2: Use the formula for the area of a parallelogram. Area = base × height Area = 20 cm × 9 cm Area = 180 cm². Final answer: The area of the parallelogram is 180 cm².
  • Q: The area of a triangle is 84 cm². If its base is 14 cm, find its height. A: Step 1: Write down the formula for the area of a triangle. Area = (1/2) × base × height Step 2: Substitute the known values into the formula. 84 cm² = (1/2) × 14 cm × height 84 = 7 × height Step 3: Solve for the height. height = 84 / 7 height = 12 cm. Final answer: The height of the triangle is 12 cm.

Frequently Asked Questions

What is the difference between perimeter and area?

Perimeter is the length of the boundary of a shape, like a fence around a garden. Area is the space inside the shape, like the grass in the garden. Perimeter is measured in units (cm, m), while area is measured in square units (cm², m²).

What is π (pi) and why do we use it?

Pi (π) is a special mathematical constant, approximately equal to 3.14 or 22/7. It represents the ratio of any circle's circumference to its diameter. We use it in all formulas for calculating the circumference and area of circles.

How do I find the area of a path built around a rectangular garden?

First, calculate the area of the outer rectangle (garden + path). Then, calculate the area of the inner rectangle (the garden itself). Subtract the inner area from the outer area to get the area of the path.

Does it matter if I use π = 22/7 or π = 3.14?

Most questions in your exam will specify which value of π to use. If not, a good tip is to use 22/7 if the radius or diameter is a multiple of 7, as it simplifies the calculation. Otherwise, 3.14 is a good choice.