Perimeter and Area Ex 11.3 NCERT: A Complete Guide for Class 7

Welcome to Exercise 11.3 of the 'Perimeter and Area' chapter! So far, you've worked with shapes like squares and rectangles. Now, we're moving on to a very special shape: the circle. This exercise is all about understanding and calculating the two most important properties of a circle: its circumference and its area.

Why is this important? Circles are everywhere! Think about the wheels of your bicycle, a wall clock, a pizza, or a circular park. Knowing how to calculate the distance around a circle (circumference) or the space it covers (area) is a super useful skill. In this guide, we'll break down the formulas, introduce the magical number Pi (π), and walk you through problems step-by-step. By the end, you'll be able to solve any question from Exercise 11.3 with confidence.

Understanding Circles: Radius, Diameter, and Pi (π)

Before we can calculate anything, we need to know the basic parts of a circle. Imagine a dot in the center. A circle is a set of all points that are at the same distance from this center.

  • Radius (r): This is the distance from the center of the circle to any point on its edge. It's the most important measurement for all our calculations.
  • Diameter (d): This is the distance from one edge of the circle to the other, passing through the center. You'll notice that the diameter is always exactly twice the length of the radius. So, the key relationship is: d = 2 × r or r = d / 2.
  • Pi (π): Pi is a special constant in mathematics. It's the ratio of a circle's circumference to its diameter. Its value is approximately 3.14159..., but it goes on forever without repeating! For our calculations in Class 7, we use approximations. The two common values are π ≈ 22/7 or π ≈ 3.14. The question will usually tell you which one to use.

How to Calculate Circumference and Area of a Circle

  1. Step-by-Step: Finding the Circumference (Perimeter) — The circumference is the distance around the circle. Think of it as 'unwrapping' the circle and measuring its length. The formula is C = 2πr. 1. Identify the radius (r). If the diameter (d) is given, first calculate the radius: r = d/2. 2. Choose the value of π (usually 22/7 or 3.14, as given in the question). 3. Substitute the values into the formula C = 2 × π × r. 4. Calculate the result. The answer will be in units like cm, m, etc.
  2. Step-by-Step: Finding the Area — The area is the space enclosed inside the circle. The formula is A = πr². 1. Identify the radius (r). 2. Calculate the square of the radius (r²), which means r × r. 3. Choose the value of π. 4. Substitute the values into the formula A = π × r². 5. Calculate the result. Remember, the answer will be in square units like cm², m², etc.

Solved Problems for Ex 11.3

  • Question: Find the circumference and area of a circle with a radius of 14 cm. (Use π = 22/7). Solution: Given: radius (r) = 14 cm. Circumference (C): Formula: C = 2πr Substitute values: C = 2 × (22/7) × 14 Calculate: C = 2 × 22 × (14/7) = 2 × 22 × 2 = 88 cm. Area (A): Formula: A = πr² Substitute values: A = (22/7) × 14² = (22/7) × 196 * Calculate: A = 22 × (196/7) = 22 × 28 = 616 cm².
  • Question: The circumference of a circular sheet is 154 m. Find its radius and also its area. (Use π = 22/7). Solution: Given: Circumference (C) = 154 m. Find the radius (r): Formula: C = 2πr 154 = 2 × (22/7) × r Rearrange to solve for r: r = (154 × 7) / (2 × 22) = (154 × 7) / 44 Simplify: r = (7 × 7) / 2 = 49/2 = 24.5 m. Find the Area (A): Formula: A = πr² Substitute values: A = (22/7) × (24.5)² = (22/7) × 24.5 × 24.5 Calculate: A = 22 × 3.5 × 24.5 = 77 × 24.5 = 1886.5 m².

Important Tips for Your Exam

  • Radius vs. Diameter: This is the most common trap! Always read the question carefully to see if you are given the radius (r) or the diameter (d). If it's the diameter, your first step should always be to divide it by 2 to find the radius.
  • Correct Formula: Memorize the formulas and don't mix them up. Circumference is a length (2πr), while Area is a space (πr²). The 'squared' in the area formula is a good hint that the unit will be squared (cm², m²).
  • Units Matter: Forgetting to write the correct units can cost you marks. Circumference is in simple units (m, cm), but Area MUST be in square units (m², cm²).
  • Value of π: Always use the value of π specified in the question (e.g., "Use π = 3.14"). If nothing is mentioned, check if the radius/diameter is a multiple of 7. If it is, using π = 22/7 will make your calculations much easier.

Practice Questions with Solutions

  • Q: A circular garden has a diameter of 9.8 m. Find its area. (Use π = 22/7) A: Step 1: Find the radius from the diameter. Radius (r) = Diameter / 2 = 9.8 m / 2 = 4.9 m. Step 2: Use the formula for the area of a circle, A = πr². Step 3: Substitute the values: A = (22/7) × (4.9)² = (22/7) × 4.9 × 4.9. Step 4: Calculate the result: A = 22 × (4.9/7) × 4.9 = 22 × 0.7 × 4.9 = 75.46 m². Final answer: The area of the garden is 75.46 m².
  • Q: Find the perimeter of a semi-circular protractor whose diameter is 14 cm. (Use π = 22/7) A: Step 1: The perimeter of a semi-circle consists of the curved arc and the straight diameter. Step 2: Calculate the length of the curved arc, which is half the circumference of a full circle. Arc length = (1/2) × 2πr = πr. Step 3: Find the radius. Given diameter = 14 cm, so radius (r) = 14 / 2 = 7 cm. Step 4: Calculate the arc length: Arc length = (22/7) × 7 = 22 cm. Step 5: Add the length of the straight diameter to the arc length. Total Perimeter = Arc length + Diameter = 22 cm + 14 cm = 36 cm. Final answer: The perimeter of the protractor is 36 cm.
  • Q: The cost of fencing a circular field at the rate of ₹24 per metre is ₹5280. Find the radius of the field. (Use π = 22/7) A: Step 1: Find the total length of the fence, which is the circumference of the field. Circumference = Total Cost / Rate per metre = ₹5280 / ₹24 = 220 m. Step 2: Use the circumference formula, C = 2πr, to find the radius. Step 3: Substitute the known values: 220 = 2 × (22/7) × r. Step 4: Rearrange the formula to solve for r: r = (220 × 7) / (2 × 22) = (220 × 7) / 44. Step 5: Simplify the calculation: r = 5 × 7 = 35 m. Final answer: The radius of the field is 35 m.
  • Q: A wire is bent into the shape of a circle with a radius of 28 cm. If it is re-bent into a square, what is the length of the side of the square? (Use π = 22/7) A: Step 1: The length of the wire does not change. First, find the length of the wire by calculating the circumference of the circle. Step 2: Circumference of circle (Length of wire) = 2πr = 2 × (22/7) × 28 cm. Step 3: Calculate the circumference: C = 2 × 22 × (28/7) = 2 × 22 × 4 = 176 cm. Step 4: This length of 176 cm is now the perimeter of the square. The perimeter of a square is 4 × side. Step 5: Equate the perimeters: 4 × side = 176 cm. Solve for the side: side = 176 / 4 = 44 cm. Final answer: The side of the square is 44 cm.

Frequently Asked Questions

What is the difference between the circumference and area of a circle?

Circumference is the length of the boundary line of the circle, like a fence around a circular park. Area is the total space enclosed within that boundary, like the grass inside the park. Circumference is measured in units (cm, m), while area is measured in square units (cm², m²).

When should I use π = 22/7 and when should I use π = 3.14?

You should always use the value of π that is specified in the question. If no value is given, it's a good strategy to use π = 22/7 if the radius or diameter is a multiple of 7, as it makes calculations easier. Otherwise, π = 3.14 is a safe choice.

How do I calculate the area if the diameter is given instead of the radius?

It's a simple two-step process. First, you must find the radius by dividing the diameter by 2 (r = d/2). Then, use this radius value in the area formula, A = πr².