Circumference and Area of a Circle

The circumference and area are fundamental concepts related to circles in geometry. The circumference is the distance around the outside of a circle, while the area is the amount of space enclosed within the circle. Understanding these concepts is crucial for solving various geometric problems and real-world applications.

Formulae

The following formulae are essential for calculating the circumference and area of a circle:

• Circumference (C): $C = 2\pi r$ where $r$ is the radius of the circle and $\pi$ (pi) is approximately 3.14159.
• Area (A): $A = \pi r^2$ where $r$ is the radius of the circle and $\pi$ (pi) is approximately 3.14159.

Examples

Let’s look at a couple of examples:

Example-1: A circle has a radius of 5 cm. Calculate its circumference and area.

1. Circumference: $C = 2\pi r = 2 \times \pi \times 5 \approx 31.42$ cm
2. Area: $A = \pi r^2 = \pi \times 5^2 = 25\pi \approx 78.54$ cm²

Example-2: A circle has a diameter of 10 inches. Calculate its circumference and area.

1. Radius: Remember that the radius is half the diameter, so $r = 10/2 = 5$ inches
2. Circumference: $C = 2\pi r = 2 \times \pi \times 5 \approx 31.42$ inches
3. Area: $A = \pi r^2 = \pi \times 5^2 = 25\pi \approx 78.54$ inches²

Common mistakes by students

• Confusing Radius and Diameter: Using the diameter instead of the radius in the formulas, or vice-versa. Always ensure you’re using the correct value for ‘r’.
• Incorrect Units: Forgetting to include the correct units (e.g., cm, m²) in the final answer.
• Approximating $\pi$: Using a rounded value of $\pi$ (like 3.14) prematurely, leading to minor inaccuracies. Using the $\pi$ button on your calculator will provide the most accurate results.
• Confusing Circumference and Area: Using the formula for circumference when calculating area, and vice versa.

Real Life Application

Understanding the circumference and area of circles has many real-world applications:

• Construction: Calculating the amount of fencing needed for a circular garden (circumference).
• Engineering: Determining the amount of material needed to build a circular storage tank (area).
• Design: Planning the layout of a circular room or the size of a pizza.
• Sports: Calculating the distance a runner covers on a circular track (circumference).

Fun Fact

The number $\pi$ (pi) is an irrational number, meaning it cannot be expressed as a simple fraction. Its decimal representation goes on forever without repeating. Mathematicians have calculated $\pi$ to trillions of digits!

Recommended YouTube Videos for Deeper Understanding

Q.1 A circular garden has a radius of 7 meters. What is the area of the garden?

Check Solution

Ans: B

Area of circle = $\pi r^2 = \pi (7)^2 = 49\pi \approx 154 m^2$

Q.2 The circumference of a circle is $20\pi$ cm. What is the radius of the circle?

Check Solution

Ans: B

Circumference = $2\pi r$, so $20\pi = 2\pi r$. Dividing both sides by $2\pi$, we get $r = 10$ cm.

Q.3 A circle has an area of $36\pi$ square inches. What is the circumference of the circle?

Check Solution

Ans: B

Area = $\pi r^2 = 36\pi$, so $r^2 = 36$ and $r = 6$ inches. Circumference = $2\pi r = 2\pi(6) = 12\pi$ inches.

Q.4 If the radius of a circle is doubled, how does the area of the circle change?

Check Solution

Ans: C

Let the original radius be $r$. Original area = $\pi r^2$. New radius = $2r$. New area = $\pi (2r)^2 = 4\pi r^2$. The area is multiplied by 4.

Q.5 What is the area of a circle with a diameter of 10 cm?

Check Solution

Ans: B

Diameter = 10 cm, so radius = 5 cm. Area = $\pi r^2 = \pi (5)^2 = 25\pi cm^2$

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