NCERT Solutions for Class 10 Maths Chapter 12 Exercise 12.1: Areas Related to Circles

Core Concepts: Circumference and Area of a Circle

To solve any problem in class 10 maths areas related to circles ex 12 1, we must first revisit two core formulas from geometry. For a circle with a radius $r$:

1. Circumference ($C$): The total boundary length of the circle, calculated as $C = 2\pi r$ (or $C = \pi d$, where $d$ is the diameter).
2. Area ($A$): The total planar surface enclosed by the circle, calculated as $A = \pi r^2$.

In Exercise 12.1, NCERT asks you to compare, add, or equate the circumferences or areas of different circles. A key mathematical skill you must develop here is algebraic simplification. For instance, if the circumference of a new circle is the sum of the circumferences of two other circles ($C = C_1 + C_2$), we write $2\pi R = 2\pi r_1 + 2\pi r_2$. Dividing both sides by $2\pi$ yields a simple relation: $R = r_1 + r_2$. Notice how we never had to calculate or approximate the value of $\pi$ to find the new radius! This elegant approach saves time and keeps your calculations error-free.

Key Geometrical Terms

Circumference

The distance around the boundary of a circle; mathematically represented as 2pir.

Area of a Circle

The regional space occupied by a circular shape on a two-dimensional plane, equal to pi*r^2.

Pi (π)

A mathematical constant representing the ratio of a circle's circumference to its diameter, approximately equal to 22/7 or 3.14159.

Radius (r)

The constant distance from the center of a circle to any point on its outer boundary.

How to Solve Sum of Circumferences and Areas Problems

1. Step 1: Identify Given Variables — Write down the given radii or diameters. Let the radii of the given circles be $r_1$ and $r_2$, and the radius of the new circle be $R$.
2. Step 2: Set up the Equation — Formulate the algebraic relation based on the question. For circumference sums, use $2\pi R = 2\pi r_1 + 2\pi r_2$. For area sums, use $\pi R^2 = \pi r_1^2 + \pi r_2^2$.
3. Step 3: Eliminate Common Constants — Factor out and cancel $\pi$ (or $2\pi$) from both sides. This reduces the equations to $R = r_1 + r_2$ or $R^2 = r_1^2 + r_2^2$.
4. Step 4: Solve for the Unknown — Substitute the numerical values of $r_1$ and $r_2$ into your simplified equation to find the value of $R$ directly.

Pro Board Exam Tips & Common Mistakes

• Don't Substitute $\pi$ Too Early: This is the most common mistake students make. Substituting $\pi = 22/7$ in early steps leads to complex decimal calculations and rounding errors. Always carry $\pi$ as a symbol and look for opportunities to cancel it out across the equation.
• Verify Unit Agreements: Ensure all units are identical before calculation. If one radius is in cm and another is in meters, convert them to a common unit first.
• Understand 'Revolutions' Problems: Remember that in one complete revolution, a wheel covers a linear distance equal to its circumference ($2\pi r$). Distance = Number of revolutions $\times$ Circumference.

Practice Questions with Solutions

• Q: The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has a circumference equal to the sum of the circumferences of the two circles. A: Step 1: Let the radius of the first circle be $r_1 = 19\text{ cm}$ and the second circle be $r_2 = 9\text{ cm}$. Let the radius of the required circle be $R$. Step 2: According to the question, the circumference of the new circle is equal to the sum of the circumferences of the two given circles. $2\pi R = 2\pi r_1 + 2\pi r_2$ Step 3: Divide both sides of the equation by $2\pi$: $R = r_1 + r_2$ Step 4: Substitute the given values of $r_1$ and $r_2$: $R = 19 + 9 = 28\text{ cm}$ Final answer: The radius of the new circle is 28 cm.
• Q: The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles. A: Step 1: Let the radii of the two given circles be $r_1 = 8\text{ cm}$ and $r_2 = 6\text{ cm}$. Let the radius of the new circle be $R$. Step 2: According to the condition given, the area of the new circle equals the sum of the areas of the two circles. $\pi R^2 = \pi r_1^2 + \pi r_2^2$ Step 3: Divide both sides by $\pi$ to simplify the equation: $R^2 = r_1^2 + r_2^2$ Step 4: Substitute the values of $r_1$ and $r_2$: $R^2 = 8^2 + 6^2 = 64 + 36 = 100$ Taking the square root on both sides: $R = \sqrt{100} = 10\text{ cm}$ Final answer: The radius of the circle is 10 cm.
• Q: The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour? A: Step 1: Find the radius of the wheel, $r = \text{Diameter} / 2 = 80 / 2 = 40\text{ cm} = 0.4\text{ m}$. Step 2: Calculate the distance covered by the wheel in one revolution (its circumference): $\text{Distance per revolution} = 2\pi r = 2 \times \frac{22}{7} \times 40 = \frac{1760}{7}\text{ cm}$. Step 3: Calculate the total distance traveled by the car in 10 minutes. $\text{Speed} = 66\text{ km/h} = \frac{66 \times 1000 \times 100}{60}\text{ cm/min} = 110,000\text{ cm/min}$. $\text{Total Distance in 10 mins} = 110,000 \times 10 = 1,100,000\text{ cm}$. Step 4: Find the number of revolutions ($n$): $n = \frac{\text{Total Distance}}{\text{Distance in 1 revolution}} = \frac{1,100,000}{\frac{1760}{7}} = \frac{1,100,000 \times 7}{1760} = 4375$. Final answer: Each wheel makes 4375 complete revolutions.
• Q: If the perimeter and the area of a circle are numerically equal, then what is the radius of the circle? A: Step 1: Let the radius of the circle be $r$. Step 2: Set up the equation according to the condition where numerical values of perimeter (circumference) and area are equal. $2\pi r = \pi r^2$ Step 3: Cancel the common term $\pi$ from both sides: $2r = r^2$ Step 4: Divide both sides by $r$ (since radius $r$ cannot be 0): $r = 2\text{ units}$ Final answer: The radius of the circle is 2 units.

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