CBSE Class 10 Mensuration: Surface Areas & Volumes — practice questions

To find the area of a circle, we use the standard geometric formula that relates the radius of the circle to the space it occupies on a 2D plane. 1. **Identify the Radius**: The radius is given as $r$, which is the distance from the center of the circle to its edge. 2. **Recall the Formula**: The area $A$ of a circle is calculated by taking the square of the radius ($r^2$) and multiplying it by the constant $\pi$ (approximately 3.14159). 3. **Set up the Equation**: $$A = \pi \times r^2$$ 4. **Conclusion**: The correct expression for the area is $\pi r^2$. Note: $2\pi r$ is the formula for the **circumference** (the distance around the circle), not the area.

To find the area of a sector of a circle, we follow these steps: 1. **Total Area**: The total area of a circle with radius $r$ is given by the formula: $$A_{total} = \pi r^2$$ 2. **Total Angle**: A complete circle subtends an angle of $360^\circ$ at its center. 3. **Proportionality**: A sector is a fraction of the entire circle. The ratio of the area of the sector to the total area of the circle is the same as the ratio of the angle subtended by the sector ($\theta$) to the total angle of the circle ($360^\circ$). $$\frac{\text{Area of Sector}}{\text{Total Area}} = \frac{\theta}{360^\circ}$$ 4. **Derivation**: Multiply both sides by the total area: $$\text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2$$ Thus, the area of the sector is $(\theta/360) \times \pi r^2$.

To find the volume of a cone, we compare it to a cylinder with the same radius $r$ and height $h$. 1. **Base Area**: The base of a cone is a circle. The area of this base is given by the formula: $$A = \pi r^2$$ 2. **Relationship to a Cylinder**: A cylinder with the same dimensions has a volume of $V_{cylinder} = \text{Base Area} \times \text{Height} = \pi r^2 h$. 3. **The Cone Factor**: In geometry, it is proven (through calculus or physical displacement experiments) that a cone occupies exactly one-third of the volume of a cylinder with the same base and height. 4. **Final Formula**: Multiplying the cylinder volume by $1/3$, we get: $$V = \frac{1}{3} \pi r^2 h$$ Therefore, the correct option is **(1/3) π r² h**.