Volume of Cylinders and Cones

The volume of a cylinder is given by the formula: $$V = \text{Base Area} \times \text{Height} = \text{Area of Circle} \times h = \text{π} \times r^2 \times h$$ .

The volume of a cone is given by the formula: $$V = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times \text{π} \times r^2 \times h$$ .

The area of a circle is calculated using the formula: $$A = \text{π} \times r^2$$ .

The radius is the distance from the center of the circle to any point on its circumference, while the diameter is twice the radius, extending from one side of the circle to the other through the center.

Using the formula for the volume of a cylinder: $$V = \text{π} \times r^2 \times h = \text{π} \times (3)^2 \times 5 = 45\text{π} \text{ cm}^3 \text{ (approximately 141.4 cm}^3)$$ .

Using the formula for the volume of a cone: $$V = \frac{1}{3} \times \text{π} \times r^2 \times h = \frac{1}{3} \times \text{π} \times (4)^2 \times 9 = 48\text{π} \text{ cm}^3 \text{ (approximately 150.8 cm}^3)$$ .

The volume of a cone is one-third the volume of a cylinder with the same base and height: $$V_{cone} = \frac{1}{3} V_{cylinder}$$ .

To convert diameter to radius, divide the diameter by 2: $$r = \frac{d}{2}$$ .

First, find the radius: $$r = \frac{10}{2} = 5 \text{ cm}$$ . Then use the volume formula: $$V = \frac{1}{3} \times \text{π} \times (5)^2 \times 12 = 100\text{π} \text{ cm}^3 \text{ (approximately 314.2 cm}^3)$$ .

The lateral surface area of a cylinder is given by: $$A = 2\text{π}rh$$ .