Mensuration: volume of solids — KCSE Mathematics

In this lesson, we will focus on calculating the volumes of pyramids, cones, and spheres, as well as composite solids. The formulas you need to remember are:

• Volume of a pyramid: ( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} )
• Volume of a cone: ( V = \frac{1}{3} \times \pi r^2 h )
• Volume of a sphere: ( V = \frac{4}{3} \pi r^3 )

To find the volume of composite solids, calculate the volume of each solid separately and then sum them up.

Example: Calculate the volume of a cone with a radius of 3 cm and a height of 5 cm.

1. Use the cone formula: ( V = \frac{1}{3} \times \pi (3^2) (5) )
2. Calculate: ( V = \frac{1}{3} \times \pi (9) (5) = \frac{45}{3} \pi = 15\pi \approx 47.12 , ext{cm}^3 )

When dealing with composite solids, ensure to break them down into recognizable shapes, calculate their volumes, and then combine them for the final answer.

In this lesson, we will solve problems involving capacity and density. Capacity refers to the volume of a substance that a container can hold, measured in cubic centimeters (cm³) or liters (L). Remember that 1 L = 1000 cm³. Density is defined as mass per unit volume, expressed as:

• Density (ρ) = Mass (m) / Volume (V)
• Rearranging gives: Mass = Density × Volume

To convert between units, use the following relationships:

• 1 m³ = 1,000,000 cm³
• 1 L = 1000 cm³

Example Problem:
A container holds 2500 cm³ of water. What is its capacity in liters?

• Solution:
  • Convert cm³ to L:
  • Capacity in liters = 2500 cm³ ÷ 1000 = 2.5 L.

Another Example:
If a substance has a density of 2 g/cm³ and the volume is 5 cm³, what is the mass?

• Solution:
  • Mass = Density × Volume
  • Mass = 2 g/cm³ × 5 cm³ = 10 g.

In mensuration, the volume of solids refers to the amount of space occupied by a three-dimensional object. Different solids have specific formulas for calculating their volume. Here are the key formulas:

• Cube: Volume = side³
• Rectangular Prism: Volume = length × width × height
• Cylinder: Volume = π × radius² × height
• Sphere: Volume = (4/3) × π × radius³
• Cone: Volume = (1/3) × π × radius² × height

To solve problems involving volume, identify the solid, use the correct formula, and substitute the values accurately. Remember to express your final answer in cubic units (e.g., cm³, m³).

For example, to find the volume of a cylinder with a radius of 3 cm and a height of 5 cm:

1. Identify the formula: Volume = π × radius² × height.
2. Substitute the values: Volume = π × (3 cm)² × 5 cm.
3. Calculate: Volume = π × 9 cm² × 5 cm = 45π cm³.

Thus, the volume of the cylinder is 45π cm³.