A rectangular parallelepiped is a box-shaped object with six rectangular faces. Examples include classrooms, matchboxes, and books.

Surface Area of a Rectangular Parallelepiped

A rectangular prism has 6 faces. The surface area is the sum of the areas of all six faces.

Formula: $\text{Surface Area} = 2(lw + lh + wh)$

Where:

• l = length
• w = width
• h = height

Volume of a Rectangular Parallelepiped

Formula: $\text{Volume} = \text{length} \times \text{width} \times \text{height} = l \times w \times h$

A cube is a special rectangular prism where all sides are equal. Examples include dice, sugar cubes, and gift boxes.

Surface Area of a Cube

Since all six faces are equal squares:

Formula: $\text{Surface Area} = 6 \times s^2$

Where s = length of one side

Volume of a Cube

Formula: $\text{Volume} = s \times s \times s = s^3$

A cylinder has two circular bases (top and bottom) and one curved surface. Examples include water tanks, drums, and poles.

Surface Area of a Cylinder

When we "unroll" a cylinder, we get two circles and one rectangle. The width of the rectangle equals the circumference of the circle.

Formula: $\text{Surface Area} = 2\pi r^2 + 2\pi rh = 2\pi r(r + h)$

Where:

• r = radius of the base
• h = height of the cylinder
• π = 22/7 or 3.14

Volume of a Cylinder

The volume is found by multiplying the area of the circular base by the height.

Formula: $\text{Volume} = \pi r^2 h$

Example 1: Rectangular Parallelepiped

A classroom is 8 m long, 6 m wide, and 3 m high. Find: a) The surface area b) The volume

Solution:

a) Surface Area = 2(lw + lh + wh) $= 2(8 \times 6 + 8 \times 3 + 6 \times 3)$ $= 2(48 + 24 + 18)$ $= 2(90) = 180 \text{ m}^2$

b) Volume = l × w × h $= 8 \times 6 \times 3 = 144 \text{ m}^3$

Example 2: Cube

A sugar cube has sides of 2 cm. Find: a) The surface area b) The volume

Solution:

a) Surface Area = 6s² $= 6 \times 2^2 = 6 \times 4 = 24 \text{ cm}^2$

b) Volume = s³ $= 2^3 = 8 \text{ cm}^3$

Example 3: Cylinder

A cylindrical water tank has a radius of 7 cm and a height of 10 cm. Find: a) The surface area b) The volume (use π = 22/7)

Solution:

a) Surface Area = 2πr(r + h) $= 2 \times \frac{22}{7} \times 7 \times (7 + 10)$ $= 44 \times 17 = 748 \text{ cm}^2$

b) Volume = πr²h $= \frac{22}{7} \times 7^2 \times 10$ $= \frac{22}{7} \times 49 \times 10$ $= 22 \times 7 \times 10 = 1,540 \text{ cm}^3$

In Tanzania, vendors at markets often need to calculate the volume of cylindrical containers like water tanks or grain silos to know how much they can hold. For example, a farmer storing maize in a cylindrical silo with radius 3.5 meters and height 8 meters can calculate the volume using V = πr²h to determine how many bags of maize the silo can hold, helping them plan storage and sales.