Example (1): A regular square pyramid

• A regular square pyramid has a base of side 6 cm and a slant height of 5 cm. Find its lateral area and its total area.
• Solution:

1. Perimeter of the base P = 4 × 6 = 24 cm.
2. Lateral area = ½ × 24 × 5 = 60 cm².
3. Area of the base = 6² = 36 cm².
4. Total area = 60 + 36 = 96 cm².

Example (2): A regular triangular pyramid

• A regular triangular pyramid has an equilateral base of side 6 cm and a slant height of 5 cm.
• Find its total area, giving the answer to two decimal places.
• Solution:

1. Perimeter of the base P = 3 × 6 = 18 cm.
2. Lateral area = ½ × 18 × 5 = 45 cm².
3. Area of the base = (√3 ÷ 4) × 6² = 9√3 ≈ 15.59 cm².
4. Total area ≈ 45 + 15.59 = 60.59 cm².

Example (3): Regular square pyramid with a given height

• A regular square pyramid has a base of side 8 cm and a height of 3 cm. Find its total area.
• Solution:

1. The distance from the centre of the base to the midpoint of one side = 8 ÷ 2 = 4 cm.
2. Slant height l = √(3² + 4²) = √25 = 5 cm.
3. Lateral area = ½ × 32 × 5 = 80 cm².
4. Area of the base = 8² = 64 cm².
5. Total area = 80 + 64 = 144 cm².

Example (4): A right circular cone of radius 3 cm and slant height 5 cm

• Find the lateral area and the total area of a right circular cone whose base radius is 3 cm and whose slant height is 5 cm.
• Leave the answer in terms of π.
• Solution:

1. Lateral area = π × 3 × 5 = 15π cm² ≈ 47.12 cm².
2. Area of the base = π × 3² = 9π cm².
3. Total area = 15π + 9π = 24π cm² ≈ 75.40 cm².

Example (5): A right circular cone of radius 6 cm and slant height 10 cm

1. Find the total area of the cone in terms of π.
2. Solution:

1. Total area = π r (r + l) = π × 6 × (6 + 10) = 96π cm².
2. 96π ≈ 301.59 cm².