Example (1): Slant height of a regular square pyramid

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

1. The distance from the centre of the base to the midpoint of one side: m = 8 ÷ 2 = 4 cm.
2. l² = h² + m² = 3² + 4² = 9 + 16 = 25.
3. l = 5 cm.

Example (2): Slant height of a right circular cone

• A right circular cone has base radius 3 cm and height 4 cm. Find its slant height.
• Solution:

1. l² = r² + h² = 3² + 4² = 25.
2. l = 5 cm.

Example (3): Finding the height from the slant height and the radius

• A right circular cone has base radius 5 cm and slant height 13 cm. Find its true height.
• Solution:

1. h² = l² − r² = 13² − 5² = 169 − 25 = 144.
2. h = √144 = 12 cm.

Example (4): A regular triangular pyramid

• A regular triangular pyramid has an equilateral triangle as its base, of side 6 cm, and its slant height is 5 cm.
• Find its true height.
• Solution:

1. The distance from the centre of an equilateral triangle to the midpoint of a side is m = a ÷ (2√3) = 6 ÷ (2√3) = √3 ≈ 1.73 cm.
2. h² = l² − m² = 25 − 3 = 22.
3. h = √22 ≈ 4.69 cm.