A solid right pyramid has a square base with a side length of 6\text{ cm} and a vertical h… — JAMB UTME

A solid right pyramid has a square base with a side length of $6\text{ cm}$ and a vertical height of $4\text{ cm}$ . What is the total surface area of the pyramid in square centimeters?

Answer: 96 cm^2

Answer

The total surface area of the pyramid is

96\text{ cm}^2

To determine the total surface area of a square pyramid, sum the base area and the total area of the four triangular faces. The base area is

6 \times 6 = 36\text{ cm}^2

. The slant height of each triangular face is found via the Pythagorean theorem using half the base side length (

3\text{ cm}

) and the vertical height (

4\text{ cm}

), yielding

\sqrt{3^2 + 4^2} = 5\text{ cm}

. The area of one triangular face is

\frac{1}{2} \times 6 \times 5 = 15\text{ cm}^2

, making four faces equal to

60\text{ cm}^2

. Adding the base area gives

36 + 60 = 96\text{ cm}^2

Step-by-Step Solution

Calculate the area of the square base

Base area =

36\text{ cm}^2

The base is a square of side

6\text{ cm}

, so

\text{Area} = 6^2 = 36\text{ cm}^2

Find the slant height of each triangular lateral face

Slant height

l = 5\text{ cm}

The slant height forms the hypotenuse of a right-angled triangle inside the pyramid with legs equal to the vertical height (

4\text{ cm}

) and half the base edge (

3\text{ cm}

l = \sqrt{4^2 + 3^2} = 5\text{ cm}

Calculate the combined area of the four triangular faces

Lateral area =

60\text{ cm}^2

Each triangle has base

6\text{ cm}

and height

5\text{ cm}

, giving an area of

\frac{1}{2} \times 6 \times 5 = 15\text{ cm}^2

. For 4 identical faces, the total is

4 \times 15 = 60\text{ cm}^2

Calculate total surface area

Total Surface Area =

96\text{ cm}^2

Sum the base area and the total lateral area:

36 + 60 = 96\text{ cm}^2

Key Concept

Total Surface Area of a Right Pyramid

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