A solid right triangular prism has a base that is a right-angled triangle with legs of len… — JAMB UTME

A solid right triangular prism has a base that is a right-angled triangle with legs of length $6\text{ cm}$ and $8\text{ cm}$ . If the height of the prism is $15\text{ cm}$ , what is the total surface area of the prism, in $\text{cm}^2$ ?

Answer: 408 cm²

Answer

The total surface area of the right triangular prism is

408\text{ cm}^2

To find the total surface area of a right triangular prism, compute the sum of the areas of its 2 triangular bases and its 3 rectangular side faces. The legs of the right-angled triangle are

6\text{ cm}

and

8\text{ cm}

, so the hypotenuse is

\sqrt{6^2 + 8^2} = 10\text{ cm}

. The combined area of the two bases is

2 \times (\frac{1}{2} \times 6 \times 8) = 48\text{ cm}^2

. The perimeter of the base is

6 + 8 + 10 = 24\text{ cm}

, making the lateral surface area

24 \times 15 = 360\text{ cm}^2

. Adding the base areas and lateral area yields

48 + 360 = 408\text{ cm}^2

Step-by-Step Solution

Determine the length of the hypotenuse of the triangular base.

Hypotenuse =

10\text{ cm}

The base is a right-angled triangle with legs

6\text{ cm}

and

8\text{ cm}

. By Pythagoras:

c = \sqrt{6^2 + 8^2} = 10\text{ cm}

Calculate the total area of the two parallel triangular bases.

Base area total =

48\text{ cm}^2

The area of one right triangle is

\frac{1}{2} \times 6 \times 8 = 24\text{ cm}^2

, so two bases have an area of

2 \times 24 = 48\text{ cm}^2

Calculate the lateral surface area of the three rectangular faces.

Lateral surface area =

360\text{ cm}^2

The lateral surface area is equal to the perimeter of the base times the height:

(6 + 8 + 10) \times 15 = 24 \times 15 = 360\text{ cm}^2

Sum the base area total and lateral surface area.

Total Surface Area =

408\text{ cm}^2

Total surface area =

48 + 360 = 408\text{ cm}^2

Key Concept

Total Surface Area of a Right Triangular Prism

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