How to Find Length of Triangular-Prism by its Surface Area and Side-Length of Base

QUESTION: A triangular prism has surface area 816cm². If side-length of each of the equilateral triangle is 12cm, what is the length of the prism?

SOLUTION:

A triangular prism is made up of two triangular bases and three rectangular sides. The two bases are parallel and congruent to each other.

Surface Area of a triangular prism = Area of 2 triangles + Area of 3 rectangles ———————(A)

Here, it is given that

Surface Area of a triangular prism = 816 cm² ——————————(1)

Side-length of an equilateral triangular base = 12cm

Formula of area of an equilateral triangle with side-length $a$ = $\frac{\sqrt3}4a^2$

Here $a$ = 12cm

So,

Area of an equilateral triangle with side-length of 12 cm = $\frac{\sqrt3}{4}12^2$ ⇒ 62.35 cm²

Area of 2 equilateral triangles = 2 × 62.35 = 124.7 cm²——————————-(2)

Formula of Area of rectangle = Length × width

In triangular prism, (you can see in the image above also)

Width of the rectangle = Side-length of the triangle

Here so width of the rectangle = 12 cm

Area of rectangle = Length × 12

Area of 3 rectangles = Length × 12 × 3 = 36 × Length ————————-(3)

Put values from equations (1), (2) and (3) in equation (A)

Surface Area of a triangular prism = Area of 2 triangles + Area of 3 rectangles

816 = 124.7 + 36 × Length

36 × Length = 816 – 124.7

36 × Length = 691.3

Length = 691.3/36 = 19.2 cm

Thus the length of the rectangles or length of prism is 19.2 cm

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