surface area of a prism with some missing lengths

Question

Question:

A prism has an equilateral triangular base with a perimeter of $12c$m. If the length of the prism is $24cm$, determine the total surface area of the prism.

hint: What is the area of 1 triangle?

I have draw a diagram and tried a few methods to figure out the area of 1 triangle, but it just doesn't seem to work.

Any help would be appreciated.

Answer 1

The length of an arete of an equilateral triangle with a perimeter of $12$ is $a=4$. Let us determine the height $h$ of a triangle : it is given by the trigonometric relation$$h=a\sin\left(\frac{\pi}{3}\right)=a\frac{\sqrt{3}}{2} $$ since the angles of an equilateral triangle are all $\frac{\pi}{3}$. So the area of $1$ triangle is $$A=\frac{ah}{2}=a^{2}\frac{\sqrt{3}}{4}=4\sqrt{3}. $$ Now the total surface area of the prism is the sum of $2$ such triangle areas and the $3$ surfaces of the lateral faces.

Answer 2

Divide the problem in simpler tasks:

• Look the a triangular prism is made.

• You can observe that its surface is made of 3 rectangles and 2 triangles.

• Make questions about data you have in order to solve the problem:

• What is the total surface given 3 rectangles and 2 triangles? The sum of their surfaces!
• What is the surface of a generic rectangle? Base $\cdot$ height

• What is the surface of a generic triangle? Also Base $\cdot$ height

  • How can I calculate it? Pythagorean theorem. The height of a triangle intersects the base in the middle. So the height = $\sqrt{\text{side}^2 - (\frac{\text{side}}{2})^2 } $

• So the remaining part of the remaining part of the problem is to put all data together...

the part that I leave to you...