The net of a triangular prism whose bases are equilateral triangles is shown below. What is the total surface area of the triangular prism?
The Correct Answer and Explanation is:
To find the total surface area of a triangular prism with equilateral triangle bases, we can break the problem into simpler steps by calculating the areas of the faces that make up the surface area. A triangular prism consists of:
- Two equilateral triangle bases:
The area of an equilateral triangle with side length sss is given by the formula: A=34s2A = \frac{\sqrt{3}}{4} s^2A=43s2 So, the area of each base is 34s2\frac{\sqrt{3}}{4} s^243s2. Since there are two such triangles (one on the top and one on the bottom), the total area for the bases is: Abases=2×34s2=32s2A_{\text{bases}} = 2 \times \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{2} s^2Abases=2×43s2=23s2 - Three rectangular lateral faces:
The lateral faces are rectangles that connect the corresponding sides of the two equilateral triangles. Each rectangle has a width equal to the side length sss of the triangles, and the height is the length of the prism, denoted as hhh. The area of one rectangular face is: Arect=s×hA_{\text{rect}} = s \times hArect=s×h Since there are three rectangular faces, the total area for the lateral faces is: Alateral=3×(s×h)=3shA_{\text{lateral}} = 3 \times (s \times h) = 3shAlateral=3×(s×h)=3sh - Total surface area:
Finally, to find the total surface area, we add the area of the two triangular bases and the area of the three rectangular lateral faces: Atotal=Abases+Alateral=32s2+3shA_{\text{total}} = A_{\text{bases}} + A_{\text{lateral}} = \frac{\sqrt{3}}{2} s^2 + 3shAtotal=Abases+Alateral=23s2+3sh
Thus, the total surface area of the triangular prism is:Atotal=32s2+3shA_{\text{total}} = \frac{\sqrt{3}}{2} s^2 + 3shAtotal=23s2+3sh
This formula gives the surface area of a triangular prism with equilateral triangle bases, where sss is the side length of the equilateral triangle and hhh is the height (or length) of the prism.
