Max bought a light prism for his mom for Mother's Day. He wants to figure out how much wrapping paper is needed to wrap it. To do this, he needs to figure out the surface area of the gift. The triangular end has a base of 3 cm and height of 4 cm. The length of each side is 6 cm and the height of the prism (length of the rectangular side) is 8 cm. What is the surface area of the gift that Max bought for his mom?
In this concept, you will learn how to calculate the surface area of a triangular prism.
Finding the Surface Area of a Triangular Prism
Area is the space that is contained in a two-dimensional figure. Surface area is the total area of all of the sides and faces of a three-dimensional figure. To find the surface area, the area of each face is calculated and then add these areas together.
One way to do this is to use a net, since a net is a two-dimensional representation of a three-dimensional solid, or an unfolded picture of a solid.
What is the surface area of the figure below?
The net for this triangular prism is as follows:
Now, let's fill in the measurements for the sides of each face in order to calculate their area. Triangular prisms have their own formula for finding surface area because they have two triangular faces opposite each other.
The formula A=12bh is used to find the area of the top and bases triangular faces, where A = area, b = base, and h = height. The formula A=lw is used to find the area of the three rectangular side faces, where A = area, l = length, and w = width.
Plugging in the measurements that are given in the net, calculate the area of each face. Remember to use the correct area formula for the triangles and rectangles.
=Bottom faceA=12bh12(8)(6) +24+ 524 cm2Top faceA=12bh12(8)(6) +24+sideA=lw17×10 +170 +sideA=lw17×10 +170 +sideA=lw17×8136
When you add these values together, you get a surface area of 524 square centimeters for this triangular prism.
SA=bh+(s1+s2+s3)HYou can also use one formula to calculate the surface area of a triangular prism which can save time over the process of using a net to derive the areas:
where b = base; h = height of the triangle; s1, s2, and s3 = the length of each side of the triangle base, and H = the height of the prism (which is the length of the rectangles).
First, find the area of the two triangular faces. Each face will have an area of 12bh. Remember, you can use a formula to calculate the area of a pair of faces. Therefore, you can double this formula to find the area of both triangular faces at once which results in the formula 2(12bh). The 2 multiplied by the 12 equals 1, or cancels each other out, and you're left with bh.
Next, you need to calculate the area of each of the rectangular side faces. The length of each rectangle is the same as the height of the prism, so call this H. The width of each rectangle is actually the same as the sides of the triangular base. Instead of multiplying the length and width for each rectangle, you can combine this information. Since there are 3 rectangular widths (all equal to the sides of the triangles), multiply the perimeter of the triangular base by the height of the rectangles, H, which will give the area of all three rectangles.
If you put these pieces together—the area of the bases and the area of the side faces—you get this formula:
SA=bh+(s1+s2+s3)H
where bh = the area of the triangle top and base, and (s1 + s2 + s3)H = the area of the rectangular side faces.
Remember that the height of the triangular base (h) is not necessarily the same as the height of the prism (H).
