Work out the surface area of the prism.
The surface area of any prism is the sum of the areas of all that prism’s faces. The prism we’ve been given is a triangular prism because its cross section is a triangle. That’s the orange face at the front of the prism. A triangular prism has five faces. There’re two triangular faces at the front and back of the prism. There’s a rectangular face on the base of the prism. There’s another rectangular face which, on this prism, forms the vertical face. And there’s then a third rectangular face which forms the sloping face of the prism. In our diagram, that’s the face that’s sort of hidden from view at the back.
Let’s consider each of these areas. Faces one and two, first of all, then which are triangles, we can use the formula area equals a half multiplied by base multiplied by perpendicular height. From the diagram we can see that the base of our triangles is three units. And the perpendicular height is four units. We know this is the perpendicular height because a right angle has been marked between these two sides on each face. So, for each of these faces, the area is one-half multiplied by three multiplied by four.
We can cancel a factor of two from the two in the denominator and the four in the numerator, leaving one over one multiplied by three multiplied by two. This is just equivalent to three multiplied by two which is six. So the area of each triangular face is six square units. Remember though that there are two of them.
Next, let’s consider the third face which we said was the rectangular face on the base of this prism. The area of a rectangle can be found by multiplying its length by its width. From the diagram, we see that the length of the base is 10 units and the width is three units. The area is therefore 10 multiplied by three which is 30 square units.
Next, let’s consider the fourth face which is the vertical rectangular face. As this is a rectangle, we can again use the formula length multiplied by width to find its area. But the dimensions of this rectangle are different from the one we’ve just calculated. The length of this rectangle is still 10 units. But its width is not three units. It’s four units, the vertical height of this prism. So the area of this face is 10 multiplied by four which is 40 square units.
Finally, we come to the fifth face which is the rectangular sloping face at the back of this prism, which is sort of hidden from our view. As this is a rectangle, we can again use the formula length multiplied by width to find its area.
But what are the dimensions of this rectangle? Well, the length of this rectangle is still 10 units. But the width is five units. That’s the sloping height of this prism. So the area of this face is 10 multiplied by five which is 50 square units. To find the total surface area of this prism then, we add together the five areas we have found for the five faces.
Remember there are two triangular faces. So we need to include that value of six twice. We have six plus six plus 30 plus 40 plus 50. We can add the values in pairs or triples. Six plus six is 12. And 30 plus 40 plus 50 is 120. Then 12 plus 120 is 132. We weren’t given any units for the measurements in the original diagram. So the units for this area will just be generic square units. We’ve worked out that the total surface area of this triangular prism is 132 square units.