Surface Area of Prisms, Cubes, and Boxes 05:42 minutes

Video Transcript

Transcript Surface Area of Prisms, Cubes, and Boxes

Sam Torpintino is an aspiring movie director who is looking for his big break. He’s pitching his latest idea to a big time studio executive: Attack of the Really, Really Big Slugs from Outer Space, but the studio executive needs a bit of convincing. In Torpintino’s new movie, a famous scientist, Navy SEAL, and men’s fashion model is elected President, but on his first day in office, the world is attacked by an onslaught of really, really big space slugs. In order to save the city, he hatches a plan to cover all the buildings in a thin layer of salt, which will protect the city from the slugs’ slime. However, to get the math just right, the director needs our help with calculating the surface area of prisms, cubes and boxes. The calculations for the surface area of cubes and boxes are not so different from each other. Each of these shapes has 6 total sides. There’s a top and bottom, a right and left face, and a front and back face. Surface area is just the sum of the areas of each of the faces. The area of each face is calculated by multiplying the two dimensions that make up the face, length times height, height times width, or width times length. If we just look at this face of the building, the length is 85 feet and the height is 190 feet. 85 times 190 is 16,150 square feet. But this was just the front face, so we need to multiply it by two to get the back face as well. This gives us 32,300 square feet. By rotating the box a bit, we see that there are two more areas that need to be calculated. Multiplying a height of 190 feet times a width of 50 feet, we get 9,500 square feet. Again, we multiply this result by two to get 19,000 square feet, since there is another face that's the same size. The only thing that’s left is the top and bottom. This time, we multiply the width of 50 feet times the length, 85 feet, giving us 4,250 square feet. Again, we multiply this result by two, providing us with an additional 8,500 square feet. By summing all the areas, we get the total surface area of the building. But, since we don't have to cover the bottom of the building in salt, we can subtract the surface area of the base, giving us 55,550 square feet. The process of finding surface area for a box is often represented by the formula: Surface area equals two times the length times width, plus two times the height times length, plus two times the width times height. But this can be written shorter by using the first letter of each dimension like this. Finding the surface area of a cube is very similar, and a bit easier. We can start with the formula we found for the surface area of a box. But since the length, width, and height of a cube are all the same, we can use the same value to calculate the area of each face. Let's just call that value "side". A cube has the same number of faces as a box, but each has the area of side times side, so we can simply multiply 6 times the length of a side squared, to obtain the total surface area. Not all of the buildings in the city, however, are shaped like boxes and cubes. There are a few more eccentric structures that have triangular shapes on the top and bottom, like this. Fortunately, calculating the surface area of these structures, known as prisms, is very similar to what we’ve already done. But in this case, we just have three sides and a top and bottom. So we can calculate the surface area by calculating 3 times the height times the triangle side length for the rectangular faces and adding two times the area of the triangular top and bottom, which, if you remember, is one-half the base of the triangle times the height of the triangle. Normally, we'd need to multiply the area of the triangle face by two to get the full surface area, but like before, we don't need to cover the base of the building in salt, so we leave that out of our expression. This weird-looking building is 100 feet tall and has a rectangular side length of 36.5 feet. The roof and floor of the building are equilateral triangles with a side length of 36.5 ft. The height of this equilateral triangle is 31.61 feet. So, after multiplying and combining like terms, if our hero collects 11,526.8825 square feet of salt for this building, he’ll be able to keep the slugs at bay. Let’s check back in with Sam Torpintino and see how the pitch is going. It looks like Sam Torpentino is showing off what he wants to do for the climactic scene of his movie, but the studio executive doesn’t look impressed. Yeah, but can you really blame him? Who would ever greenlight a movie all about surface area and giant slugs…??