How to Find the Area of an Obelisk
This video shows how to find the area of an obelisk. An obelisk has a square or rectangular body and a pyramid-shaped head. First find the area of the rectangular prism of the body, then the area of the pyramid shape at the top. Note that the pyramid shape does not have a bottom; it is attached to the rectangular prism by the sides.
Area of an Obelisk
A stone obelisk has the shape of a square prism with a pyramid on top. The height of the prism is 40 feet, the height of the pyramid is 4 feet, and the width of both is 6 feet. What is the total surface area of the obelisk, not counting the base? We’re going to split this into 2 parts.
First, I want to find the area of the Prism with the square base, and it said to find the surface area not counting the base, so that just means the very bottom that it’s sitting on. Just this square prism has 4 faces, and each one of the faces has the same dimensions.
Each face is 40 feet by 6 feet, (40 feet by 6 feet, 40 feet by 6 feet, 40 feet by 6 feet) so there are 4 faces that all have the same area. Since these are the faces around the sides, those are called the lateral faces. I’m going to say that I’m finding the lateral area of the prism (the lateral area of the prism).
Like I said, there are 4 faces that all have the same area, so 4 times (and then, since all of these faces are rectangles, the area of a rectangle is base times height, or length times width) 6 feet, times 40 feet (so 4 times 6 feet, times 40 feet).
Since we are multiplying we can multiply in any order, so I’m going to multiply 6 feet times 40 feet first. 6 times 4 is 24, add your 0, that’s 240 feet times 4, so 240, 240, 480. 480 and 480, 960 feet squared for the lateral area of the prism.
Now to find the area of the pyramid that is on top. Again, we won’t be finding the area of the base of the pyramid, since it’s really not one of the surfaces of the obelisk, it’s attached to the square prism on the bottom, so all we’re finding is the lateral area (again) of the pyramid.
That just means we need to find the area of all the triangles. There are 4 triangles, 4 congruent triangles, because each one of these triangles has the same dimensions. 6 feet by, well, and that’s the problem. To find the area of a triangle (the area of a triangle) is 1/2 base times height, but we don’t know the height of the triangles, we know the height of the pyramid.
We can use that height of the pyramid to find the height of our triangles. The height of a pyramid is perpendicular to the base, and it goes right to the center of the pyramid, so if we draw a line across to the side that’s perpendicular.
This segment right here goes halfway across our square base, and the length all the way across the square base is 6 feet, so halfway across the square base would be 3 feet (so this length is 3 feet). Then we need to find this perpendicular height, (right there) and this segment is the height that’s perpendicular to the base of the triangle, (which is 6 feet down here).
We need to find this perpendicular height of the triangle, so that we can find the area of the triangle. To do that you would just use Pythagorean theorem. A squared plus B squared equals C squared, were A and B are the legs of the triangle–and sometimes the picture gets a little busy, so what helps is if you’ll take this triangle and just pull it out of your picture (that’s your right angle, this is the halfway across your square part, so that’s 3 feet, the 4 feet is the height of the pyramid).
Then this line here, it’s called the slant height of a pyramid, and we use an L for that, but it’s also the height of the lateral face. A and B are the legs of your triangle, 3 feet squared plus 4 feet squared equals, and I’m going to use an L for slant height.
Then we simplify, 3 feet squared is 9 feet squared, plus 4 feet squared is 16 feet squared, is equal to slant height squared. 9 feet squared plus 16 feet squared is 25 feet squared, is equal to slant height squared. Then, to solve for slant height we need to get it alone, so we do the opposite of squaring, which is to square root both sides.
The square root of 25 feet squared is 5 feet, so 5 feet is the slant height. Now we know the height of each one of our lateral faces. Just like the prism on the bottom had 4 congruent rectangular faces, the pyramid on top has 4 congruent triangular faces.
If we find the area of 1 triangular face, then we just need to multiply that times 4. The area of the pyramid is there are 4 triangular faces that all have an area of 1/2 times the base (which is the 6 feet) times the height, (which is what we just found, 5 feet) and then we simplify.
4 times 1/2 is 2, 6 feet times 5 feet is 30 feet, squared. 30 feet squared times 2 is 60 feet squared. Now that we found the area of each part of the obelisk, we need to add those together to find the total area. The total area is 960 feet squared plus 60 feet squared, which gives us a total of 1,020 feet squared.