A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #7 # and #5 # and the pyramid's height is #4 #. If one of the base's corners has an angle of #pi/3#, what is the pyramid's surface area?
1 Answer
Explanation:
Let's first try and visualize the pyramid. It would look something like this from an angled top-down perspective:
Let's start with the area of the parallelogram (the base). It looks like this: To be able to solve for the area of this parallelogram, we need to solve for it's height,
If we solve for h, we get that it is equal to:
The area of a parallelogram is equal to
Now that we have the base finished, let's have a look at the triangles in the upper part of the pyramid. We will start with the ones marked in blue:
To do this, we will need to solve for the height of the triangle,
The bottom left corner is the center of the base parallelogram in the pyramid, so the side with length
Using the pythagorean theorem,
Solving for
Now we can calculate the area of one of the blue triangles. It will be equal to the height of the blue triangle multiplied by
Let's then multiply this by two to get the area of both of the blue triangles:
Now, we can go through a similar process to solve the red triangles. This time we will consider a triangle like this:
Here,
We can once again just multiply this
Then we need to multiply by two because we have
Now all we need to to is add all of the areas together: