# Find the base, lateral area, surface area and volume of the figure below: http://www.flickr.com/photos/93084714@N07/8673220753/

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Hi, user4481582 . I hope this will assist.

I assume you are looking for the base area when you say "base". The base is a square. The area of a square is s^2, s is the length of the side. Since s=3 ft, the area of the base is:

**A = s^2 = 3^2 = 9 square feet**

For the lateral area, we need to find the lateral area of the cube and the lateral area of the pyramid on top. For the cube, there are a variety of ways. I like to consider is as finding the total areas of the vertical faces. There are 4 faces, all the same shape, a square, exactly like the bottom. So, the total lateral area of the cube is:

**LA = 4*9 = 36 square feet**

For the pyramid, there are 4 faces, all the same shape, a triangle. So, we can find the area of one triangle, then multiply it by 4. The formula for the area of a triangle is b*h/2, b is the length of the base, h is the height of the triangle. For ours, b = 3 and h = 5. So, the area of one triangle is:

A = b*h/2 = 3*5/2 = 7.5 square feet

For all sides:

**7.5*4 = 30 square feet.**

For the total lateral area, we add these two lateral areas:

**30+36 = 66 square feet**

Similarly, for the volume, we would find the volume of the cube and the pyramid separately, then add them together. For the cube, the formula for the volume is s^3. For ours:

V = s^3 = 3^3 = 27 cubic feet.

For the pyramid, the volume is 1/3 * B * h, B is the area of the base, h is the height. B for us is the area of one of the faces of the cube; we found that to be 9 square feet. We need to calculate the height. To do that, we can solve a right triangle, as shown by the attachment. We are solving for the red line in the figure on the attachment. Using Pythagorean Theorem:

a^2 + b^2 = c^2

1.5^2 + h^2 = 5^2

2.25 + h^2 = 25

h^2 = 22.75

h = 4.77 feet, rounded to 2 decimal places

Using this for our volume formula for the pyramid:

V = (1/3)*B*h

V = (1/3)*9*4.77

V = 14.31 cubic feet

So, for the total volume:

27+14.31 = 41.31 cubic feet.

Good luck, user4481582. I hope this helps.

Steve