# 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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steveschoen | Certified Educator

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

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