Find the base, lateral area, surface area and volume of the figure below: http://www.flickr.com/photos/93084714@N07/8674300862/

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To find the area `B` of a hexagon:

`A=((3sqrt(3))/(2))*x^2`

Here, `x` represents the side length. Simply plug in the length of the side given, which is 5ft.

`A=((3sqrt(3))/(2))*5^2`

`B ~~ 64.95190528`

So now we know the base area.

To find the lateral area of the haxagonal pyramid, you must first find the height of each triangular side. The picture gives you a side length. You can use Pythagorean Theorem to solve for the midding height, Divide the base given by 2 in order to get the base length. So:

`3.5^2 + b^2 = 10^2`

`12.25 + b^2 = 100`

`b^2 = 87.25`

`sqrt(b) = sqrt(87.25)`

`b ~~ 9.301`

Alright, so now we know the height of each triangle on the hexagonal pyramid. Next, we can simply use the formula for area of a triangle and multiply it by 6, since there are 6 triangles on the pyramid.

`LatA=6((Bh)/(2))`

So plug in 5ft for the base of the triangle, and plug in the height we found earlier.

`LatA=6(((5)*(9.341))/(2))`

`LatA= 6(23.3525)`

`LatA=140.115ft^2`

So that is our lateral area.

After finding the lateral surface area, the total surface area is easy! We simply add on the base area that we found earlier.

`140.115ft^2+64.9519053ft^2=205.00669053`

`SurfA~~205.01`

So that is our surface area.

Finally, the volume. So, firstly, in order to find this, we need to find the apothem of the hexagon. To find the apothem, we use the formula:

`apothem=((s)/(2tan(180/n)))`

Here, `` is the length of any side, `` is the number of sides, and *tan *is the tangent function calculated in degrees.

So, we can plug in the numbers we have into this formula.

`apothem=((5)/(2tan(180/6)))` `->` `apothem~~4.33`

Alright, so now that we have our apothem, we need to find the height of the hexagonal pyramind. We have all the numbers we need to use Pythagorean Theorem. Here, the apothem is `a,`

the height given (10ft) is `c,` and the height is the unknown `b.`

`4.33^2+b^2=10^2`

`18.7489 + b^2 = 100`

`b^2=81.2511`

`sqrt(b)=sqrt(81.2511)`

`b~~9.014ft`

Alright. So we have the height of our hexagonal pyramid. We can use the formula:

`V=1/3(Bh)`

Now we can simply plug in the numbers we found earlier.

`V=1/3((64.951905528)(9.014))`

`V=1/3(585.4764764)`

`V=195.1588255ft^3`

So, we now have all of the required measurments. :) Good luck!

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