### Video Transcript

A regular pentagonal pyramid has
base side length 41 centimeters and height 71 centimeters. Compute, to one decimal place, the
volume of the pyramid.

Let’s start by sketching the
pyramid with the given dimensions. So, here, we have the pyramid
drawn. Because we’re told that it’s a
pentagonal pyramid, that means the base will have five sides. And because we’re told that it’s a
regular pyramid, we know that all the lengths on the base will be the same at 41
centimeters. We’re given that the height is 71
centimeters and that means that it’s the perpendicular height. Remember that the slant height
would be the height of one of the triangles that makes up the lateral sides.

We can remember that to calculate
the volume of a pyramid, we work out one-third times the area of the base times the
perpendicular height. We’re not given the area of the
base of this pyramid, but we can calculate it using the information that this is a
regular pentagonal pyramid. And so, the base of this pyramid
will be a five-sided polygon with all the sides the same length. The area of a regular 𝑛-sided
polygon of side length 𝑥 is given by the area of a polygon is equal to 𝑛𝑥 squared
over four times the cot of 180 degrees over 𝑛. Therefore, substituting in the
values that the number of sides 𝑛 is equal to five and the side length 𝑥 is 41
gives us that the area of the polygon is equal to five times 41 squared over four
times the cot of 180 degrees over five. This simplifies to 8405 over four
times the cot of 36 degrees.

Since multiplying by the cotangent
is equal to dividing by the tangent, we can write this as 8405 divided by four times
the tan of 36 degrees. Using a calculator, we can find the
decimal equivalent as 2892.122 and so on square centimeters. Because we haven’t quite finished
with this value, we’re not going to round it yet. And when we use it in the next part
of the calculation, we can keep the long decimal or alternatively use this
fractional value in the step before.

Now we can work out the volume of
the pyramid, remembering that the area of the base is the area of the polygon that
we’ve just calculated. So, we can write that the volume of
the pyramid is equal to one-third times 2892.122 and so on multiplied by the
perpendicular height, which is 71. This gives us a value of 68446.899
and so on. And because it’s a volume, we’ll be
working in cubic units, which means that the units will be cubic centimeters. We’re asked to round our answer to
one decimal place, so we can give the volume of the pyramid here as 68446.9 cubic
centimeters.