Question Video: Finding the Volume of a Regular Pentagonal Pyramid given Its Height and Its Base Length | Nagwa Question Video: Finding the Volume of a Regular Pentagonal Pyramid given Its Height and Its Base Length | Nagwa

# Question Video: Finding the Volume of a Regular Pentagonal Pyramid given Its Height and Its Base Length Mathematics • Second Year of Secondary School

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

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### 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.

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