Find the volume of the following
regular pyramid rounded to the nearest hundredth.
As we’re told that this is a
regular pyramid, this means that the base of the pyramid is a regular polygon. And so, we know that the lengths of
all the sides on the triangle at the base will be 14 centimeters. To find the volume of a pyramid, we
calculate a third multiplied by the area of the base multiplied by the height.
Let’s begin by working out the area
of this triangle at the base of the pyramid. To find the area of a triangle, we
work out a half times the base times the height. If we take a closer look at this
triangle, we know that it will have all three sides of 14 centimeters. Which means that we know the base
length, but we don’t know the height of this triangle. As we have a right-angled triangle,
we could apply the Pythagorean theorem.
The Pythagorean theorem tells us
that the square of the hypotenuse is equal to the sum of the squares on the other
two sides. So, let’s use the Pythagorean
theorem to work out the height of our triangle. We can define the height here to be
the unknown value of 𝑥. So, the hypotenuse here, 𝑐
squared, will be 14 squared, which is equal to 𝑥 squared, plus seven squared since
seven is half of 14. And we’re using the smaller half of
the original base triangle.
Evaluating the squares then, we
have 196 equals 𝑥 squared plus 49. We then rearrange by subtracting 49
from both sides of the equation, giving us 147 equals 𝑥 squared. And we then take the square root of
both sides to give us the square root of 147 equals 𝑥. We’re going to keep our answer in
this square root form as we continue on through the question.
So, now, we’ve worked out that the
height of the triangle is the square root of 147. We can work out the area of this
triangle using the formula. So, our area is equal to half times
the base, which is 14, and multiplied by the height, which is the square root of
147. We can then simplify this
calculation to seven multiplied by the square root of 147. We could at this stage evaluate
this using a calculator, but as we still need to work out the volume, we can keep it
in this format of seven root 147.
Now we’ve worked out the area of
the triangle — that’s the area of the base of the pyramid — we can now go ahead and
work out the volume of the pyramid. So, our volume is equal to
one-third multiplied by the area of the base, which is seven root 147, multiplied by
the height of the pyramid, which is 17 centimeters. Using our calculator, we can
evaluate this as 480.93277 and so on. And rounding to the nearest
hundredth means that we check our third decimal digit to see if it is five or
more. And as it is not, then our answer
stays as 480.93. And our units here will be cubic