Video Transcript
If a sphere is inscribed in a
cylinder and the surface area of the sphere is 16π centimeters squared, what is the
total surface area of the cylinder?
We begin by drawing out all the
relevant information weβve been given in the question. And the first thing to consider is
that the sphere is inscribed in the cylinder. What this means geometrically is
that the sphere is touching the top and base of the cylinder and that its
circumference is touching the central horizontal circular cross section of the
cylinder. This means that the sphere and the
cylinder have the same diameter and also that the height of the cylinder must equal
its diameter. And since the radius is half the
diameter, the radius must also equal half the height.
Our next bit of useful information
is that the surface area of the inscribed sphere is 16π centimeters squared. Now, we know that the surface area
of a sphere equals four ππ squared, where π is its radius. So, if we equate this with the
given surface area of the inscribed sphere, we can solve for its radius π. Leaving out the units for the
moment, this gives four ππ squared equals 16π. Now, dividing both sides by four π
gives π squared equal to four. And taking the positive square root
on both sides, since π is a length which is always positive, we find π equals
two.
So now remember, we want to find
the surface area of the cylinder. And recalling that the formula for
the surface area of a cylinder is two ππβ plus two ππ squared, with the radius
of our cylinder equal to the radius of the sphere, which is two centimeters and with
the radius equal to half the height, we can deduce that the height is equal to two
times the radius so that the height of our cylinder equals two times two, which is
four.
We need this and the value of π to
find the surface area of the cylinder using the formula. So now substituting π equals two
and β equals four into our formula for the surface area of a cylinder, we have two
π times two times four plus two π times two squared. Thatβs 16π plus eight π, which is
equal to 24π. Hence, the surface area of the
cylinder is 24π centimeters squared.
