Find the surface area of a sphere to the nearest tenth if the area of the great circle is 441𝜋 inches squared.
The great circle is the circle who has the same center and the same radius as the sphere. And if we know the area of the great circle, we have enough information to find its radius. Since the area of the great circle is 441𝜋 inches squared, we can say 441𝜋 is equal to 𝜋 times the radius squared. From there, we divide both sides of our equation by 𝜋. And we find that the radius squared equals 441.
If we take the square root of both sides of this equation, we’re only interested in the positive square root of 441 since we’re talking about distance. And that is positive 21. And so we can say that the radius of our sphere is 21 inches, which means we have enough information to now calculate the surface area of our sphere, which is equal to four times 𝜋 times the radius squared. The surface area will be equal to four times 𝜋 times 21 squared. 21 squared is 441. Four times 441 is 1764. So we have a surface area of 1764𝜋.
If we wanna round to the nearest tenth, we’ll need to use a calculator or an approximation to multiply 1764 by 𝜋. When we do that, we get 5541.7694 continuing. To round to the nearest tenth, we’ll look to the digit to the right of the tenths place, which tells us that we should round up to 5541.8. And since we measured our radius in inches, the surface area, a measure of area, will be inches squared.
But before we leave this question, let’s observe something else. We know that our great circle and the surface area have to share the same radius. And that means the great circle has an area of 𝜋𝑟 squared. And the same 𝜋𝑟 squared is found in the surface area formula. The surface area of the sphere is actually equal to four times the area of the great circle. And that means if we started with a great circle area of 441𝜋 inches squared, the surface area of the sphere is going to be four times that 441𝜋.
If we have the area of the great circle and our only goal is finding the surface area, we don’t need to specifically know the radius. We can simply multiply the area of the great circle by four to find the surface area. The only thing to remember is that this only works with the great circle. It doesn’t work with any other circles in the sphere. This is because any smaller circles in our sphere will not have the same radius.
For this sphere, we’ve found that the surface area must be 5541.8 inches squared.