Find, to the nearest tenth, the curved surface area of a hemisphere, given that the circumference of its great circle is four 𝜋 centimeters.
So in this question, what we’re looking at is a hemisphere, and we want to find its curved surface area. Well, if we want to find the curved surface area of a hemisphere, we can actually work this out from the surface area of a sphere. Well, the surface of a sphere is equal to four 𝜋𝑟 squared. Well, we also know then from this that the curved surface area of a hemisphere must be equal to two 𝜋𝑟 squared, half of this. But how are we gonna work this out because we don’t have the radius?
But we are told that the circumference of the great circle of the hemisphere is four 𝜋 centimeters. And the great circle is in fact the circle at the top or bottom of our hemisphere, because it’s the circle that’s formed if you cut the sphere in half. Well, we know that the circumference is equal to two 𝜋𝑟 or 𝜋𝑑. We’re gonna write it as two 𝜋𝑟 because what we’re looking for is the radius. So therefore, what we can say is that four 𝜋, because that’s the circumference of the great circle, is equal to two 𝜋𝑟.
Well, therefore, if we divide through by two 𝜋, what we get is that four 𝜋 over two 𝜋 is equal to 𝑟, our radius. Well, 𝜋’s cancel. And then we’re gonna divide the numerator and denominator by two. So we just get two over one. So therefore, we can say that the radius is equal to two centimeters.
So great, what can do now is use this to work out what the curved surface area of our hemisphere is. So what we’re gonna get is a surface area is equal to two multiplied by 𝜋 multiplied by two squared. So the surface area is gonna be equal to eight 𝜋. That’s because we’ve got two multiplied by two squared, which gives us eight, so eight 𝜋.
Well, this would be the best way to leave our answer if we wanted to give an exact value. However, the question asks us to find to the nearest tenth the curved surface area of the hemisphere. So therefore, what we’re gonna do is calculate this. We’re gonna multiply eight by 𝜋. So the surface are is gonna be equal to 25.1327 et cetera. But because we only want it to the nearest tenth, we’re gonna round it. So therefore, because the number which is the deciding number, which is the decimal after the tenth, is a three, we’re gonna round down. So we can say that, to the nearest tenth, the curved surface area of the hemisphere is 25.1 centimeters squared.