Question Video: Finding the Total Surface Area of a Hemisphere given Its Radius Mathematics • 8th Grade

Video Transcript

Find the total surface area of a hemisphere. Round the answer to the nearest tenth.

First of all, let’s recall the formula for finding the surface area of a sphere. It’s four 𝜋𝑟 squared, where 𝑟 is the radius of the sphere. Now this question isn’t about a sphere, it’s about a hemisphere or half a sphere. So we need to think of it, how we can adapt the formula for the sphere to help us find the surface area of the hemisphere. As the hemisphere is half of the sphere, its curved surface area will be half of that of the sphere. So half of four 𝜋𝑟 squared is two 𝜋𝑟 squared. But by cutting the sphere in half to form a hemisphere, we’ve also created an extra face, a circular face on the base of the hemisphere. We need to make sure we include the surface area for this face as well. As this face is a circle, it will have an area of 𝜋𝑟 squared. So our formula for the surface area of the hemisphere is two 𝜋𝑟 squared, for the curved surface area, plus 𝜋𝑟 squared, for the circle on the base. This of course simplifies to three 𝜋𝑟 squared.

Now we know the radius of this hemisphere; it’s given in the diagram, 18 centimeters. Therefore, our calculation for the surface area is three multiplied by 𝜋 multiplied by 18 squared. Evaluating this constant gives 972 𝜋. The question has asked for the answer to the nearest tenth, so we need to evaluate this as a decimal. This value is 3053.6280, and the decimal continues.

Finally, we need to round the answer to the nearest tenth, as requested. So we have that the total surface area of the hemisphere is 3053.6 and the units for this are centimeters squared.