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

Video Transcript

Find the total surface area of a quarter sphere of radius two centimeters. Round your answer to two decimal places.

In this example, we’re looking for the total surface area of a quarter sphere, where the radius is two centimeters. We know that the outer surface area of a whole sphere with radius 𝑟 is given by four 𝜋 times 𝑟 squared. The outer surface area of a quarter sphere, which we’ll call 𝐴 𝑄, is then one-quarter of this, which is 𝜋 times 𝑟 squared.

Now recalling that the area of a circle is also given by 𝜋 times 𝑟 squared, then the area of a semicircle must be half this. So that’s 𝜋𝑟 squared over two. And noting that the surface of a quarter sphere comprises its outer quarter spherical shell and two semicircles, where the outer shell and the two semicircles all have the same radius, we have the surface area of the quarter sphere, which we’ll call 𝑆 𝑄, is 𝜋 times 𝑟 squared, the outer shell, plus two times 𝜋𝑟 squared over two. That’s the two semicircles. This is 𝜋𝑟 squared plus 𝜋𝑟 squared, which is two times 𝜋 times 𝑟 squared.

In our case, the radius of all three elements is equal to two, so we have two times 𝜋 times two squared. And that’s equal to eight 𝜋. Eight 𝜋 evaluates to 25.1327 and so on. And we’re asked to round our answer to two decimal places. The radius is given in centimeters. So our units of area will be square centimeters.

Hence, the total surface area of the quarter sphere of radius two, to two decimal places, is 25.13 square centimeters.