Question Video: The Surface Area of Composite Solids Containing Spheres Mathematics • 7th Grade

Video Transcript

This shape is composed of a cylinder and a hemisphere. Find the surface area of the shape, leaving your answer in terms of 𝜋.

To find the surface area of a three-dimensional shape, we find the area of each face on the shape and add them together. Here, we’re going to find the surface area of the hemisphere, the area of the cylinder, and the area of the circle at the bottom.

So, let’s begin by finding the surface area of our hemisphere. And here, we’ll need to use the formula for the surface area of a sphere, which is four 𝜋𝑟 squared, where 𝑟 is the radius. Since we know that a hemisphere is half of a sphere, to find the formula for the surface area of a hemisphere, we take half of our value four 𝜋𝑟 squared, which will leave us with two 𝜋𝑟 squared. And since we’re asked in this question to leave our answer in terms of 𝜋, we don’t need to use a numerical value for 𝜋 in our calculations.

We know that the radius of our hemisphere will be five centimeters, since the cylinder has parallel sides, and the hemisphere sits perfectly on top of the cylinder. We can substitute our value five for the radius into two 𝜋𝑟 squared, which gives us two times 𝜋 times five squared. And since five squared is the same as five times five, we can calculate this as 25. And we can simplify two 𝜋 times 25 as 50𝜋. The area units here will be centimeters squared, since the length measurements are given in centimeters.

Moving on then to finding the surface area of our cylinder, if we imagine that we cut open a cylinder, the area of this would be a rectangle where the length of this is the circumference of the circle on which it sits. And the width would be the height of the cylinder. We can use the formula the surface area of the cylinder equals two 𝜋𝑟ℎ, where two 𝜋𝑟 is the circumference and ℎ is the height. In this case, that would be 12. Substituting in our values five for the radius and 12 for the height gives us two 𝜋 times five times 12. Multiplying the values two, five, and 12 gives us 120. So, our surface area is 120𝜋 centimeters squared.

And finally, the last area that we need to calculate is the area of the circle at the bottom of the shape. We use the formula the area of a circle equals 𝜋𝑟 squared. And since we have a radius of five, that will give us 𝜋 times five squared, which simplifies to 25𝜋 centimeters squared. So, to find the total surface area, we add together the areas of the three faces that we find, giving us 50𝜋 plus 120𝜋 plus 25𝜋, giving us final answer 195𝜋 centimeters squared.