Lesson Video: Surface Areas of Spheres Mathematics • 8th Grade

Video Transcript

In this video, we will learn how to calculate the surface area of a sphere using either its radius or its diameter. We’ll first introduce the standard formula for doing so and then see how we can apply this to some examples. We’ll also see how we can work backwards from knowing the surface area of a sphere to determining its radius or diameter, which will require some skill in solving equations.

First, we recall that a sphere is a three-dimensional shape. Its size is completely determined by one measurement, its radius, which is the distance from the center of the sphere to any point on its surface. The formula we need for calculating the surface area of any sphere is this, four 𝜋𝑟 squared. And we must remember that it is only the radius that is being squared, not the factor of four or the factor of 𝜋. Remember that, in two dimensions, the formula for finding the area of a circle is 𝜋𝑟 squared. So the formula for calculating the surface area of a sphere differs only in that there is an extra factor of four.

In some cases, we may be given the diameter rather than the radius of the sphere. That’s the distance between two opposite points on the sphere’s surface, passing through the center of the sphere. In this case, we must make sure that we calculate the radius of the sphere before attempting to find its surface area, which we can do by recalling that the diameter is twice the radius or, equivalently, the radius is half the diameter. Now that we’ve identified the key formulae we need, let’s have a look at some examples.

Find the surface area of the given sphere to the nearest tenth.

Firstly, we recall the formula we need for finding the surface area of a sphere. It’s four 𝜋𝑟 squared, where 𝑟 represents the sphere’s radius. That’s the distance between the center of the sphere and any point on the sphere’s surface. From the figure, we can see that the radius of this sphere is six centimeters. So we can substitute this value of 𝑟 directly into our formula, giving four 𝜋 multiplied by six squared. And we must remember that it is only the radius that we need to square. Now, we can simplify six squared is 36 and four multiplied by 36 is 144. So the exact surface area of this sphere is 144𝜋.

However, the question asks us to give this answer to the nearest tenth. So we can use a calculator to evaluate this as a decimal. And it gives 452.3893. If we’re rounding to the nearest tenth, then our deciding digit is the eight in the hundredths column which tells us that we need to round up. So we have a value of 452.4. As the units for the radius were centimeters, the units for the surface area of the sphere will be square centimeters. So we have our answer to the problem. The surface area of this sphere to the nearest tenth is 452.4 square centimeters.

In this question, we saw how to find the surface area of a sphere, given its radius. In our next example, we’ll see how to calculate the surface area of a sphere when the measurement we’ve been given is its diameter.

Find the surface area of a sphere whose diameter is 12.6 centimeters. Use 𝜋 equals 22 over seven.

The formula we need for calculating the surface area of a sphere is this, four 𝜋𝑟 squared, where 𝑟 is the radius of the sphere. In this question though, we haven’t been given the radius; we’ve been given the length of the sphere’s diameter. That’s no great problem, though, because we know the relationship that exists between the radius and diameter of the sphere. The radius is half the length of the diameter. So if the diameter is 12.6 centimeters, the radius is half of this. That’s 6.3 centimeters. So we can substitute this value for the radius directly into our formula for the surface area, giving four 𝜋 multiplied by 6.3 squared. And remember, it’s only the radius we’re squaring.

Now, the question actually asks us to use 22 over seven as an approximation for 𝜋. So this suggests we haven’t got access to a calculator in this question. Our surface area becomes four multiplied by 22 over seven multiplied by 6.3 squared. And let’s see how we could work this out without a calculator. First, we can write 6.3 squared as 6.3 multiplied by 6.3. Now, we should spot that seven is a factor of 63. So we can divide 6.3 by seven relatively easily. Using a short division or bus-stop method, firstly, there are no sevens in six, so we put a zero and carry the six. And then there are nine sevens in 63. So 6.3 divided by seven is 0.9. So our calculation becomes four multiplied by 22 multiplied by 0.9 multiplied by 6.3.

We can work this calculation out in pairs. Four multiplied by 22, first of all, is 88. And to work out 0.9 multiplied by 6.3, we can first work out 63 times nine, which is 567, and then recall that we need to divide by 10 twice in order to give the answer to the decimal calculation. So dividing 567 by 100 gives 5.67. This also makes sense from an estimation point of view. We’re multiplying 6.3 by something a little less than one, so the answer we get should be a little less than 6.3, and 5.67 is reasonable.

Finally, we can work out 5.67 multiplied by 88 by first working out 567 multiplied by 88 which is 49896 and then dividing this value by 100, which gives 498.96. As the units for the diameter were centimeters, the units for the surface area will be square centimeters. And so we have our answer to the problem. Using 22 over seven as an approximation for 𝜋, we found that the surface area of the sphere whose diameter is 12.6 centimeters is 498.96 square centimeters. Remember, the key point in this question was that we needed to calculate the radius of the sphere first before we could calculate its surface area.

In our next example, we’ll see how we can work backwards from knowing the surface area of a sphere to calculating its radius or diameter.

What is the diameter of a sphere whose surface area is 36𝜋 square centimeters?

In this question, we’ve been given the surface area of a sphere and asked to use this to determine its diameter. We recall that the general formula for finding the surface area of a sphere is four 𝜋𝑟 squared. So by equating these two pieces of information, we can form an equation that will enable us to determine firstly the radius of the sphere. We have the equation four 𝜋𝑟 squared equals 36𝜋. And we can now solve this equation. Firstly, we can cancel a factor of 𝜋 on each side. We can then divide each side of the equation by four to leave 𝑟 squared on the left-hand side and nine on the right-hand side. So we now have the equation 𝑟 squared is equal to nine.

We solve this equation by square rooting. And we’re only going to take the positive value here because 𝑟 has a physical meaning as the radius of the sphere. Nine is a square number, and its square root is three. So we found that the radius of the sphere is three centimeters. We must be careful, though, because it wasn’t the radius of the sphere that we were originally asked to find. It was the diameter. But that’s no problem because we know that the diameter of a sphere is twice its radius. So if the radius is three, the diameter will be six. We’ve solved the problem, and the diameter of the sphere whose surface area is 36𝜋 square centimeters is six centimeters.

Let’s now take a bit of time to consider hemispheres, which is simply half a sphere. We can therefore adapt the formula for the surface area of a sphere to finding a formula for the surface area of a hemisphere. But we must be careful as the hemisphere has an extra surface. In addition to the curved or lateral surface area, which will be half the total surface area of the sphere, a hemisphere also has an additional flat circular base, which is called the great circle of the sphere. Formally, a great circle of a sphere is the intersection of the sphere and any plane ⁠— that’s a two-dimensional slice ⁠— that passes through the center of the sphere. But we can just think of it as any circle which divides the sphere up into two identical hemispheres.

The curved or lateral surface area of the hemisphere will be half the surface area of the full sphere. That’s half of four 𝜋𝑟 squared, which is two 𝜋𝑟 squared. And the area of the circular base is simply 𝜋𝑟 squared. Overall then, the total surface area of the hemisphere is three 𝜋𝑟 squared. We must be careful in questions, though, to make sure we’re clear whether we’ve been asked to find the total surface area or simply the curved or lateral surface area.

Find, to the nearest tenth, the curved surface area of a hemisphere, given that the area of the great circle is 441𝜋 square millimeters.

Let’s begin by sketching this hemisphere out. Remember that the great circle of a sphere is the circle which divides the sphere up into two hemispheres. So it is the flat circular base of our hemisphere. It’s this face here. We’re told that the area of this great circle is 441𝜋 square millimeters. And we also know that the general formula for calculating the area of a circle is 𝜋𝑟 squared. So we can use these two pieces of information to form an equation. 𝜋𝑟 squared equals 441𝜋. We can solve this equation to find the radius of our hemisphere. First, we can divide through by 𝜋 to give 𝑟 squared equals 441. We then take the square root of each side of the equation. And as 441 is a square number, its square root is 21. So we have that the radius of this hemisphere is 21.

We then recall that the curved surface area of a hemisphere is half the surface area of a full sphere. It’s two 𝜋𝑟 squared. So we can substitute our value for the radius into this formula. That gives two 𝜋 multiplied by 21 squared. We know from our earlier working that 𝑟 squared or 21 squared is 441. So we have two 𝜋 multiplied by 441 which is 882𝜋. The question asks us to give this value to the nearest tenth. So we can evaluate 882𝜋 on a calculator, and it gives 2770.884. The deciding digit in this case is the eight in the hundredths column. So we’re rounding up, which gives 2770.9, and the units for this will be square millimeters.

Now you may’ve spotted that, actually, we didn’t need to calculate the radius of this hemisphere at all. If the area of the great circle is 𝜋𝑟 squared and the curved surface area of the hemisphere is two 𝜋𝑟 squared, then we could have simply doubled the area we were given for the great circle. That would give two multiplied by 441𝜋, which is 882𝜋. And so we’d have arrived at the same stage of calculation as we had here in our previous method. Both of these methods would be equally fine and give the same answer of 2770.9 square millimeters.

Let’s now consider one final example with a slightly greater emphasis on problem solving.

A water feature can be modeled as a hemisphere with its base set onto a square patio. If the diameter of the hemisphere is four feet and the patio has a side of length six feet, what would the visible area of the patio be? Give your answer accurate to two decimal places.

Let’s begin with a sketch of this water feature. It is a hemisphere, so that’s half a sphere, with a diameter of four feet. And it’s sitting on a square patio, which has a side length of six feet. The visible area of the patio will be all of the patio’s area which isn’t covered by the circular base of this hemisphere. It’s the area of the square minus the area of the circle, which is in fact the great circle of this hemisphere. We know how to find the areas of each of these two-dimensional shapes. The area of a square is simply a side length squared; that’s six squared. And the area of a circle is 𝜋𝑟 squared.

Now, we need to be a little careful here because we were given the diameter of the hemisphere, which is the diameter of this circle. It’s four feet. So the radius is half of this; it is two feet. We have then six squared minus 𝜋 multiplied by two squared. That simplifies to 36 minus four 𝜋. And we could leave our answer in this form if we wanted. But this question actually asked us to give our answer accurate to two decimal places. Evaluating on a calculator gives 23.43362, and then rounding to the required two decimal places gives 23.43. As the units for the length in this question were given in feet, the units for area will be square feet. And so we have our answer to the problem. The visible area of the patio to two decimal places is 23.43 square feet.

Let’s now summarize some of the key points that we’ve seen in this video. Firstly, the surface area of a sphere can be found using the formula four 𝜋𝑟 squared, where 𝑟 is the radius of the sphere. We must make sure we check carefully in any problem whether we’ve been given the radius or the diameter of a sphere. We could use the equivalent formula four 𝜋 multiplied by 𝑑 over two squared, or we could simply halve the diameter to find the radius before substituting into the formula.

We also saw that we can work backwards from knowing the surface area of a sphere to calculate its radius or diameter by forming and solving an equation. We then saw that the great circle of a sphere, which formally is the intersection of the sphere with any plane that passes through its center, divides the sphere up into two hemispheres. A sphere has infinitely many great circles, depending on the angle of the plane we draw. And finally, for a hemisphere, we saw that its lateral or curved surface area is given by two 𝜋𝑟 squared and its total surface area, which includes its circular base, is given by three 𝜋𝑟 squared.