Question Video: Finding the Surface Area of a Sphere Given Its Diameter | Nagwa Question Video: Finding the Surface Area of a Sphere Given Its Diameter | Nagwa

Question Video: Finding the Surface Area of a Sphere Given Its Diameter Mathematics • Second Year of Preparatory School

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Find the surface area of a sphere whose diameter is 12.6 cm. Use 𝜋 = 22/7.

03:23

Video Transcript

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.

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