The solid cone in the diagram has a volume of 12𝜋 centimetres cubed. The height of the cone is four centimetres. Using the following equations, find the surface area of the cone in terms of 𝜋. The volume of a cone is equal to one-third 𝜋𝑟 squared ℎ. The surface area of a cone is equal to 𝜋𝑟 multiplied by 𝑟 plus 𝑙, where 𝑙 is the slant height of the cone.
So we’ve been told the volume and the height of this cone and asked to calculate its surface area. We’ve been given the formula. But in order to use it, we need to know the radius of the cone and the slant height.
Let’s think about how to calculate the radius first of all. We’re told that the volume of the cone is 12𝜋 centimetres cubed. And we’re given the formula for this, which means that one-third multiplied by 𝜋 multiplied by 𝑟 squared multiplied by ℎ is equal to 12𝜋. The height of this cone is four, so we can substitute this value of ℎ.
And now we have an equation that we can solve in order to find the value of 𝑟. Both sides of the equation have a factor of 𝜋, which we can therefore cancel out. We can then multiply both sides of the equation by three and divide by four, to give 𝑟 squared is equal to 12 multiplied by three over four. 12 multiplied by three is equal to 36, and 36 divided by four is equal to nine. So 𝑟 squared is equal to nine. 𝑟 is, therefore, equal to the positive square root of nine, as 𝑟 is the radius of the cone. So 𝑟 is three.
We’ve used the information given about the volume of the cone to find its radius. And now we need to consider how to find the value of 𝑙, the slant height of the cone. The slant height is this height that I’ve marked in pink. Note that it’s different from the perpendicular height of the cone, which is four centimetres.
The radius of a cone, the perpendicular height, and the slant height form a right-angled triangle. And we can, therefore, calculate any of these lengths if we know the other two, as we can apply Pythagoras’s theorem.
Pythagoras’s theorem tells us that, in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side, the hypotenuse. If I label the two shorter sides as 𝑎 and 𝑏 and the hypotenuse as 𝑐, we have the equation 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared. Remember, the longest side, the hypotenuse, is always the side directly opposite the right angle.
Substituting the values in this question, we have that three squared plus four squared is equal to 𝑙 squared. Three squared is nine, and four squared is 16. And summing these values together, we have that 25 is equal to 𝑙 squared. 𝑙 is, therefore, equal to the square root of 25, which is five.
You may have been able to spot this without going through all of the working out, as three, four, five is a commonly known Pythagorean triple, that is, a right-angled triangle in which all three of the sides are integers.
We can now substitute the calculated values of 𝑟 and 𝑙 into the formula we’re given for the surface area of the cone. It’s equal to 𝜋 multiplied by three multiplied by three plus five. Three plus five is eight, and eight multiplied by three is 24. So the surface area is 24𝜋.
We’ve been asked to give our answer in terms of 𝜋 because we don’t have access to a calculator to evaluate what 24𝜋 is equal to as a decimal. The units for surface area are square units, so in this question that’s centimetres squared. The surface area of the cone in terms of 𝜋 is 24𝜋 centimetres squared.