### Video Transcript

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 ππ squared β. The surface area of a cone is equal to ππ multiplied by π plus π, where π is the slant height of the cone.

So weβve been told the volume and the height of this cone and asked to calculate its surface area. Weβve been given the formula. But in order to use it, we need to know the radius of the cone and the slant height.

Letβs think about how to calculate the radius first of all. Weβre told that the volume of the cone is 12π centimetres cubed. And weβre given the formula for this, which means that one-third multiplied by π multiplied by π squared multiplied by β is equal to 12π. The height of this cone is four, so we can substitute this value of β.

And now we have an equation that we can solve in order to find the value of π. 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 π 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 π squared is equal to nine. π is, therefore, equal to the positive square root of nine, as π is the radius of the cone. So π is three.

Weβve 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 π, the slant height of the cone. The slant height is this height that Iβve marked in pink. Note that itβs 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 Pythagorasβs theorem.

Pythagorasβs 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 π and π and the hypotenuse as π, we have the equation π squared plus π squared is equal to π 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 π squared. Three squared is nine, and four squared is 16. And summing these values together, we have that 25 is equal to π squared. π 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 π and π into the formula weβre given for the surface area of the cone. Itβs 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.

Weβve been asked to give our answer in terms of π because we donβt 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 thatβs centimetres squared. The surface area of the cone in terms of π is 24π centimetres squared.