### Video Transcript

A conical lampshade is 31
centimetres high and has a base circumference of 145.2 centimetres. Find the curved surface area of the
outside of the lampshade. Give your answer to the nearest
square centimetre.

The lampshade is in the shape of a
cone with a height of 31 centimetres as shown. The circumference of the base is
equal to 145.2 centimetres. We have been asked to calculate the
curved surface area of the lampshade. The curved or lateral surface area
of a cone is equal to πππ. We multiply π by the radius by the
slant height π. Currently, we donβt know either of
these values. We donβt know the slant height, and
we donβt know the radius. The circumference of a circle can
be calculated using the formula two ππ. We can use this to calculate the
radius in this question.

145.2 is equal to two ππ. Dividing both sides of this
equation by two π gives us π is equal to 145.2 divided by two π. This means that π is equal to
23.1092 and so on. For accuracy, weβll not round this
answer at this point. As we know that the radius of the
cone is 23.10 and so on centimetres and that the height is 31 centimetres, we can
now calculate the slant height. We will do this using Pythagorasβs
theorem, which states that π squared plus π squared is equal to π squared, where
π is the longest side of a right triangle known as the hypotenuse.

Substituting in our values gives us
π squared is equal to 31 squared plus 23.1092 and so on squared. Typing this into our calculator
gives us π squared is equal to 1495.0396 and so on. Square-rooting both sides gives us
π is equal to 38.6657 and so on. We can now substitute the value of
the radius and the slant height into the formula for the curved surface area. We multiply π by the radius by the
slant height. Typing this into the calculator
gives us a curved surface area of 2807.132. We need to round this to the
nearest square centimetre, which means we need to round to the nearest whole
number. The curved surface area of the
lampshade is, therefore, equal to 2807 square centimetres.