Video: Finding the Curved Surface Area outside a Conical Lampshade

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

03:18

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 πœ‹π‘Ÿπ‘™. We multiply πœ‹ by the radius by the slant height 𝑙. Currently, we don’t know either of these values. We don’t know the slant height, and we don’t know the radius. The circumference of a circle can be calculated using the formula two πœ‹π‘Ÿ. We can use this to calculate the radius in this question.

145.2 is equal to two πœ‹π‘Ÿ. Dividing both sides of this equation by two πœ‹ gives us π‘Ÿ is equal to 145.2 divided by two πœ‹. This means that π‘Ÿ is equal to 23.1092 and so on. For accuracy, we’ll 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 Pythagoras’s theorem, which states that π‘Ž squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the longest side of a right triangle known as the hypotenuse.

Substituting in our values gives us 𝑙 squared is equal to 31 squared plus 23.1092 and so on squared. Typing this into our calculator gives us 𝑙 squared is equal to 1495.0396 and so on. Square-rooting both sides gives us 𝑙 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œ‹ 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.

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