Video: Finding the Total Surface Area of a Cone given Its Slant Height and Base Radius

Determine, to the nearest hundredth, the total surface area of the cone shown.

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Video Transcript

Determine to the nearest hundredth the total surface area of the cone shown.

First, let’s recall the formula for calculating the surface area of a cone. It is πœ‹π‘Ÿ squared plus πœ‹π‘Ÿπ‘™, where π‘Ÿ represents the radius of the cone and 𝑙 represents the slant height. The first term πœ‹π‘Ÿ squared gives the area of the circle on the base of the cone, the second term πœ‹π‘Ÿπ‘™ gives the curved surface area of the cone, and together these give the total surface area.

We’ve been given the values of π‘Ÿ and 𝑙 in the diagram: π‘Ÿ is 19 centimetres and 𝑙 is 40 centimetres. Remember 𝑙 represents the slant height of the cone, not the perpendicular height, and it is the slant height that we’ve been given. So let’s substitute the values of π‘Ÿ and 𝑙 into our formula for the total surface area.

We have that the surface area is equal to πœ‹ multiplied by 19 squared plus πœ‹ multiplied by 19 multiplied by 40. Evaluating each of these constants gives 361 πœ‹ plus 760 πœ‹. Combining these two terms gives a total surface area of 1121 πœ‹. The question remember has asked for the surface area to the nearest hundredth, so we need to evaluate this as a decimal and then round. The decimal is 3521.7253 and rounding this to the nearest hundredth will give us our answer: 3521.73, and the units of this surface area are centimetres squared.

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