Video: Finding the Total Surface Area of a Right Cone

Find the total surface area of the right cone approximated to the nearest two decimal places.

03:05

Video Transcript

Find the total surface area of the right cone approximated to the nearest two decimal places.

We’re told on the diagram that the height of the cone is 14.5 centimetres. And its slant height is 16.5 centimetres. The radius is currently unknown. We can calculate the length of the radius by using Pythagoras’s theorem. This 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 π‘Ÿ squared plus 14.5 squared is equal to 16.5 squared. Subtracting 14.5 squared from both sides gives us π‘Ÿ squared is equal to 16.5 squared minus 14.5 squared. Square-rooting both sides of this equation gives us π‘Ÿ is equal to 16.5 squared minus 14.5 squared. This is equal to the square root of 62.

For accuracy, we will leave our answer in this form at present. We were asked to calculate the total surface area of the cone. A cone has two surfaces, a curved surface and a base. Therefore, the total surface area is equal to the area of the curved surface plus the area of the base. The area of the curved or lateral surface is equal to πœ‹π‘Ÿπ‘™. We multiply πœ‹ by the radius by the slant height. As the base is a circle, we work out the area of the base by multiplying πœ‹ by the radius squared. Substituting in our values for the radius and slant height gives us πœ‹ multiplied by the square root of 62 multiplied by 16.5 plus πœ‹ multiplied by the square root of 62 squared.

The square root of 62 squared is just equal to 62. As we need to calculate this to two decimal places and not in terms of πœ‹, we can type this calculation into our calculator. This gives us an answer of 602.93801 and so on. The eight in the thousandths column is the deciding number. When this digit is greater than or equal to five, we round up. The total surface area of the cone to two decimal places is 602.94 square centimetres. Any surface area will be measured in square units.

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