A right circular cone has base diameter 10 centimeters and height 12 centimeters. Determine the total surface area to the nearest tenth.
First let’s draw a sketch of this cone so we can visualize it more clearly. We’re told that the cone has a base diameter of 10 centimeters and a height of 12 centimeters. We’re asked to calculate the total surface area of the cone. The total surface area of a cone is the area of its base plus the lateral surface area. This is 𝜋𝑟 squared plus 𝜋𝑟𝑙, where 𝑟 is the radius of the base of the cone and 𝑙 is the slant height.
We know the radius of the base of the cone. If the diameter is 10 centimeters, then the radius is half of this; it’s five centimeters. So our calculation becomes 𝜋 multiplied by five squared, for the area of the base, plus 𝜋 multiplied by five multiplied by 𝑙, for the lateral surface area.
Now we haven’t been given the value of 𝑙 in the question. Remember, 𝑙 is the slant height of the cone. We’ve been given the perpendicular height, 12. It’s really important that you read questions like this carefully and determine whether you’ve been given the height or the slant height because they aren’t the same as each other.
Let’s think about how we can calculate the slant height from the information we have. The slant height, the vertical height, and the radius of the cone form a right-angled triangle. This means that we can apply the Pythagorean theorem to calculate the slant height as we know the other two sides of this right-angled triangle.
Applying the Pythagorean theorem tells us that 𝑙 squared is equal to five squared plus 12 squared. Evaluating each of these numbers gives us 𝑙 squared is equal to 25 plus 144. The sum of these two values is 169. Therefore 𝑙 is equal to the square root of 169, which is 13.
So now we know the slant height of the cone, 13 centimeters. You may have spotted this a little bit earlier as the values of five, 12, and 13 form a Pythagorean triple. At any case, we now know the value of 𝑙. Substituting into our calculation for the total surface area, we now have 𝜋 multiplied by five squared plus 𝜋 multiplied by five multiplied by 13.
Evaluating each of the constants gives 25𝜋 plus 65𝜋. This gives a total of 90𝜋. Now we we’re asked for our answer not as a multiple of 𝜋, but as a decimal to the nearest tenth. So we need to evaluate this. As a decimal, this is 282.7433 and the decimal continues.
Remember we need to round it to the nearest tenth, and so we have our answer for the total surface area of the cone: 282.7 centimeters squared.