Question Video: Finding the Slant Height of a Cone given Its Total Surface Area and Its Base Radius Mathematics • Second Year of Secondary School

Video Transcript

The surface area of a cone is 364𝜋 square centimeters, and the radius of the base is 13 centimeters. Determine the slant height of the cone.

In this problem, we’re given the radius and the surface area of a cone and asked to determine its slant height. Remember, this is the distance from the apex of the cone to any point on the circumference of its circular base. We can recall that the formula for calculating the total surface area of a cone, which we can assume we have here as the question hasn’t specified otherwise, is 𝜋𝑟 squared plus 𝜋𝑟𝑙, where 𝑟 is the radius of the cone and 𝑙 is its slant height.

Now, the first part of this formula, 𝜋𝑟 squared, gives the area of the circular base of the cone. And the second part of the formula, 𝜋𝑟𝑙, gives the lateral or curved surface area of the cone. We know the total surface area of the cone. It’s 364𝜋 square centimeters. And we know the radius of the base, so we can substitute these values to form an equation. We have 364𝜋 is equal to 𝜋 multiplied by 13 squared plus 𝜋 multiplied by 13 multiplied by 𝑙.

We can now solve this equation to determine the value of 𝑙, the slant height of the cone. We can evaluate 13 squared, which is 169, and at the same time cancel a factor of 𝜋 from every term. So the equation simplifies to 364 is equal to 169 plus 13𝑙. Next, we can subtract 169 from each side, giving 195 is equal to 13𝑙. Finally, we can divide both sides of the equation by 13 to give 15 is equal to 𝑙, or 𝑙 is equal to 15. Remember, this represents a length, so its units will be the same as the units used for the length of the radius, which are centimeters.

The slant height of the cone is 15 centimeters.