Question Video: Finding the Lateral Surface Area of a Cone given Its Height and Its Base Radius Mathematics • 8th Grade

Video Transcript

Find, in terms of 𝜋, the lateral area of a right cone with base radius nine centimetres and height 13 centimetres.

Let’s begin by drawing a diagram of the cone. We’re told that the base radius is equal to nine centimetres. The height of the cone, which goes from the apex at the top to the centre or centroid of the base, is 13 centimetres. This creates a right-angled triangle with a slant height 𝑙. The lateral area of a cone is the area of its curved surface. This is equal to 𝜋𝑟𝑙. We multiplied 𝜋 by the radius by the slant height. We know that the radius of the cone is nine centimetres. However, we don’t know the slant height at present. We can, however, calculate this by using Pythagoras’s theorem. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the hypotenuse in a right triangle.

In this question, 𝑙 squared is equal to nine squared plus 13 squared. Nine squared is equal to 81. 13 squared is equal to 169. 81 plus 169 is equal to 250. Therefore, 𝑙 squared equals 250. Square-rooting both sides of this equation gives us 𝑙 is equal to root 250. Root 250 is equal to root 25 multiplied by root 10. As root 25 is equal to five, this is equal to five root 10. The slant height of the cone is five root 10 centimetres.

We can now substitute in this value to calculate the lateral area. The lateral area is equal to 𝜋 multiplied by nine multiplied by five root 10. Nine multiplied by five root 10 is 45 root 10. As we’re asked to give our answer in terms of 𝜋, this is equal to 45 root 10𝜋. The lateral area of a right cone with base radius nine centimetres and height 13 centimetres is 45 root 10𝜋 square centimetres. Remember that our units for any area or surface area are square centimetres, square metres, et cetera.