A cone has only one base. The lateral surface area is a sector of a circle. It looks a little like a skirt or a curved triangle.

A cone has a circular base; the formula for finding area of the circle is A = πr².

The lateral surface area is a sector of a circle. The letter l is used for the slant height of the lateral surface of the cone as shown below.

The formula for the area of the skirt-like lateral surface area is A = πrl.

Answer the following questions.

• Describe how to find the total surface area.
  
  Interactive popup. Assistance may be required. Check Your Answer Find the sum of the lateral surface area and the base.
  
  
• The formula for the area of the circle is ___?___ .
  
  Interactive popup. Assistance may be required. Check Your Answer A = πr²
  
  
• The formula for the area of the circular sector is ____?_____.
  
  Interactive popup. Assistance may be required. Check Your Answer A = πrl
  
  

The total surface area of a cone is:

S = πrl + πr²

Pythagorean Theorem in the strangest and most unlikely of places…

The Pythagorean Theorem is often used to find the slant height of the cone. Recall that the Pythagorean Theorem is:

a² + b² = c²

Below, drag the labels from the Pythagorean Theorem, a, b, and c, to the appropriate place on the diagram of the cone. Either of these diagrams is correct.

How would you find the slant height, l, of this cone?

Locate the right triangle in the diagram. Locate the height, radius and slant height that forms the right triangle.

• The height and radius are the legs of the right triangle.
  
  
• Their values are ____?____ and ___?____.
  
  Interactive popup. Assistance may be required. Check Your Answer 12 cm and 5 cm
  
  
• The slant height, l, is the ____?___ of the right triangle.
  
  Interactive popup. Assistance may be required. Check Your Answer hypotenuse
  
• The length of the slant height l is ___?____.
  
  Interactive popup. Assistance may be required. Check Your Answer l² = 12² + 5² = 144 + 25 = 169; l = √169 = 13 cm

Watch this video explaining the derivation of the formula for the surface area of a cone, as well as examples of problems you can solve.