Calculating the Surface Area of a Cone

The cone is a three-dimensional object, which extends from a flat surface and tapers as it rises upwards; ultimately ending in a pointed vertex. There can be various types of cones, depending on the kind of base they have.

This type has a circular base and the axis passing through the vertex is perpendicular to the base of the cone. Viewed sideways, it looks like two identical right angled triangles, placed with one of their sides (other than hypotenuse and base), connected back-to-back and their bases aligned in a straight line.

Calculating the surface area of such a cone is made easier by the fact that its base is perpendicular to the axis passing through its vertex. That is because, the Pythagoras theorem formula can be used in calculation.

The cone has a base, perpendicular height, and slant height. The slant height is the length of the inclined portion, that joins the vertex, with the base.

There are two different components that you need to calculate, while determining the surface area.

Total Surface Area of a Cone = (Area of Circular Base of Cone) + (Area of Curved or Lateral Surface Area of the Cone)

The base of a right circular cone is a circle. Therefore, its surface area is simply the area of a circle, which is πR², where R is the radius of the base.

The curved surface area of the cone is given by πRS, where R is radius of circular base and S is the slant height. Slant height can be calculated with the knowledge of the radius of circular base and the perpendicular height of the cone. That means, to calculate the slant height 'S', one uses the Pythagorean theorem, which gives us the formula:

S = √(R² + H²)

where 'H' is the surface area of the cone. 

Cone Surface Area = πR² + πRS = πR (R + S)

While calculating, if the slant height (S) is not given, then calculate it using the Pythagorean theorem relation, mentioned earlier.

It is useful to know the formula for volume too. The volume of a cone is given by the following formula:

V_Cone = 1/3 πR²H

where H is the perpendicular height and R is the radius of circular base. To calculate volume, all you have to do is plug in these known values. 

A truncated cone or frustum is the geometrical object created, when the top of the cone is cut off, leaving a flat parallel base at top. The surface area and volume of such an object is given by the following formulas:

Surface Area of Truncated Cone = π [(R₁² + R₂²) + √{(R₁² - R₂²)² + (H(R₁ + R₂))²}]

where

• R₁ = Top Base Radius
• R₂ = Bottom Base Radius
• H = Perpendicular Height of Frustum

Now, let us have a look at the volume of a truncated right circular cone:

Volume of Truncated Cone = πH/3 (R₁² + R₂² + R₁R₂)

where again

• R₁ = Top Base Radius
• R₂ = Bottom Base Radius
• H = Perpendicular Height of Frustum

With practice, these formulas will get safely stored in your permanent memory. A simple understanding of the geometry of the object and visualization helps in solving geometry problems. Of course, there is no substitute for practice.