Surface Area of a Cone

Geometry is one of the most interesting branch of mathematics that has a broad range of practical applications. Studying and taking interest in geometry will stand you in good stead if you are planning to pursue a career in engineering or science.

Three objects in solid geometry that are mostly studied are the cone, cylinder and the sphere. All three objects are distinctly different. 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 it has.

What is a Right Circular Cone?

This type of a cone has a circular base and the axis passing through the vertex is perpendicular to the base of the cone. Viewed sideways, a right circular cone 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 a right circular cone is made easier by the fact that its base is perpendicular to the axis passing through cone 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 of cone that joins the vertex with base.

Formula

There are two different components that you need to calculated 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 base of the radius.

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 above.

Volume of a Right Circular Cone

It is useful to know the formula for volume of a right circular cone 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.

Volume and Surface Area of a Truncated Cone or Frustum

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 a truncated cone 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.