All About Cones – Surface Area, Properties, Volume, and Height

A cone is a distinctive geometric figure that has three dimensions, it has a flat surface at the bottom and another surface that points upwards. The apex is what the pointed top of a cone is called, and the surface at the bottom which is flat is known as the base of a cone. The cone looks like the cone of ice cream.

A cone finds its shape from the lines that connect each other from a single point. The lines used for giving shape to the cone are known as the Apex or Vertex, they start from the top and touch the base that is the shape of a circle. When you measure the distance from the top of the cone, straight to its base at the bottom, you can derive the height of the cone. Cone also has a slanting height, the length of which is calculated when the top touches the circumference anywhere at the bottom. Based on these measurements, formulas for the surface area of the cone and volume of cone can be easily derived. 

Please note the general properties that a cone has:

  • Single face a cone has, that is its base which is circular in shape without any edges
  • A cone consists of just one point at the top, known as the vertex or apex.
  • A cone’s volume can be calculated
  • A cone’s sum of surface area can be calculated
  • The cone has a slant height as well, which can be calculated
  • When the cone plane and the base plane intersect with each other they are parallel and form a circle
  • A right circular cone is formed when the isosceles triangle rotates around its axis at one eighty degrees.

The formulas for a cone are derived when we have the following three things in place,

  • Height of Cone

The height of the cone is calculated from the top point called the vertex to the base of the circular base, straight in the middle.

  • Slant Height of Cone 

The height that is slant is calculated from the top point called the apex to anywhere on the outer line of the circle, at the base of the cone, using the Pythagoras theorem, we derive the formula for calculating the same.

                                           l = √(r2+h2)

  • Radius of Cone

The total area of the circular base at the base of the cone is called its radius.

Basically, there are two kinds of cones

  1. Right Circular Cone

This is the normal cone of the most common cone that is used in the field of geometry, in mathematics. This has a base which is the shape of a circle and the line or axis from the top point called the apex cuts through in the center of the circle of the base. We use the word “right circular cone” because the line that falls at the base of the circle makes a right angle with it.

  1. Oblique Cone

The only difference is that the line that falls at the base from the Vertex, does not fall at the center, but elsewhere. The apex is not positioned at the middle of the base. This makes the shape of the cone, slant or tilted.

Area of Surface of a Cone

This area is the entire area occupied by the surface of a cone. The overall surface area of  cone equals the total sum of its area at the surface and base area of the circle.

So, if we have a radius asr and height as h of a cone, the formula would be:

Area = πr(r+√(h2+r2))

Area =πr(r+L), where the slant height is L

Calculation of Volume of Cone

Generally, a cone has the shape of a pyramid. It is easy to find the volume if we have the height and radius measurements.

Cone volume = (1/3) πr2h cubic units

You need to practice the formulas well and Cuemath provides the right guidance. Do refer to the worksheets from them and you will come out with flying colors.

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