Cone

What is a Cone?

A cone is one of the complex geometrical figures. In many textbooks, you will find that a cone is a solid of revolution generated by a right triangle that rotates around one of its legs. Here, at Superprof, we make things simple and easy for students to understand. Imagine a circle, the radius of the circle diminish by a small factor as you move the z-axis. A time will come when the circle will become a point, that point is known as Apex. Basically, a cone is a 3-dimensional geometrical figure that has a circular base and a pointy top. Did you notice the flat surface at the bottom of the cone? That is the base and the distance between the base and the apex is called the height of the cone. In addition, the distance from the circumference of the base to the apex is called slant height. A cone looks like this: You might be confused with the new terms, not to worry because we are going to talk about them.

Axis

The best way to define an axis is that it is the fixed leg about which the triangle turns. If the axis is perpendicular to the circle, it means that the cone is a right circular cone. However, if the axis isn't perpendicular to the circular base then the cone is an oblique cone.

Base

The bottom part of the cone is the base. The base of a cone is always a circle that forms the other leg.

Height

The height means how tall is the cone. It is the distance from the base to the apex.

Slant Height

The slant height is the distance from any point on the circumference to the apex of the cone.

You can calculate the slant height using the Pythagorean theorem. We all know that the Pythagorean theorem is applied to triangles and cone is not a triangle then how we are applying this theorem? Of course, a cone isn't a triangle but can transform a part of the cone into a right-angle triangle as we did in the above image. This is only applicable to the right circular cone. By the Pythagorean theorem, the slant height of the cone is equal to:

Unfold of a Cone

How about unfolding a cone? Let's grab the scissor and make an incision from the circumference to the apex. Below is the shape that you will get after the cut. The shape looks something like a circle. Of course, it is not a full circle but a part of it. We call it a pie-shaped circle. There is no telling whether the pie shape circle is a quarter circle or something else but one thing is assured that you will get a sector. To calculate the area of an unfolded cone, you will need the sector's formula:

Formulas for Cone

A handful of formulas can help in solving cone related problems. We have compiled a list of formulas that can be used to find different parameters of a cone.

Lateral Area of a Cone

Surface Area of a Cone

Volume of a Cone

Examples

For a party, Louis has made about 10 conical hats out of cardboard. How much cardboard was used in total if each cap has a radius of and a slant height of ?

We will find the lateral area of the cone because a hat don't have a base.

Since Louis made 10 hats =

Calculate the lateral area, surface area and volume of a cone whose slant height is and its base radius is .

Calculate the lateral area, surface area and volume of a cone whose height is 4 cm and its base radius is 3 cm.