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# Cone

In high school geometry, we learn two and three dimensional geometric shapes. Two dimensional shapes possess length and breadth, but no height. While, the three-dimensional shapes have all three of them. General three-dimensional or solid shapes are cube, sphere, cuboid, prism, cylinder, pyramid, cone etc. In this lesson, we are going to focus on the cone only.

A cone is a three dimensional geometric shape. For example, If one end of a line is twisted about a second set line while keeping line's other end fixed, we get a cone. The point about which the line is curved is known as the vertex and the foundation or base of the cone is a circle. The vertex is directly above the centre of the bottom.

There are two types of cones

• right circular cone
• oblique cone

right circular cone is a kind of cone in which the axis passing through the center of cone is inclined exactly at right angles to the plane of base. In other words, axis and height of a right cone are same. It is demonstrated in the diagram as under.

On the other hand, an oblique cone does not have its axis perpendicular to the plane of its base. So for oblique cone, the height and axis are different as shown in the image below.

 Related Calculators Cone Calculator Area of a Cone Calculator Cone Volume Calculator Surface Area Right Cone

## Properties of a Cone

There are a number of properties of a cone.

Some of the properties of a cone are:

• Volume of a Cone
• Face Surface Area of a Cone
• Total Surface Area of a Cone

## Cone Formula

For a right circular cone of radius r, height h and slant height s, we have

Area of the curved surface or lateral surface of the cone = $\pi$ r s sq. units

Total surface area of the cone = $\pi$ r(s + r) sq. units

Volume of the cone = $\frac{1}{3}$ $\pi$ r$^2$ h cubic units.

## Cone Volume

The volume of a cone is exactly one third the volume of a cylinder of the same height.

Volume of a cone = $\frac{1}{3}$ (Volume of a cylinder)

= $\frac{1}{3}$ ($\pi$ r$^2$ h) [$\because$ Volume of a cylinder = $\pi$ r$^2$ h ]

### Cone Volume Formula:

The volume of a cone formula is as follows,

V = $\frac{\pi}{3}$ $r^2$ h

h - height of a cone.

## Area of Cone

The surface area of the cone depend upon the radius of base and slant height of the cone. The surface area of a cone whose radius of the base r, the slant height s, and the area of the lateral surface S is,

Cone surface area (S) = $\pi$ r(r + s)

## Right Circular Cone

If a right triangle is revolved about one of the sides containing the right angle, then the solid thus generated, is called a right circular cone.

In this figure: Right angled triangle POQ is revolved about PO, it generates a right circular cone.

## Truncated Cone

If we cut a right circular cone with a plane parallel to the base of the cone, then the solid shape formed between the base of the cone and plane is called the truncated cone.

Formulas for the truncated cone:
For a truncated cone of radius r and R, height h and slant height s.
Slant Height, s = $\sqrt{(R - r)^2 + h^2}$

Lateral Surface Area, LSA = $\pi$(R + r) S sq. units

Total Surface Area, SA = $\pi$[(R + r)s + R$^2$ + r$^2$] square units

Volume, V = $\frac{1}{3}$ $\pi$ h[Rr + R$^2$ + r$^2$] cubic units

## Cone Example Problems

Given below are some of the examples on cone.

### Solved Examples

Question 1: Find the volume of the cone if r = 3 cm and h = 5 cm.
Solution:

V = $\frac{\pi}{3} r^2 h$

= $(\frac{3.14}{3}) \times (3^2) \times (5)$

= $(\frac{3.14}{3}) \times (9) \times (5)$

= 47.1

The volume of a cone is 47.1 cm3

Question 2: What is the face surface area of the cone with the radius = 5 cm and height = 7 cm?
Solution:

Formula for face surface area is as follows:

Surface Area = $\pi$$\times$ r $\times$ $\sqrt{r^2 + h^2}$

3.14 $\times$ 5 $\times$ $\sqrt{5^2 + 7^2}$

= 135.06

The face surface area of a cone is 135.06 cm2

Question 3: The radius and perpendicular height of a cone are 10 cm and 15 cm respectively. Find its curved surface area.
Solution:

Here, r = 7 cm, h = 24 cm

So, slant height, s = $\sqrt{r^2 + h^2}$ = $\sqrt{7^2 + 24^2}$

=  $\sqrt{49 + 576}$

= 25

Slant height of cone = 25 cm

Now, curved surface area of cone = $\pi$ r s

= 3.14 * 7 * 25

= 549.50

$\therefore$ curved surface area of a cone is 549.50 cm2

 More topics in Cone Surface Area of a Cone Volume of a Cone Right Circular Cone Altitude of a Cone Altitude of a Cylinder Slant height of a cone
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