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

A **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.

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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

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.

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 ]

The volume of a cone formula is as follows,

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

Where, r - radius

h - height of a 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)

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.

→ Read More 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:

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

Given below are some of the examples on cone.

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 cm^{3}

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 cm^{2}

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 cm^{2}

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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