# Cone

A cone is a unique three-dimensional shape with a flat circular face at one end and a pointed tip at another end. The word ‘cone’ is derived from the Greek word ‘konos’, meaning a peak or a wedge.

A traffic signal cone, an ice-cream cone, or a birthday hat are some common examples of a cone.

• Its circular face is the base.
• Above the circular base is the curved surface that narrows to a pointed tip called the vertex (or apex).
• Has no edge.
• Since it has a curved surface, it is not a polyhedron.
• The shape is achieved by rotating a triangle. So it can be said to be a rotated triangle.

## Parts of a Cone

1. Radius (r) – It is the distance between the center of its circular base to any point on the circumference of the base.
2. Height (h) – It is the perpendicular distance between its vertex to the base center. We consider the height of a right circular cone as its axis.
3. Slant Height (s) – It is the distance from its vertex to the point on the outer edge of its circular base.

## Types of Cones – Right vs. Oblique

Based on the position of the vertex with respect to its base, a cone is of two types, as shown in the figure.

### Right Circular Cone

It has its vertex aligned right above the center of the base. The axis coincides with the height and makes a right angle at the base center.

### Oblique Cone

It does not have its vertex aligned perpendicular to its base. It looks tilted.

Since, in practical life, a cone means a right circular cone, here, we will learn the formulas related to it.

## Formulas

### Surface Area of a Cone

The formula of the surface area (or total surface area) of a right circular cone is:

Surface Area (SA) = πr2 + πrs, here r = radius, s = slant height, π = 3.141, πr2 = base area, πrs = lateral (curved) surface area of a cone (LSA)

So, we can rewrite the formula as SA = πr2 + LSA

Find the surface area of a right circular cone with a slant height of 6 cm and radius of 4 cm.

Solution:

As we know,
Surface Area (SA) = πr2 + πrs, here r = 4 cm, s = 6 cm, π = 3.141
∴ SA = 3.141 × 42 + 3.141 × 4 × 6
= 125.7 cm2

Calculate the lateral surface area of a cone with a radius of 3 in and slant height of 8 in.

Solution:

As we know,
Lateral Surface Area (LSA) = πrs, here r = 3 in, s = 8 in, π = 3.141
∴ LSA = 3.141 × 3 × 8
= 75.3 cm2

Find the base area of a cone with a radius of 7.5 cm.

Solution:

As we know,
Base area = πr2, here π = 3.141, r = 7.5 cm
= 3.141 × (7.5)2
= 176.7 cm2

### Volume

The formula is:

Volume ${\left( V\right) =\dfrac{1}{3}\pi r^{2}h}$, here r = radius, h = height, π = 3.141

[Volume (V) = 1/3 πr2h]

We can relate the formula of the volume of a cylinder which is πr2h

Therefore, the volume of a cone is precisely one-third of that of a cylinder.

Calculate the volume of a cone with a radius of 7 mm and height of 12 mm.

Solution:

As we know,
Volume ${\left( V\right) =\dfrac{1}{3}\pi r^{2}h}$, here r = 7 mm, h = 12mm, π = 3.141
∴ ${V=\dfrac{1}{3}\times \pi \times 7^{2}\times 12}$
= 615.7 mm3

### Slant Height

The formula is:

Slant Height ${\left( s\right) =\sqrt{r^{2}+h^{2}}}$, here r = radius, h = height

Find the slant height of a cone with a radius of 6 mm and height of 11 mm.

Solution:

As we know,
Slant Height ${\left( s\right) =\sqrt{r^{2}+h^{2}}}$, here r = 6 mm, h = 11 mm
${\therefore s=\sqrt{6^{2}+11^{2}}}$
= 12.53 mm

Finding the HEIGHT of a cone when the VOLUME and RADIUS are known

Find the height of a cone with a volume of 65.9 mm3 and a radius of 3 mm..

Solution:

Volume ${\left( V\right) =\dfrac{1}{3}\pi r^{2}h}$, here r = radius, h = height, π = 3.141
So, by rearranging the equation, we get,
Height ${h=\dfrac{3V}{\pi r^{2}}}$, here V = 65.9 mm3, r = 3 mm
${= \dfrac{3\times 65\cdot 9}{3.141\times 3^{2}}}$
= 6.99 mm

Finding the RADIUS when the VOLUME and HEIGHT are known

Find the radius of a cone with a volume of 165.4 cm2 and a height of 7.8 mm.

Solution:

Volume ${\left( V\right) =\dfrac{1}{3}\pi r^{2}h}$, here r = radius, h = height, π = 3.141
So, by rearranging the equation, we get,
Radius ${\left( r\right) =\sqrt{\dfrac{3V}{\pi h}}}$, here V = 165.4 cm2, h = 7.8 cm
${= \sqrt{\dfrac{3\times 165\cdot 4}{3\cdot 141\times 7\cdot 8}}}$
= 4.5 cm

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