What is a Cone?

A cone is a three-dimensional solid shape having a flat base and a pointed edge at the top. The flat base of the cone tapers smoothly to form the pointed edge known as the apex. The flat base of the cone can either be circular or elliptical. A cone is drawn by joining the apex to all points on the base, using segments, lines, or half-lines, provided that the apex and the base both are in different planes.

Terminologies

• Radius - The radius of a cone is the distance between the center of the base to any circumferential point on the base.
• Height - The height of a cone is the perpendicular distance between the apex and base.
• Slant height - The slant height is the distance between the apex and any point on the circumference of the base.
• Axis - The axis of a cone is a straight line passing through the apex and center of the base of a cone, about which the base of a cone has circular symmetry.

Types of a cone

There are three types of cones: right circular cone, oblique cone, and elliptic cone.

• Right circular cone - When the base of a cone is circular and the apex of the cone is exactly above the center of the base, the cone is known as a right circular cone. In a right circular cone, the axis will be perpendicular to the radius of the base.
• Oblique Cone - When the base of a cone is circular, but the apex of the cone is not exactly above the center of the base and is sideways, the cone is known as an oblique cone. In an oblique cone, the axis will not be perpendicular to the radius of the base.
• Elliptic Cone - When the base of a cone is elliptical, the cone is known as an elliptic cone.

Properties of a cone

• A cone has one vertex, one face, and no edges.
• A cone has a circular or an elliptical base and one continuous curve.

Volume and Area of a cone

The calculations of a cone are primarily done for the three parameters: Volume, curved surface area, and total surface area. The slant height also needs to be calculated in some of the cases.

The Volume of a cone

The volume of a cone is the total space occupied by the cone. The formula for the volume of a cone is derived using Cavalieri's principle.

Consider a cone having a height = $h$, and the area of base = $A$. Then the volume of any cone is given as,

$Volume = \frac{1}{3}\times A\times h$

For a right circular cone and an oblique cone, consider the radius of the base = $r$, height = $h$. Hence, the volume of the right circular cone can be given as,

$$\begin{gathered} Volume of a cone= \frac{1}{3}\times \mathrm{\pi }\times {\mathrm{r}}^2\times \mathrm{h} \\ \mathrm{Volume} \mathrm{of} \mathrm{a} \mathrm{cone} =\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h} \end{gathered}$$

The Curved Surface Area of a cone

The curved surface area of a cone is the area occupied by the curved surface of a cone.

Consider a cone with a radius of the base =$r$, height = $h$, and slant height = $l$. Hence, the curved surface area of a cone can be given as,

$$\begin{gathered} Curved surface area = \mathrm{\pi }\times \mathrm{r}\times \mathrm{l} \\ \mathrm{CSA} = \mathrm{\pi rl} \\ \mathrm{Where} \\ \mathrm{l}=\sqrt{{\mathrm{r}}^2+ {\mathrm{h}}^2} \end{gathered}$$

The Total Surface Area of a cone

The total surface area of a cone is the total area occupied by the cone. The total surface area of a cone is the sum of the curved surface area of a cone and the area of the base of the cone.

Consider a cone having a radius of the base =$r$, height = $h$, and slant height = $l$. Hence, the total surface area of a cone can be given as,

$$\begin{gathered} Total surface area = area of the base + curved surface area \\ TSA = \pi r^2 + \pi rl \\ TSA = \mathrm{\pi r}(r+l) \end{gathered}$$

Frustum of Cone and Bicone

When a cone is cut by a plane into two parts, the bottom part of the cone is known as a frustum. A frustum is formed when the cutting plane is kept parallel to the base of the cone. Consider a frustum having a smaller radius = $r_2$, bigger radius = $r_1$, and height = $h$

The volume of a frustum is given as-

$Volume of a frustum = \frac{\mathrm{\pi h}}{3}.({r_1}^2+r_1.r_2+{r_2}^2)$

The curved surface area of a frustum is given as-

$Curved surface area of a frustum = \mathrm{\pi }.({\mathrm{r}}_1+{\mathrm{r}}_2).\sqrt{{({\mathrm{r}}_1-{\mathrm{r}}_2)}^2+{\mathrm{h}}^2}$

The total surface area of a frustum is given as-

$TSA of a frustum =\mathrm{\pi }.\left((r_1+r_2).\sqrt{{(r_1-r_2)}^2+h^2}+{r_1}^2+{r_2}^2\right)$

A bicone is a three-dimensional solid formed when two equal right circular cones are joined to each other from their bases. A bicone is also formed when a rhombus is rotated around one of its axis of symmetry. The image below shows a bicone-

Formulas

$$\begin{gathered} Volume of a cone = \frac{1}{3}\pi r^{2}h \\ Curved surface area of a cone = \pi rl , where l =\sqrt{r^2+ h^2} \\ Total surface area of a cone = \pi r(r+l) \\ Volume of a frustum = \frac{\mathrm{\pi h}}{3}.({r_1}^2+r_1.r_2+{r_2}^2) \\ CSA of a frustum = \mathrm{\pi }.({\mathrm{r}}_1+{\mathrm{r}}_2).\sqrt{{({\mathrm{r}}_1-{\mathrm{r}}_2)}^2+{\mathrm{h}}^2} \\ TSA of a frustum =\mathrm{\pi }.\left((r_1+r_2).\sqrt{{(r_1-r_2)}^2+h^2}+{r_1}^2+{r_2}^2\right) \end{gathered}$$

Context and Applications

Cones are very useful in day-to-day life. Various objects that we see daily such as ice-cream cones, funnels, party hats, traffic cones, waffle cones, Christmas trees, and so on, are conical in shape.

The cones are useful for students studying in schools, various undergraduate and postgraduate courses such as:

• Bachelors in Engineering (Mechanical and Civil)
• Masters in Engineering (Mechanical and Civil)
• Bachelors in Science (Mathematics)
• Masters in Science (Mathematics)

Practice Problems

Q1. Which of the following is a straight line passing through the apex and center of the base of a cone?

1. Radius
2. Height
3. Axis
4. Slant height

Answer– Option c

Explanation – Axis is a straight line passing through the apex and center of the base of a cone.

Q2. Which of the following is the distance between the apex and any point on the circumference of the base?

1. Radius
2. Height
3. Axis
4. Slant height

Answer – Option d

Explanation – Slant height is the distance between the apex and any point on the circumference of the base.

Q3. Which of the following is a property of the cone?

1. A cone has one vertex, one face, and no edges.
2. A cone has 4 sides
3. A cone has 5 sides
4. A cone has infinite edges.

Answer – Option a

Explanation – A cone has one vertex, one face, and no edges is a property of the cone.

Q4. Which of the following is the area occupied by the curved surface of a cone?

1. The volume of a cone
2. The curved surface area of a cone
3. Slanted area of a cone
4. Linear area of a cone

Answer – Option b

Explanation – Curved surface area of a cone is the area occupied by the curved surface of a cone.

Q5. Which of the following is a three-dimensional solid formed when two equal right circular cones are joined to each other from their bases?

1. Frustum
2. Sphere
3. Pyramid
4. Bicone

Answer – Option d

Explanation – Bicone is a three-dimensional solid formed when two equal right circular cones are joined to each other from their bases.