Right Circular Cone: Surface Area & Volume

Right Circular Cone: Curved Surface Area, Total Surface Area, Volume

This section covers the fundamental concepts related to right circular cones, including their surface areas and volume. Understanding these concepts is crucial for various applications in mathematics and real-world scenarios.

A right circular cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex. The line segment connecting the apex to the center of the base is perpendicular to the base, hence the term “right” cone. Key components include the radius ($r$) of the base, the height ($h$) (the perpendicular distance from the apex to the base), and the slant height ($l$) (the distance from the apex to a point on the circumference of the base).

Formulae

Here are the key formulae for calculating the surface area and volume of a right circular cone:

• Curved Surface Area (CSA): $CSA = \pi r l$
• Total Surface Area (TSA): $TSA = \pi r l + \pi r^2 = \pi r (l + r)$
• Volume (V): $V = \frac{1}{3}\pi r^2 h$
• Relationship between r, h, and l: $l^2 = r^2 + h^2$ (by the Pythagorean theorem)

Examples

Example-1: A right circular cone has a radius of 7 cm and a slant height of 15 cm. Calculate its curved surface area and total surface area.

Solution:

CSA = $\pi r l = \pi \times 7 \times 15 \approx 329.87 \text{ cm}^2$

TSA = $\pi r (l + r) = \pi \times 7 \times (15 + 7) = \pi \times 7 \times 22 \approx 483.80 \text{ cm}^2$

Example-2: A cone has a radius of 3 cm and a height of 4 cm. Find the volume of the cone.

Solution:

First, we calculate the slant height: $l = \sqrt{r^2 + h^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm}$ (Although slant height is not required for this problem, its good to know for other calculations).

Now, the volume is calculated using the formula: $V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi \times 3^2 \times 4 = 12\pi \approx 37.7 \text{ cm}^3$

Common Mistakes by Students

• Confusing the height ($h$) and the slant height ($l$). Remember to use the Pythagorean theorem ($l^2 = r^2 + h^2$) to correctly relate these values.
• Using the wrong formula. Always double-check the formula for CSA, TSA, and Volume before plugging in values.
• Forgetting the units. Make sure to include the correct units (e.g., cm², cm³) in your final answers.

Real Life Application

Right circular cones are prevalent in real-life scenarios:

• Ice Cream Cones: The shape of the cone itself.
• Traffic Cones: Used for traffic management.
• Tents: Conical tents for camping.
• Lampshades: Certain lampshades are cone-shaped.
• Funnel: Conical funnels for pouring liquids and powders.

Fun Fact

The volume of a cone is exactly one-third the volume of a cylinder with the same base radius and height. This relationship is a fundamental concept in solid geometry!

Recommended YouTube Videos for Deeper Understanding

Q.1 A right circular cone has a base radius of $5$ cm and a slant height of $13$ cm. What is the curved surface area of the cone?

Check Solution

Ans: A

The curved surface area of a cone is given by $\pi r l$, where $r$ is the radius and $l$ is the slant height. So, the curved surface area is $\pi \times 5 \times 13 = 65\pi$ cm².

Q.2 A right circular cone has a radius of $6$ cm and a height of $8$ cm. What is the total surface area of the cone?

Check Solution

Ans: B

First find the slant height, $l = \sqrt{r^2 + h^2} = \sqrt{6^2 + 8^2} = 10$ cm. The total surface area is $\pi r l + \pi r^2 = \pi (6)(10) + \pi (6^2) = 60\pi + 36\pi = 96\pi$ cm².

Q.3 The volume of a right circular cone is $100\pi$ cm³ and its height is $12$ cm. What is the radius of the base of the cone?

Check Solution

Ans: A

The volume of a cone is given by $\frac{1}{3}\pi r^2 h$. So, $100\pi = \frac{1}{3} \pi r^2 (12)$. This simplifies to $100 = 4r^2$, or $r^2 = 25$, hence $r = 5$ cm.

Q.4 A cone and a cylinder have the same base radius and the same height. If the volume of the cylinder is $300$ cm³, what is the volume of the cone?

Check Solution

Ans: A

The volume of a cone is $\frac{1}{3}$ the volume of a cylinder with the same base and height. Therefore, the volume of the cone is $\frac{1}{3} \times 300 = 100$ cm³.

Q.5 A right circular cone has a base diameter of $10$ cm and a slant height of $13$ cm. Find the volume of the cone.

Check Solution

Ans: A

The radius is $5$ cm. First, find the height $h = \sqrt{l^2 – r^2} = \sqrt{13^2 – 5^2} = 12$ cm. The volume is $\frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (5^2)(12) = 100\pi$ cm³.

Next Topic: Sphere & Hemisphere: Surface Area & Volume