Right Circular Cone Basics

Right Circular Cone Basics
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Try imagining a triangle rotating.

The 3D shape so formed will be a cone!

 cone is formed from a triangle

Any triangle will form a cone when it is rotated, taking one of its two short sides as the axis of rotation.

A cone is a 3D geometric figure that has a flat circular surface and a curved surface that meets at a point toward the top.

The point formed at the end of the cone is called the apex or vertex, whereas the flat surface is called the base.

Other parts of the cone are shown in the below diagram:

cone and its parts

Let's learn more amazing facts and properties related to this interesting  3D-shape as we scroll down.  

Lesson Plan

What Is a Right Circular Cone? 

A right circular cone is one whose axis is perpendicular to the plane of the base.

A right circular cone is generated by a revolving a right triangle about one of its legs.

Right circular cone meaning


Right Circular Cone vs Oblique Cone

A right circular cone or regular cone's axis is perpendicular to its base, whereas the oblique cone appears to be tilted and its axis is not perpendicular to the base.

Right vs oblique cone

Another way to check if a cone is a right circular cone is to check its cross-section in a horizontal plane.

A right circular cone will give a circular cross-section, whereas an oblique cone will give an oval cross-section.

Right circular vs oblique cone


Surface Areas of a Cone (Curved and Total)

To learn about the surface area of a cone, we cut and open up the cone.

The curved surface forms a sector with radius \( s\), as shown below.

curved surface area of cone

The curved surface area of a cone = Area of the sector with radius length equal to the slant height i.e., \(s\)

Curved Surface Area of a Cone = \(\pi rs\) = \(\pi r\sqrt{r^{2}+h^{2}}\)

The total surface area of a cone = Area of circular base + Curved surface area of a cone (sector's area)

Surface areas of cone

Total Surface Area of a Cone = \(\pi r^{2} + \pi rs\)
Total Surface Area of a Cone = \(\pi r^{2} + \pi r\sqrt{r^{2}+h^{2}}\)

Curved surface area is also known as the lateral surface area.

The surface area is measured in terms of square units.


Volume of a Right Circular Cone/ Volume of a Cone

The volume of a cone that has a circular base with radius \(r\) and height \(h\) will be equal to one-third of the product of the area of the base and its height.

Therefore,

\[\begin{align*}\text{Volume of a Cone} (V) &= \dfrac{1}{3} \times \text{Area of Circular Base} \times \text{Height of the Cone}\\ V &= \dfrac{1}{3} \times \pi r^{2} \times h\end{align*}\]

Volume of a cone

\(\text{Volume of a Cone} = \dfrac{1}{3} \times \pi r^{2}h\)


Right Circular Cone Calculator

 

Here's a calculator to calculate the surface area and volume of a cone.

 

Enter the values for the base and height and it will do the desired calculations for you.

 
Challenge your math skills
Challenging Questions
  • Two children planned to dig a pit in the ground in the shape of a cone.
    They dug out \(48\pi\) cubic units of mud to form the pit.
    If the depth of the pit was 9 units, find the radius of the pit.
  • Sam knows that the ratio of the volume of a cone and the volume of a cylinder with the same height and radius is 1:3.
    How will you help him prove that?

Properties of a Right Circular Cone

  • It has a circular base. The axis is a line that joins the vertex to the center of the base.
  • The slant height of the cone is measured from the vertex to the edge of the circular base. It is denoted by \(l\) or \(s.\)
  • The altitude or height of a right cone coincides with the axis of the cone and is represented by \(h.\)

cone and its parts

  • If a right triangle is rotated with the perpendicular side as the axis of rotation, a right circular cone is constructed. The surface area generated by the hypotenuse of the triangle is the curved surface or the lateral surface area.
  • Any horizontal section of the right circular cone parallel to the base produces the cross-section of a circle.
  • A plane that passes through the vertex and any two points of the base of a right circular cone generates an isosceles triangle as given below:

Isosceles triangle with right circular cone

 
important notes to remember
Important Notes
  1. A cone has one circular face, no edge, and one apex (vertex).
  2. A right circular cone is one whose axis is perpendicular to the plane of the base.
  3. Curved surface area of a cone \(= \pi rs = \pi r\sqrt{r^{2}+h^{2}}\)
  4. Total surface area of a cone \(= \pi r^{2} + \pi r\sqrt{r^{2}+h^{2}}\)
  5. Volume of a cone \(= \dfrac{1}{3} \times \pi r^{2}h\)

Solved Examples

Example 1

 

 

Jane went camping over the weekend.

At the campsite, she notices a conical tent and approximates that the height of the tent is three times the radius (r) of the tent.

Jane needs your help in finding the approximate volume of the tent in terms of its radius.

A girl and a conical tent

Solution

Given that the height is three times the radius.

\(h = 3r\) 

The volume of a right circular cone can be calculated as follows:

\[\begin{align} Volume~of~a~Cone(V) &= \frac{1}{3} \pi r^2 h \\ V &= \frac{1}{3} \pi r^2 (3r) \\ V &= \frac{1}{3} \times 3 \times \pi r^3 \\ V &= \pi r^3\end{align}\]

\(\therefore \) The volume of the tent is \(\pi r^3 \) cubic units.
Example 2

 

 

Mary uses a thick sheet of paper and prepares a birthday cap in the shape of a cone.

The radius of the cap is 3 inches and the height is 4 inches.

How can Mary find the slant height of the birthday cap?

Solution

Given radius \((r)\) = 3 inches and height \((h)\) = 4 inches

Required slant height \((s)\) = ?

\[\begin{align} (Slant~height)^2 &=(radius)^2 + (height)^2 \\ s^2 &= r^2 + h^2 \\  &=3^2 + 4^2 \\ &=9 + 16 \\ s^2 &= 25 \\ s &= \sqrt {25} \\ s &= 5 \end{align}\]

\(\therefore \) The slant height is 5 inches.
Example 3

 

 

Sam and Jacob went to a circus.

On their way back home, a clown gifted them a conical cap.

Find the volume and the curved surface area of the cap if the height of the cap is 6 \(in\), the radius is 4 \(in\), and the slant height is 8 \(in\).

Surface area of the cone-shaped cap

Solution

Given dimensions are:

Radius = 4 \(in\)

Height = 6 \(in\)

Slant height = 8 \(in\)

Substituting the values in the volume of cone formula \[\begin{align}&\text{The volume of a cone}\\ &= \frac{1}{3}\pi\times r^2\times h\ cubic\ units\\ &=\frac{1}{3}\pi \times (4)^2\times 6\\ &= 32\pi\ in^3\end{align}\]

Substituting the values in the formula \[\begin{align}&\text{The curved surface area of a cone}\\ &= \pi r l\ sq.\ units\\ &=\pi \times 4\times 8\\ &= 32\pi\ in^2\end{align}\]

\(\therefore\) Volume of the cone is \(\ 32\pi\ in^3\) 

Surface area of the cone is \(\ 32\pi\ in^2\)


Interactive Questions on Right Circular Cone

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 


Let's Summarize

The mini-lesson targeted the fascinating concept of right circular cone basics. The math journey around right circular cone basics starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs)

1. What is a right circular cone?

A right circular cone is one whose axis is perpendicular to the plane of the base.

A right circular cone is generated by revolving a right triangle about one of its legs.

2. What is the volume of a right circular cone?

\(\text{Volume of a Cone} = \dfrac{1}{3} \times \pi r^{2}h\)

where, \(r\) is the radius and \(h\) is the height of the cone.

3. What is the surface area or total surface area of a right circular cone?

\(\text {Total Surface Area of a Cone} = \pi r^{2} + \pi rs \)

\(\text {Total Surface Area of a Cone} = \pi r^{2} + \pi r\sqrt{r^{2}+h^{2}}\)

where, \(r\) is the radius, \(s\) is the slant height, and \(h\) is the height of the cone.

4. What is the curved surface area of a right circular cone?

\(\text{Curved Surface Area of a Cone} = \pi rs = \pi r\sqrt{r^{2}+h^{2}}\)

where, \(r\) is the radius, \(s\) is the slant height, and \(h\) is the height of the cone. 

5. How can you find the radius of a cone?

The radius of a cone refers to the radius of its circular base.

The radius of a cone could be found using its volume and height. 

6. How many vertices are there in a right circular cone?

There is only one vertex in a right circular cone.

7. Does a cone have two faces?

A cone has only one circular face. 

8. How do you find the radius of a right circular cone?

We can find the radius of a right circular cone using the formula in terms of its height and radius.

\(s = \sqrt {r^2 + h^2}\)

Here \(r\) is the radius, \(s\) is the slant height, and \(h\) is the height of the cone. 

9. How do you find the side length/ slant height of a right cone?

The height of the cone, the slant height, and the radius of the base form a right triangle. So, we can use the Pythagorean theorem to find the slant height.

10. Is it possible to find the right circular cone with the same height and slant height?

A right circular cone cannot have the same height as its slant height. If the slant height is considered as the hypotenuse of the right triangle, then we know that the length of the hypotenuse is greater than the lengths of the remaining two sides of the triangle.

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