## Volume of a Cone

Given the radius and h, the volume of a cone can be found by using the formula:

Formula: V_{cone} = 1/3 × b × h

b is the area of the base of the cone. Since the base is a circle, area of the base = pi × r^{2}

Thus, the formula is V_{cone} = 1/3 × pi × r^{2} × h

Use pi = 3.14

**Example #1:**

Calculate the volume if r = 2 cm and h = 3 cm

V_{cone} = 1/3 × 3.14 × 2^{2} × 3

V_{cone} = 1/3 × 3.14 × 4 × 3

V_{cone} = 1/3 × 3.14 × 12

V_{cone} = 1/3 × 37.68

V_{cone} = 1/3 × 37.68/1

V_{cone} = (1 × 37.68)/(3 × 1)

V_{cone} = 37.68/3

V_{cone} = 12.56 cm^{3}

**Example #2:**

Calculate the volume if r = 4 cm and h = 2 cm

V_{cone} = 1/3 × 3.14 × 4^{2} × 2

V_{cone} = 1/3 × 3.14 × 16 × 2

V_{cone} = 1/3 × 3.14 × 32

V_{cone} = 1/3 × 100.48

V_{cone} = 1/3 × 100.48/1

V_{cone} = (1 × 100.48)/(3 × 1)

V_{cone} = 100.48/3

V_{cone} = 33.49 cm^{3}

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## Volume of a Cone | Formula for Volume of a Cone

Volume of a Cone is a measurement of the occupied units of a Cone. The volume of a Cone is represented by cubic units like cubic centimeter, cubic millimeter and so on. Volume of a Cone is the number of units used to fill a Cone.

Cone is a three dimensional shape which has a circular base and tapers to a point known as the vertex of the cone. The cone axis is a straight line which passes through the apex of the cone. Cone has a rotational symmetry. In elementary geometry, cone is used as a right circular object.

The volume of cone formula is given as,

**V = `(pir^2h)/(3)` **

Where,

r = radius of the circular base

h = height of the cone.

Value of ? = 3.14

Using this formula, we can find the volume of any cone whose radius and height are given.

Given below are some solved examples on how to find the volume of a cone.

**Example 1: **Find the volume of cone, whose radius is 6 cm and height, is 8 cm.

**Solution:**** **

The volume formula of cone is,

V =`(pir^2h)/3`

V = `((3.14)*(62)*(8))/3`

= `((3.14)*(36)*(8))/3`

= `((113.04)*(8))/3`

= `904.32 / 3` cm3.

The volume of the cone = 301.44cm3.

**Example 2:** Find the volume of cone, whose radius is 8 cm and height, is 5 cm.

**Solution:**

** **

The volume of a cone formula is,

V = `(pir^2h)/3`

V = `((3.14)*(82)*(5))/3`

= `((3.14)*(64)*(5))/3`

= `((200.96)*(5))/3`

= `(1004.8)/3` cm^{3}.

Volume of the cone = 334.93

**Example 3:** Find the volume of cone, whose diameter is 8 cm and height, is 11 cm.

**Solution:**** **

The formula for volume of a cone is,

V = `(pir^2h)/3`

radius = `D/2`

= 8/2

Radius = 4 cm.

V = `((3.14)*(42)*(11))/3`

= `((3.14)*(16)*(11))/3`

= `((50.24)*(11))/3`

= 552.64 /3 cm^{3}.

Volume of the cone = 184.21

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## Volume of a right cone

Definition:

The number of cubic units that will exactly fill a cone.

Try this

Drag the orange dots to adjust the radius and height of the cone and note how the volume changes.

The volume enclosed by a cone is given by the formula

Where r is the radius of the circular base of the cone and h is its height. In the figure above, drag the orange dots to change the radius and height of the cone

and note how the formula is used to calculate the volume.

### Oblique cones

Recall that an

oblique cone

is one that ‘leans over’ — where the apex is not over the base center point.

Drag the apex left and right above to see this. It turns out that the volume formula works just the same for these. You must remember to use the perpendicular height in the formula.

To illustrate this, in the figure above, click ‘Reset’ then ‘Freeze height’. As you drag the apex left and right, watch the volume calculation and note that the volume never changes.

### Relation to a cylinder

Recall that the

volume of a cylinder is

If you compare the two formulae, you will see one is exactly a third of the other.

This means that the volume of a cone is exactly one third the volume of the cylinder with the same radius and height.

Such a cylinder is the «circumscribed cylinder» of the cone — the smallest cylinder that can contain the cone. In the

figure above, select «Show cylinder» to see the cone embedded in its circumscribed cylinder.

### Relation to pyramid

The volume of a cone and a pyramid are calculated in a similar way. They are both equal to one third the base area times the height.

See Volume of a pyramid. In fact, you can think of a cone as a pyramid with an infinite number of sides.

To see this go to Pyramid definition and keep increasing the number osides. It will begin to look a lot like a cone.

### Things to try

- In the figure above, click «hide details».
- Drag the orange dots to set the radius and height of the cone.
- Calculate the volume of the cone using the formula
- Click «show details» to check your answer.

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## Volume of a Cone

Do you remember the formula for the volume of a cylinder?

That’s right, it was V = Πr^{2}h.

But what happens when the cylinder is cut into three equal sized pieces?

You get a cone! Yes, one of the pieces is a cone. Because all three pieces are equal, the volume of the cone formed is one-third of the total volume.

You might also notice that the cone has the same height as the cylinder and the same area of the base as the cylinder. Therefore, the volume of the cone would

be one third of the volume of the cylinder.

Let’s see this formula in action!

Example 1: Determine the volume of the cone.

In this example, the radius is 6 cm and the height is 15 cm.

These values will be «plugged» into the formula and solved.

«Double Duty» on this step. Both the 6 was squared and the 15 and 3 were divided.

This is the exact answer, also referred to as «in terms of pi.»

This is an approximation to the nearest tenth.

Example 2: Determine the volume of the cone.

In this case, the diameter and the height are given. Be sure to divide the diameter in half to determine the radius of the base of the cone. 13 in ÷ 2 = 6.5 in

Now use the given information and the formula.

You can stop at cubic inches for an exact answer, or you can round your answer. In this case, the answer was rounded to the nearest tenth to get 1415.8 cubic inches.

Don’t be fooled by a cone on its side.

Be sure to locate the circular base. Look to the base to find either the radius or diameter of the cone. This cone gives the radius of 9 ft.

The other measurement of 24 ft. from the base to the tip is the height of the cone.

Now we are ready to determine the volume of the cone.

The volume of this cone is exactly 243Π cubic feet which is approximately 763.4 cubic feet.

**Let’s Review**

The process for determining the volume of a cone is very similar to determining the volume of a cylinder. A cone has one-third the volume of a cylinder with the same base. So we can determine the volume of the corresponding cylinder and divide by 3. That gives the volume formula as

Once you begin to use the formula, you can either decide to keep the answer exact by leaving it in terms of pi or you can give an approximate answer by rounding the answer to a given place value.

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## What is the formula for finding the volume of a cone

The volume of a sphere is V = (4/3)ÏR 3 . A formal way to obtain it. Take the origin of a system of axes in the center of the sphere, then use spherical coordinates: (x, y, z) = (Ï cosÎ¸ cosÏ, Ï cosÎ¸ sinÏ, Ï sinÎ¸), with Ï in [0, R], Î¸ in [- Ï/2, Ï/2] …and Ï in [0, 2Ï]. Now you can build the Jacobian matrix of the coordinate change, and find its determinant: it is Ï 2 cosÎ¸. Thus the volume is V := â« V d 3 r = â« [0, R] â« [- Ï/2, Ï/2] â« [0, 2Ï] Ï 2 cosÎ¸ dÏ dÎ¸ dÏ = (R 3 /3) (1 — (-1)) 2Ï = (4/3)ÏR 3 . I don’t think so! And I’ll demonstrate to you: if you take V := â« V d 3 r then you have V := â« (â« V d 3 r) d 3 r and then V := â«(â«(â« V d 3 r) d 3 r) d 3 r and etc.. so it’s recursive! For the eternity! This is because Ï is irrational! The only way to solve this question is to deep a sphere in water and measure the volume of the moved water. Well, probably my notation is not clear. Notice that in V := â« V d 3 r the second time «V» appears it is smaller: i meant with this that you have to integrate d 3 r on the volume V of the sphere (that is, in fact, the definition of the volume). In V := â« V d 3 r the little «V» is at the foot of the «â«» symbol. An easier way to obtain the answer: Imagine an oragasm which is divided into an infinite amount of prisms with a common vertex at the centre of the sphere. By calculating the volume of all these prisms, one can obtain the volume of the sphere. The formula for the volume of a prism is (1/3)bh. If we apply this formula to the infinite number of pyramids, the total area of the bases (b) would be the SA of the sphere, (4ÏR^2), the height (h), would be the distance from the surface area to the centre, which is the radius (R). This means the formula for finding the Volume of a sphere would then be (1/3)bh which is (1/3)(R)(4ÏR^2) which could then be simplified to (4/3)ÏR^3 (MORE)

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## Math Labs with Activity — Volume of a Right-Circular Cone Formula

**Math Labs with Activity – Volume of a Right-Circular Cone Formula**

**OBJECTIVE**

To demonstrate a method to derive a formula for finding the volume of a right-circular cone.

**Materials Required**

- A hollow right-circular cone of known height and base radius
- A hollow cylinder with open top having the same height and base radius as those of the cone
- A few packs of table salt

**Procedure** **Step 1:** Take a hollow right-circular cone of a known height h and base radius r. Also, take a hollow right-circular cylinder of the same height h and base radius r such that it is open at the top and closed at the bottom.

**Step 2:** Fill the cone with salt up to the brim, as shown in Figure 40.2.

Pour the entire quantity of salt from the cylinder. Fill the cone again with salt. Pour this salt again into the cylinder. Once again fill the cone with salt and then pour the salt into the cylinder (see Figure 40.3).

**Observations and Calculations**

We observe that the volume of a cylinder is three times the volume of a cone of the same height h and the same base radius r. Since the volume of a cylinder with height h and base radius r is given by πr²h, therefore the volume of a cone with height h and base radius r will be 1/3 πr²h.

**Result**

The volume of a right-circular cone of the height h and base radius r is given by 1/3 πr²h.

Math Labs with ActivityMath LabsScience Practical SkillsScience Labs

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## All formulas for volume of geometric solids

— side

Calculate the volume of a cube if given side ( * V* ) :

, , — sides of a parallelepiped

Calculate the volume of a rectangular prism if given sides (* V *) :

— radius

— center of a sphere

3,14

Calculate the volume of a sphere if given radius (* V *) :

— radius

— height of a spherical segment

3,14

— center of the sphere

Calculate the volume of a spherical segment if given radius and height (* V *) :

— radius of the upper base

— radius of the lower base

— height of a spherical zone

3,14

— center of the sphere

Calculate the volume of a spherical zone if given radii and height (* V *) :

— radius of a sphere

— height of a spherical segment

3,14

— center of the sphere

Calculate the volume of a spherical sector (* V *) :

— radius of a base

— height of a cylinder

3,14

Calculate the volume of a cylinder if given radius and height (* V *) :

— radius of the base

— height

3,14

Calculate the volume of a cone if given radius and height (* V *) :

— radius of the upper base

— radius of the lower base

— height of a truncated cone

3,14

Calculate the volume of a truncated cone if given radii and height (* V *) :

— area of the base *abcde*

— height

Calculate the volume of a pyramid if given height and base area (* V *) :

— area of the lower base

— area of the upper base

— height of a truncated pyramid

Calculate the volume of a truncated pyramid if given base area and height (* V *) :

— side of a base

— number of sides of the base

— height of a pyramid

Calculate the volume of a regular pyramid if given height, side of a base and number of sides (* V *) :

** A pyramid, which base is a regular polygon and which lateral faces are equal triangles, is called regular.*

— side of a base

— height of a pyramid

Calculate the volume of a regular triangular pyramid if given height and side of a base (* V *) :

** A pyramid, which base is an equilateral triangle and the lateral faces are equal isosceles triangles, is called a regular triangular pyramid. *

— side of a base

— height of a pyramid

Calculate the volume of a regular quadrangular pyramid if given height and side of a base (* V *) :

** A pyramid, which has a square base and equal isosceles triangles as the lateral faces, is called a regular quadrangular pyramid. *

— edge of a tetrahedron

Calculate the volume of a regular tetrahedron if given length of an edge (* V *) :

** Regular tetrahedron is a pyramid in which all the faces are equilateral triangles. *

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