How to Find the Volume of a Cone in 4 Easy Steps

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How to find the Volume of a Cone | Mathcation

Key Points about Volume of a Cone

  • The volume of a cone is calculated using the formula (1/3)πr²h or (1/3)π(d/2)²h.
  • The volume of a cone is an important concept in mathematics and engineering.
  • The formula for calculating the volume of a cone can be used to find the volume of any cone, regardless of its size or dimensions.

The Short Explanation for finding the Volume of a Cone

Finding Volume of a Cone can be completed easily by using the formula and following correct order of operations. The first step in finding Volume of a Cone is understanding that you are multiplying the area of the base by the height of the cone. The base happens to be a circle so you must use the Area of a Circle formula to calculate the area of the base. Be sure to remember that the radius is equal to half of the diameter. Once you know the area of the base, you must multiply it by the height of the cone. Next, you divide the solution by three. The final step for finding Volume of a Cone is to follow order of operations and simplify the equation.

A cone is a three-dimensional geometric shape that has a circular base and a single vertex or apex. It is one of the most common shapes found in everyday life, from traffic cones to ice cream cones. The volume of a cone is an important concept in mathematics and engineering, as it is used to calculate the amount of space inside a cone-shaped container or object.

The formula for calculating the volume of a cone is (1/3)πr²h, where r is the radius of the circular base and h is the height of the cone. This formula can be used to find the volume of any cone, regardless of its size or dimensions. The volume of a cone can also be calculated using the diameter of the base instead of the radius, by using the formula (1/3)π(d/2)²h.

Common Core Standard: 8.G.C
Related Topics: Pythagorean Theorem, Parallel Lines Cut by a Transversal, Triangle Angle Sum, Exterior Angle of a Triangle, Volume of a Cylinder, Volume of a Sphere
Return To: Home, 8th Grade

How to find the Volume of a Cone

Volume of a Cone Formula: How to Use

The volume of a cone is the amount of space inside a cone. The formula for the volume of a cone is determined by the height of the cone, the radius of the base, and pi. The formula is given by:

V = (1/3) * π * r^2 * h

Where V is the volume of the cone, r is the radius of the base, h is the height of the cone, and π is a mathematical constant approximately equal to 3.14159.

Calculating the Volume

To calculate the volume of a cone, you need to know the height of the cone and the radius of its base. The formula for the area of the base of a cone is given by:

A = π * r^2

Where A is the area of the base of the cone, r is the radius of the base, and π is a mathematical constant approximately equal to 3.14159.

Once you have the area of the base, you can apply the formula for the volume of a cone. To calculate the volume of a cone, you can use the following steps:

  1. Calculate the area of the base of the cone using the formula A = π * r^2.
  2. Multiply the area of the base by the height of the cone.
  3. Divide the result by 3 to get the volume of the cone.

The volume of a cone is usually measured in cubic units such as cubic inches or cubic feet.

In summary, the volume of a cone can be calculated using the formula V = (1/3) * π * r^2 * h, where V is the volume of the cone, r is the radius of the base, h is the height of the cone, and π is a mathematical constant approximately equal to 3.14159. To calculate the volume of a cone, you need to know the height of the cone and the radius of its base.

 

Volume of a Cone with Diameter

Calculating the volume of a cone with diameter is a straightforward process. The formula to find the volume of a cone is (1/3) x π x r² x h, where r is the radius of the base and h is the height of the cone. However, when given the diameter of the base instead of the radius, the formula needs to be adjusted accordingly.

The volume of a cone with diameter can be calculated using the formula (1/12) x π x d² x h, where d is the diameter of the base and h is the height of the cone. This formula is derived by substituting the value of radius r with d/2 in the original formula for the volume of a cone.

To better understand the formula, consider the following example. Suppose a cone has a diameter of 8 cm and a height of 12 cm. To find the volume of the cone, the formula can be used as follows:

Volume of cone = (1/12) x π x (8 cm)² x 12 cm

Volume of cone = 201.06 cm³ (rounded to two decimal places)

It is worth noting that the diameter of the base is twice the radius, so the volume of a cone with diameter is one-third of the volume of a cone with the same height and radius. This relationship can be expressed mathematically as:

Volume of cone with diameter = (1/3) x Volume of cone with radius

In conclusion, calculating the volume of a cone with diameter is a simple process that involves substituting the diameter for the radius in the formula for the volume of a cone. By following the formula, anyone can easily find the volume of a cone with diameter.

 

How to find the Volume of a Cone Solution

3 Simple Volume of a Cone Examples

Calculating the volume of a cone is a common math problem that involves using the formula V = (1/3)πr²h, where V is the volume, r is the radius of the base, and h is the height of the cone.

  1. In order to solve for Volume of a Cone you must know that the volume is equal to the base times the height divided by three.
  2. The base is equal to a circle so you use the area of a circle formula to find the area of the base.
  3. The height is equal to how tall the cone is.
  4. You substitute in the radius of the cone for r, and the height of the cone for h.
  5. Multiple those together in your calculator.
  6. Finally divide by three and be sure to use the correct units.

Here are a few examples of how to calculate the volume of a cone:

Example 1

Suppose a cone has a radius of 4 cm and a height of 10 cm. To find the volume of the cone, plug in the values of r and h into the formula:

V = (1/3)π(4²)10 V = (1/3)π(16)10 V = (1/3)π(160) V = 167.55 cm³

Therefore, the volume of the cone is approximately 167.55 cubic centimeters.

Example 2

Suppose a cone has a radius of 6 cm and a height of 8 cm. To find the volume of the cone, plug in the values of r and h into the formula:

V = (1/3)π(6²)8 V = (1/3)π(36)8 V = (1/3)π(288) V = 301.59 cm³

Therefore, the volume of the cone is approximately 301.59 cubic centimeters.

Example 3

Suppose a cone has a radius of 5 cm and a height of 15 cm. To find the volume of the cone, plug in the values of r and h into the formula:

V = (1/3)π(5²)15 V = (1/3)π(25)15 V = (1/3)π(375) V = 392.7 cm³

Therefore, the volume of the cone is approximately 392.7 cubic centimeters.

In each of these examples, the formula for the volume of a cone is used to find the volume of a cone with given dimensions. By plugging in the values of r and h, the volume can be calculated easily.

 

5 QUICK Volume of a Cone Practice Problems

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Volume of a Cone Quiz

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Find the Volume of the Cone.

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Find the Volume of the Cone.

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Find the Volume of the Cone.

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Find the Volume of the Cone.

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Find the Volume of the Cone.

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Volume of a Cone Equation with Radius

The volume of a cone is the amount of space inside a cone. It is measured in cubic units such as cubic inches or cubic feet. The formula for calculating the volume of a cone is V = 1/3(πr^2h), where V is the volume, r is the radius of the base, h is the height of the cone, and π is a mathematical constant that represents the ratio of the circumference of a circle to its diameter, approximately equal to 3.14159.

Calculating the Volume

To calculate the volume of a cone, you need to know the height of the cone and the radius of its base. Here are the steps to calculate the volume of a cone:

  1. Measure the radius of the base of the cone. If you only have the diameter, divide it by 2 to get the radius.
  2. Measure the height of the cone.
  3. Use the formula V = 1/3(πr^2h) to calculate the volume of the cone.

For example, if the radius of the base of a cone is 4 inches and the height of the cone is 8 inches, the volume of the cone can be calculated as follows:

V = 1/3(π(4^2)(8)) V = 1/3(π(16)(8)) V = 1/3(π(128)) V = 1/3(401.92) V = 133.97 cubic inches

Therefore, the volume of the cone is approximately 133.97 cubic inches.

In summary, the volume of a cone can be calculated using the formula V = 1/3(πr^2h), where V is the volume, r is the radius of the base, h is the height of the cone, and π is a mathematical constant. To calculate the volume of a cone, measure the height of the cone and the radius of its base, and use the formula to find the volume in cubic units.

 

How to Find the Volume of a Cone FAQ

What is the formula for the volume of a cone?

The formula for the volume of a cone is V = (1/3)πr²h, where V is the volume, r is the radius of the circular base, and h is the height of the cone.

Why is there a 1/3 in the formula for the volume of a cone?

The 1/3 in the formula for the volume of a cone comes from the fact that the volume of a cone is one-third the volume of a cylinder with the same base and height.

What is the TSA of a cone?

The total surface area (TSA) of a cone is the sum of the area of the base and the lateral surface area. The lateral surface area is given by πrl, where r is the radius of the circular base and l is the slant height of the cone.

How many times the volume of a cylinder is equal to the volume of a cone?

The volume of a cone is one-third the volume of a cylinder with the same base and height.

What is the Volume of a Cone Formula?

The volume of a cone formula is V = (1/3)πr²h, where V is the volume, r is the radius of the circular base, and h is the height of the cone.

What is the Volume of Cone?

The volume of a cone is the amount of space occupied by the cone and is given by the formula V = (1/3)πr²h, where V is the volume, r is the radius of the circular base, and h is the height of the cone.

Is cylinder twice the volume of a cone?

No, a cylinder is not twice the volume of a cone. The volume of a cone is one-third the volume of a cylinder with the same base and height.

How do you find the height of a cone without the volume?

To find the height of a cone without the volume, you need to know the radius of the base and the slant height of the cone. The height can be found using the Pythagorean theorem, which states that the square of the height is equal to the square of the slant height minus the square of the radius.

 

Volume of a Cone Worksheet: Video Explanation

Watch our free video on Formula for Volume of Cone. This video shows how to solve problems that are on our free Volume of a Cone worksheet that you can get by submitting your email above.

Watch the free formula of a cone video on YouTube here: How to Find the Volume of a Cone

Video Transcript:

This video is about how to find the volume of a cone and we are at our volume of cone worksheet in order to show you some example problems.

The first step in finding the volume of a cone is to figure out the cone volume formula. In order to figure out the volume of cone formula you have to use a formula that you should already know which is volume of a cylinder. We already know that volume of a cylinder is the area of the base times the height, and in the case of a cylinder and a cone the base is a circle.  We use PI R squared because that is how you find the area of a circle. This right here is how to find the volume of a cylinder.

If we were to draw an imaginary cylinder around this cone it would look something like this. Our blue line is our imaginary cylinder you will notice that the cone takes up less space than the cylinder does. In order to compensate for that we have to change our volume of a cylinder formula and it just so happens that the volume of a cone is 1/3 that of a cylinder.

So in order to change our formula from volume of a cylinder to volume of a cone you just divide by 3. Here’s our new formula just to reiterate because we know that the volume of a cone is 1/3 that of a cylinder, all you have to do is take the volume of a cylinder which is in red and divide it by 3 to compensate.

Let’s do our first problem here on the volume cone worksheet. We are at number one and we just went over our explanation on formula for volume of a cone and here it is. The volume of the cone is the area of the base times the height divided by 3, so in order to solve for the volume you have to substitute in for the radius and you have to substitute in for the height because those are the two pieces we need in our formula.

When you look at our problem here our radius of our circle is 9 inches so that’s going to be nine and then the height of our cone is 17 inches so you’re gonna plug in 17 for height. Then we’ll take our radius which is nine and we will substitute it in for R and then we will also substitute in 17 for the height so that will go here and then the whole thing gets divided by three. Once you put that in your calculator you will get 14401.99 inches cubed because that’s our unit and that’s going to be our answer.

In our next example we have to find out how to find the volume of a cone that has a height of 19 and a diameter of 10. We know our formula for a cone is PI R squared times the height divided by 3 and we know that we have to find the radius and we know that we have to find the height. In order to do that the height is easiest because you can look at your cone and you can see that the height is 19 feet.

We’re gonna put 19 feet here put 19 in for H and then when we do the radius what they give us is 10 feet but the 10 feet is for the whole diameter it’s the whole length of the circle. Instead of using diameter we have to use the radius which is half of the diameter so what we’re going to do is we’re gonna take our diameter divided by 2 10 divided by 2 is 5.

We will use 5 for the radius and now we know our radius and we know our height. We’re gonna take our volume we’re gonna take our volume formula and substitute in radius and height  we know the radius is 5 that goes in for R and then we know the height is 19.

That goes in for H and then we have to divide everything by 3. We’re going to take our formula we’re going to multiply all of this together then divide it by 3 and we will get four hundred ninety seven point four two and then our unit is feet cubed. Try all the practice problems on our volume of cones worksheet above.

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