Volume of 3D Objects (Review Video & Practice Questions)

Q: How do you find the volume of a cube?

Find the volume of a cube by cubing its side length. V=s3 Ex. What is the volume of a cube with a side length of 7 cm? V = s3 = 73 = 343 cm3

How to Calculate the Volume of 3D Objects Video

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Hey, guys! Welcome to this video on the volume of three-dimensional objects.

Let’s start off by defining volume. Volume is the measurement of how much space a liquid or gas takes up, or how much space a liquid or gas takes up within a given object.

You may not know it, but people use volume every day. Volume is used to calculate the drinking amounts. The amount of water you can hold in a cup is dependent on the volume of the cup. There are several other ways that volume is used.

Now, let’s look at how to calculate the volume of a triangular prism, a rectangular prism, a sphere, and a cone.

The Volume of a Triangular Prism

The area of a triangle is $A=12bh<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>b</mi><mi>h</mi></math>$ . Essentially, to find to the volume of the triangular prism, you are multiplying the area of the triangle times the length or depth. So, the formula for the volume of a triangular prism would be $V=12bhl<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>b</mi><mi>h</mi><mi>l</mi></math>$ .
Let’s take a look:

triangular prism

We have a triangular prism with a height of 8 meters, a base of 13 meters, and a length of 4 meters. All we have to do is plug our numbers into our formula then solve. So we have $V=12bhl<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>b</mi><mi>h</mi><mi>l</mi></math>$ . And once we solve, we get our answer, which is $208 m 3 <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>208</mn><msup><mi>m</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></math>$ . It’s important to know that, when dealing with volume, we will always have cubic units because we are multiplying the units by themselves 3 times.

Volume of a Cube or Rectangular Prism

To find the same volume of a cube or rectangular prism, you will use the same formula. Just like with the triangular prism, you want to find the area of one side, then multiply it times the length.

However, it’s important to know that the formula you use to find the area of a triangle is not the same formula you use to find the area of a square or a rectangle. The formula for the area of both a square and a rectangle is $A = b h <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>b</mi><mi>h</mi></math>$ . So, to find the volume of a cube or rectangular prism, you would find the area of the square or rectangle then multiply it times the length. Which, makes the formula $V = b h l <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi>b</mi><mi>h</mi><mi>l</mi></math>$ .

Here is an example:

cube

Here we have a cube, which is a rectangular prism, but all the sides are perfect squares. Because it’s a cube, we know that all of the sides are the same distance. So all we need to do is multiply 10 times itself 3 times. This gives us 1,000 meters cubed. Let’s try another:

rectangular prism

Here, we have a rectangular prism with sides that are different in distance. We have a base of 12 cm, a height of 8 cm, and a length of 6 cm. Now, all we need to do is plug those numbers into our formula, and once we solve, we get 576 cm cubed.

Volume of a Sphere

Now if you remember the area of a circle is $A = π r 2 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>π</mi><msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></math>$ . That is pi times the radius squared. Well, to find the volume of a sphere you will use a similar formula, but multiply it times $43<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>3</mn></mfrac></math>$ and switching the $r 2 <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></math>$ to make it $r 3 <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></math>$ . Making the formula for the volume of a sphere $V=43πr3<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></math>$ . When you do what is called a proof, to prove that this is the formula, but for now we’ll just plug the numbers into the given formula.

sphere

The sphere has a diameter of 20 meters. This is all the information we need to plug in and solve our equation. We are looking for the radius, and we know that the radius is equal to half of the diameter, which means that our radius is equal to 10 meters. When we plug 10 into our formula, and solve, we get 4,188.9 meters cubed.

Volume of a Cone

The formula for volume of a cone is very similar to the formula for the area of a circle. However, there are two things added to the formula.

To find the volume of a cone, you multiply times $13<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac></math>$ , and times the height because now you have a height (because now you are working with a three-dimensional shape). This makes the formula for the volume of a cone $V=13hπr2<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>h</mi><mi>π</mi><msup><mi>r</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></math>$ . Here is an example:

cone

Here we can see that we have a height of 5 cm and a radius of 2 cm. Once we plug in all of our numbers, we have $V=13(5cm)π(2)2<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo stretchy="false">(</mo><mn>5</mn><mi>c</mi><mi>m</mi><mo stretchy="false">)</mo><mi>π</mi><mo stretchy="false">(</mo><mn>2</mn><msup><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></math>$ . When solved, we have $V = 20.94 c m 3 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mn>20.94</mn><mi>c</mi><msup><mi>m</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></math>$ .

Great job, you guys. Learning new formulas can be hard. The important thing is to keep practicing so that you are able to recognize which formula you need to use and to memorize the formulas. I hope this was helpful. See you next time!

Frequently Asked Questions

Q

How do you find the volume of a triangular prism?

A

Find the volume of a triangular prism by multiplying the area of the base (the triangle part) by the height of the prism.
A = Bh
Ex. Find the volume of this triangular prism.

$V=Bh=12(7)(8)(12)=336m3<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi>B</mi><mi>h</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>12</mn><mo stretchy="false">)</mo><mo>=</mo><mn>336</mn><msup><mi>m</mi><mn>3</mn></msup></math>$

Q

How do you find the volume of a cube?

A

Find the volume of a cube by cubing its side length.
V=s³
Ex. What is the volume of a cube with a side length of 7 cm?
V = s³ = 7³ = 343 cm³

Q

How do you find the volume of a rectangular prism?

A

Find the volume of a rectangular prism by multiplying its length times its width times its height.
V = lwh
Ex. What is the volume of this rectangular prism?

V = 3 × 7 × 11 = 231 in³

Q

How do you find the volume of a sphere?

A

Find the volume of a sphere by cubing its radius and multiplying by $43<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>3</mn></mfrac></math>$ .
$V=43πr3<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>r</mi><mn>3</mn></msup></math>$
Ex. What is the volume of a sphere with radius 3 cm?
$V=43π(3)3=43π(27)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>π</mi><mo stretchy="false">(</mo><mn>3</mn><msup><mo stretchy="false">)</mo><mn>3</mn></msup><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>π</mi><mo stretchy="false">(</mo><mn>27</mn><mo stretchy="false">)</mo></math>$ $= 36 π c m 3 \approx 113.04 c m 3 <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>36</mn><mi>π</mi><mi>c</mi><msup><mi>m</mi><mn>3</mn></msup><mo>\approx</mo><mn>113.04</mn><mi>c</mi><msup><mi>m</mi><mn>3</mn></msup></math>$

Q

How do you find the volume of a cone?

A

Find the volume of a cone by multiplying 1/3 times the area of the base times the height, or multiplying 1/3 times pi (π) times the radius squared times the height.
$V=13Bh<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>B</mi><mi>h</mi></math>$ or $V=13πr2h<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup><mi>h</mi></math>$
Ex. What is the volume of this cone?

$V=13π(4)2(9)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>π</mi><mo stretchy="false">(</mo><mn>4</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">(</mo><mn>9</mn><mo stretchy="false">)</mo></math>$ $= 48 π i n 2 \approx 150.72 i n 2 <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>48</mn><mi>π</mi><mi>i</mi><msup><mi>n</mi><mn>2</mn></msup><mo>\approx</mo><mn>150.72</mn><mi>i</mi><msup><mi>n</mi><mn>2</mn></msup></math>$