Volume of 3D Lesson

volume of 3d objects

When we think of dimensions and geometry, we can think in terms of measurements and exponents.

A line is the first dimension and might be measured with cm or in, area of a figure might be measured in cm² or in², volume of a 3d object might be measured in cm³ or in³.

Notice the exponent matches the dimension.

How much liquid could these 3d object hold?

The volume is how we find out the answer to this question.

For most 3d objects, the volume is found by finding the area of the base and multiplying that by the height.

Some exceptions are pyramids, cones, and spheres.

We will use this as the base of a cylinder.

area of the green circle was A=πr²

Notice the area of a circle is in the formula for volume of a cylinder

we take the area of a circle and multiply it by the height of the cylinder to get the volume

a=πr² and v=πr²h

What information would we need to find the volume of this cylinder?

The radius is 8 and the height is 15.

We can use that info to substitute those values into our formula.

V=πr²H

V=π*8²*15

Multiple Choice

What is the volume(πr²h) of a cylinder with radius 3 and height 10?

• 90π

• 30π

• 13π

• 7π

Notice the volume of a cone is almost the same as a cylinder. But a cone converges to a point(called a vertex).

We use the same formula for volume as the cylinder but we multiply by 1/3 or divide by 3(same thing) to find the volume of a cone.

V=(π*r²*h) /3

Multiple Choice

What is the volume[(π*r²*H)/3] of a cone with radius 3 and height 7?

• 21π

• 63π

• 7π

• 10π

spheres have a radius just like circles

Notice the formula has a fraction bigger than 1 and the radius has an exponent of 3 now.

V=(4*π*r³) / 3

The radius of a sphere is from the center of the sphere to a point on the outside of the sphere.

What information would we need to find the volume of this sphere?

The radius is 6. That is all we need to substitute for R in our formula.

V=(4*π*r³) / 3

V=4*π*6³ / 3

Multiple Choice

What is the volume[(4*π*r³)/3] of a sphere with radius 3?

• 36π

• 12π

• 9π

• 27π

remember a square is a rectangle

area of this base is L*W

Notice the area of a rectangle is in the formula for volume of a rectangular prism

we take the area of a rectangle and multiply it by the height of the prism to get the volume

a=L*W and v=L*W*H

What information would we need to find the volume of this rectangular prism?

Width=4 Length=4 Height=6

We can use that info to substitute those values into our formula.

V=W*L*H

V=4*4*6

Multiple Choice

What is the volume(L*W*H) of this rectangular prism with width 2 and length 7 and height 4?

• 56

• 28

• 14

• 8

we will use this as the base of a triangular prism.

area of the triangle is (b*h) / 2

Notice the area of a triangle is in the formula for volume of a triangular prism

we take the area of a triangle and multiply it by the length of the prism to get the volume

a=(B*H)/2 and v=(L*B*H) / 2

What information would we need to find the volume of this triangular prism?

Length=15 Height=11 base=13

We can use that info to substitute those values into our formula.

V=(L*B*H) / 2

V=(11*13*15) / 2

Hint: you can divide by 2 early to make bigger numbers smaller if you do not have a calculator.

Multiple Choice

What is the volume[(B*H*L)/2)] of a triangular prism with base 3 and height 6 and length 8?

• 72

• 144

• 18

• 24

Notice the volume of a pyramid is almost the same as a prism. The base of the pyramid determines its name, just like with prisms.

The B is this formula stands for area of the base.

We use the same formula for volume as the prism but we multiply by 1/3 or divide by 3(same thing) due to the object converging at a single vertex.

Volume of a rectangular pyramid
V=(L*W*H) / 3
Volume of a triangular pyramid

 $\frac{1}{3}\cdot\frac{1}{2}\cdot B\cdot H\cdot L\$  

a triangular pyramid volume formula can be simplified to 
(B*H*L) / 6 

Multiple Choice

What is the volume[(L*W*H)/3] of a rectangular pyramid with width 8 and length 10 and height 6?

• 160

• 480

• 48

• 80

Multiple Choice

What is the volume[(B*H*L) /6] of a triangular pyramid with base 3 and height 4 and length 4?

• 8

• 48

• 16

• 12

You should now be able to find the volume of all these 3d objects.

Did you get all the quiz questions right?