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Volume is the amount of three-dimensional space an object occupies, in cubic units, within a container.

Volume and Unit Cubes:

Volume is measured in cubic units. Think in three-dimensions.

The volume of an object can be represented by
the number of unit cubes that can be placed within the object.

For some figures, the unit cubes fit "nicely" into the object,
while other objects hold fractional parts of a unit cube.
unitcube

pinkbox
The volume of this rectangular prism ("box") is the total of the number of unit cubes it holds. There are a total of 16 unit cubes within the solid.
The volume = 16 cubic units

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Volume as Area times Depth:

While counting the number of unit cubes within the rectangular prism,
a pattern can be seen that will help to determine the count of the cubes quickly.

Pattern: Find the number of cubes seen in one face and multiply times the number of rows of that face, going to the back of the figure.
     Volume = (cubes in face) • (rows going back)
                 
= 8 • 2 = 16 square units
pinkboxF

Pattern: A pattern similar to that shown above can be expressed using the term "area". The number of cube faces seen in the face of the figure, comprise the area of the face of the figure. The new pattern is expressed as:
     Volume = (area of face) • (depth of face)
                
 = (4 • 2) • 2 = 16 square units
pinkboxA

star This will be a popular strategy to determine the volume of many solids.
This pattern utilizes a "limiting argument" with the area being a 2-dimensional cross section with no thickness. The thickness, however, can be theoretically considered very, very, very small so as not to affect calculations. These cross sections are stacked to the height of the figure, creating the formula V = B • h, with B the area of the base and h = height.
(In the example about consider the area of the face to be the base, and the depth to be the height; just tip it over.)

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Surface area is the total area that the surface of a three-dimensional object occupies, in square units.

Surface Area using a Net:



pinkboxNet2

Finding the surface ares means finding the area of EVERY
face of this figure.

If you cut apart this box and flatten out the pieces, you will get a shape similar to the one at the right, called a net.
Several options are possible.

The advantage of examining the net is that you can see each of the faces of the figure, making computing the surface area easier.

pinkboxN
The surface area of this rectangular prism will be the sum of areas of all six shapes in the net.
         Surface Area = (2•2) + (2•4) + (2•4) + (2•4) + (2•4) + (2•2) = 40 square units.

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Formulas:



solid1p
Rectangular Solid (Prism)
solid1m
This formula assumes a "closed box", with all 6 sides.
prismform
Prism (all forms)
vbh
B = area of end face; h = height (depth)
SA = sum of all surface areas
(2 triangular end faces and 3 rectangular faces)
prismform
Pyramid
pyramidform
B = area of base; h = height
SA = sum of all surface areas
(1 base and all triangular faces)
solid2p
Cylinder
solid2m
This formula assumes a "closed container"
with a top and bottom.
solid3p
Sphere
solid3m
solid4p
Cone
solid4m
This formula assumes a "closed container", with a bottom.

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Working with different units:



1 square foot = 144 square inches
(a square foot is 1 ft. by 1 ft. which is also 12 in. by 12 in.)

1 cubic foot = 1728 cubic inches
(a cubic foot is 1 ft. by 1 ft. by 1 ft. which is also 12 in. by 12 in. by 12 in.)

1 cubic yard = 27 cubic feet
(a cubic yard is 1 yd. by 1 yd. by 1 yd. which is also 3 ft. by 3 ft. by 3 ft.)


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