Volume measures the amount of space inside a three-dimensional figure, in cubic units such as cubic centimeters or cubic feet. You can find it by counting the unit cubes that fill the solid. For a rectangular prism, V = l × w × h; for any prism, it equals the area of the base multiplied by the height.
Volume counts cubes. One layer of this box is 4 × 3 = 12 cubes, and there are 2 layers, so V = 24 cubic units. That is why V = l × w × h works.
Formal definition and notation
The volume of a solid is the number of unit cubes (cubes with edge 1) needed to fill it without gaps or overlaps, extended to fractional amounts. A unit cube has volume 1 cubic unit. For a right prism or cylinder with base area B and heighth, V = Bh; a pyramid or cone with the same base and height has V = 13Bh; a sphere of radiusr has V = 43π r3.
Symbol
Read it as
Meaning
V
V
volume
cm3
cubic centimeters
a cubic unit: a cube 1 cm on each edge
B
capital B
the area of the base of a prism, cylinder, pyramid or cone
Key facts
Volume uses cubic units, not square units: in3, ft3, m3.
Surface area is the total area of the outside faces, in square units; volume is the space inside, in cubic units. A 2 by 2 by 2 cube has surface area 24 square units and volume 8 cubic units.
Area measures a flat region (square units). Volume measures a solid (cubic units).
Volumevs.Capacity
Capacity is the volume a container can hold, often given in liters, cups or gallons instead of cubic units.
Quick check: 3 questions
Question 1 of 3
What is the volume of a cube with edges of 5 inches?
Show answer and explanation
Answer: C.V = s3 = 5 × 5 × 5 = 125. Choice A is the area of one face, B is the surface area (6 × 25), and D adds the edges.
Question 2 of 3
Which unit could measure the volume of a swimming pool?
Show answer and explanation
Answer: D. Volume uses cubic units, such as cubic meters. Meters measure length, square meters measure area, and kilograms measure mass.
Question 3 of 3
A prism has a base area of 18 square feet and a height of 7 feet. What is its volume?
Show answer and explanation
Answer: A. For any prism, V = Bh = 18 × 7 = 126 cubic feet. Choice B adds, C doubles the answer, and D treats it like a pyramid (13Bh).
Volume: frequently asked questions
What is the difference between volume and capacity?
Volume is the amount of space a solid takes up, usually in cubic units such as cm3. Capacity is how much a container can hold, usually in liters, milliliters, cups or gallons. They measure the same idea: 1 milliliter equals 1 cubic centimeter, so a 1,000 cm3 box holds 1 liter.
What is the difference between volume and surface area?
Volume is the space inside a solid, measured in cubic units. Surface area is the total area of its outside faces, measured in square units. To fill a box with sand you need its volume; to wrap it in paper you need its surface area. A 3 by 3 by 3 cube has volume 27 and surface area 54.
How do you find the volume of an irregular object?
Split it into simpler solids, find each volume and add them. For a real object, use water: put it in a measuring container and read how much the water level rises. The rise in volume equals the object’s volume, because the object pushes aside its own volume of water.
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