Volume with Fractional Edges — Challenge
Volume still counts cubes — but when the cubes are smaller than one unit, counting is not the whole answer.
What these problems cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Area — Space inside, measured in square units.
- Volume — Space inside, in cubic units.
- Base — The number being multiplied. In 53 the base is 5.
- Dimensions — Rows × columns. A 2 × 3 matrix has two rows and three columns.
- Mixed number — A whole number with a fraction, like 212.
- Height — The perpendicular distance from the base to the opposite corner.
- Cubic unit — A cube one unit on every edge.
- Edge length — The measurement along one side.
Problem 1
What is the volume of a cube with side 12?
- 18
- 16
- 32
- 12
- 14
Worked answer
18
The volume of a cube is the side cubed.
(12)3 = 18, because the top and the bottom are each cubed.
Multiplying the side by 3 instead would give 32, which is a length and not a volume.
Problem 2
A rectangular prism has side lengths 1, 1, and 7, so its volume is 7. How many faces does it have?
- 4
- 5
- 6
- 7
- 8
Worked answer
6
A rectangular prism has three pairs of opposite rectangular faces.
Therefore it has 6 faces, regardless of the side lengths.
Problem 3
What is the volume of a box measuring 2 by 3 by 12?
- 1.5
- 3
- 6
- 12
- 0.5
Worked answer
3
Multiply the three dimensions: 2· 3·12.
2· 3 = 6, and half of 6 is 3 cubic units.
Problem 4
A solid consists of a rectangular prism measuring 4 by 3 by 2, with a 1 by 1 by 1 cube removed from one corner. What is its volume?
- 24
- 23
- 25
- 22
- 12
Worked answer
23
The prism has volume 4·3·2=24.
Subtract the removed cube volume 1: 24-1=23.
Problem 5
How many cubes of side 12 exactly fill a cube of side 1?
- 2
- 4
- 6
- 8
- 12
Worked answer
8
Two small cubes fit along each edge of the large one.
So the small cubes fill it 2 deep, 2 wide and 2 high: 2· 2· 2 = 8 cubes.
Checking by volume: each small cube is 18, and eight eighths make 1.
Problem 6
A right triangle has vertices (0,0), (3,0), and (0,4). It is rotated about the y-axis. What is the volume of the resulting solid?
- 16π
- 36π
- 48π
- 4π
- 12π
Worked answer
12π
The solid is a cone with radius 3 and height 4.
Its volume is 13π(3)2(4)=12π.
Problem 7
What is the volume of a box measuring 34 by 2 by 4?
- 4.5
- 6
- 8
- 9
- 3
Worked answer
6
Multiply the three dimensions: 34· 2· 4.
2· 4 = 8, and 34 of 8 is 6 cubic units.
Problem 8
A triangular pyramid has three mutually perpendicular edges of length 12 meeting at one vertex. What is its volume?
- 144
- 576
- 864
- 288
- 1728
Worked answer
288
Two edges form a right-triangle base of area 12(12)(12)=72.
The third perpendicular edge is the height, 12.
The volume is 13(72)(12)=288.