Mathematics
Grade 10
15 min
Volume of irregular figures made of unit cubes
Volume of irregular figures made of unit cubes
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1
Introduction & Learning Objectives
Learning Objectives
Represent the base of a 3D figure made of unit cubes on a 2D coordinate plane.
Determine the volume of an irregular figure by counting the total number of unit cubes from a 2D 'top-down' view with height values.
Calculate the volume of an irregular figure by summing the volumes of its constituent vertical columns.
Deconstruct a complex irregular figure into simpler rectangular prisms to calculate its total volume.
Use coordinate points (x, y) to identify the location and height of cube stacks within a larger figure.
Justify their volume calculation method by explaining the one-to-one correspondence between a unit cube and a unit of volume.
Ever built a complex castle in a game like Minecraft and wondered about its total volume? 🏰 What if you on...
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Key Concepts & Vocabulary
TermDefinitionExample
Unit CubeA cube with side lengths of 1 unit, and thus a volume of 1 cubic unit. It is the fundamental building block for the figures in this lesson.A single block with dimensions 1cm x 1cm x 1cm has a volume of 1 cm³.
Irregular FigureA three-dimensional shape that is not a standard geometric solid (like a perfect cube, prism, or pyramid). It is composed of smaller, regular shapes like unit cubes.A figure shaped like the letter 'T' extruded into 3D.
VolumeThe amount of three-dimensional space occupied by an object, measured in cubic units.If a structure is made of 35 unit cubes, its volume is 35 cubic units.
Coordinate Plane (x-y plane)A two-dimensional plane formed by the intersection of a horizontal x-axis and a vertical y-axis. We use this to represent th...
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Core Formulas
Volume of a Unit Cube
V_{unit\_cube} = s^3 = 1^3 = 1 \text{ cubic unit}
This is the foundational principle. The total volume of any figure is simply the total count of the unit cubes it contains, as each one represents exactly one unit of volume.
Total Volume by Summation
V_{total} = \sum_{i=1}^{n} V_{cube\_i} = n \times (1 \text{ cubic unit})
Where 'n' is the total number of unit cubes. This rule states that the total volume of an irregular figure is found by counting every single unit cube that makes up the figure.
Volume from a Base Plan
V_{total} = \sum h(x, y)
When given a base plan on a coordinate plane, the total volume is the sum of all the height values, h, at each coordinate (x, y) that forms the base of the figure. Each h(x, y) represents the num...
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Challenging
A solid 4x4x3 rectangular prism is made of unit cubes. A vertical 2x1 hole is drilled all the way through the prism, centered at coordinates (2,3) and (3,3). What is the volume of the remaining figure?
A.48 cubic units
B.42 cubic units
C.45 cubic units
D.36 cubic units
Challenging
An artist combines two separate structures. Structure 1 is defined by h(1,1)=3 and h(1,2)=2. Structure 2 is defined by h(2,2)=4 and h(3,2)=1. If both structures are placed on the same coordinate plane to form a single new figure, what is its total volume?
A.10 cubic units
B.9 cubic units
C.11 cubic units
D.Cannot be determined as they might overlap
Challenging
An irregular figure is defined by the base plan h(1,1)=2, h(1,2)=4, h(2,1)=3. How many additional unit cubes must be added to this figure to form a solid 2x2x4 rectangular prism that completely encloses it?
A.16
B.9
C.7
D.4
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