Mathematics
Grade 10
15 min
Volume of cubes and rectangular prisms
Volume of cubes and rectangular prisms
What you'll learn
- Identify the length, width, and height of cubes and rectangular prisms by measuring them with a ruler to the nearest quarter inch.
- Apply the formulas V = l x w x h and V = b x h to calculate the volume of at least 8 out of 10 given cubes and rectangular prisms.
- Solve word problems involving the volume of cubes and rectangular prisms, showing all work, with 70% accuracy.
- Explain the difference between area and volume using appropriate units (square units vs. cubic units).
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Calculate the distance between two points in a three-dimensional coordinate system.
Determine the edge lengths of a cube or rectangular prism given the coordinates of its vertices.
Identify if a prism defined by coordinates is a cube by comparing its edge lengths.
Apply the correct formula to calculate the volume of a rectangular prism (V = lwh).
Apply the correct formula to calculate the volume of a cube (V = s³).
Solve multi-step problems involving the volume of prisms defined within a 3D coordinate plane.
How can the GPS coordinates of a shipping container's corners tell us exactly how much it can hold? 📦
In this lesson, you will bridge the gap between 2D coordinate geometry and 3D shapes. We will use coordinates to define cubes and rectangular...
2
Key Concepts & Vocabulary
TermDefinitionExample
Rectangular PrismA three-dimensional solid shape which has six faces that are rectangles. It has three dimensions: length (l), width (w), and height (h).A shoebox with vertices at (0,0,0), (5,0,0), (0,3,0), and (0,0,2) defining its corner.
CubeA special type of rectangular prism where all six faces are squares and all edges have the same length (s).A die where the length, width, and height are all 2 cm.
3D Coordinate SystemA system used to locate points in three-dimensional space using an ordered triple of numbers (x, y, z), representing distances along the x-axis, y-axis, and z-axis.The point P(3, 4, 5) is located 3 units along the x-axis, 4 units along the y-axis, and 5 units along the z-axis from the origin (0,0,0).
Vertex (plural: Vertices)A point where two or mo...
3
Core Formulas
Distance Formula in 3D
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
Use this formula to calculate the length of a line segment (like an edge of a prism) between two points (vertices) P1(x1, y1, z1) and P2(x2, y2, z2) in a 3D coordinate system.
Volume of a Rectangular Prism
V = l \times w \times h
Once you have determined the lengths of the three unique, perpendicular edges (length, width, and height) from the vertices, multiply them together to find the volume.
Volume of a Cube
V = s^3
If all edge lengths are equal (s), the prism is a cube. Calculate the volume by cubing the side length. This is a special case of the rectangular prism formula where l = w = h = s.
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Challenging
A cube has two opposite vertices at A(1, 2, 3) and B(5, 6, 7). What is the volume of the cube?
A.64 cubic units
B.48 cubic units
C.16 cubic units
D.100 cubic units
Challenging
A rectangular prism has a volume of 192 cubic units. Its vertices are located at coordinates (x, y, z) where x can be 1 or 5, y can be -2 or 4, and z can be 3 or k, with k > 3. What is the value of k?
A.7
B.11
C.8
D.19
Challenging
A student is given vertices A(2,2,2), B(2,7,2), C(8,2,2), and D(2,2,9) for a rectangular prism. They calculate the lengths of AB, AC, and AD to find the volume. What is the fundamental error in this method?
A.Vertex D cannot be part of the same prism.
B.The vertices A, B, C, and D are not coplanar.
C.The segment AC is a face diagonal, not an edge of the prism.
D.The volume formula requires four lengths, not three.
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Frequently asked questions
What grade level is "Volume of cubes and rectangular prisms"?
Volume of cubes and rectangular prisms is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Volume of cubes and rectangular prisms?
You'll be able to: Identify the length, width, and height of cubes and rectangular prisms by measuring them with a ruler to the nearest quarter inch; Apply the formulas V = l x w x h and V = b x h to calculate the volume of at least 8 out of 10….
Is "Volume of cubes and rectangular prisms" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Volume of cubes and rectangular prisms?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.