Cuboids
Semester 4B · Chapter 17 · Lesson 1 · MATH-4B-C17-L01
Before you begin: prerequisite check
Recall: 3D shape vocabulary. Attempt these questions independently first.
How many sides does a rectangle have?
Show answer and reasoning
A rectangle is a quadrilateral.
What are two-dimensional faces of a box shaped like?
Show answer and reasoning
A rectangular box is bounded by rectangles; a cube is a special case with squares.
What do we call a line segment where two faces meet?
Show answer and reasoning
Two faces of a solid meet along an edge.
Key vocabulary
- cuboid
- solid with six rectangular faces
- face
- flat rectangular side of a cuboid
- edge
- segment where two faces meet
- vertex
- point where edges meet
- net
- connected plane arrangement of faces that folds into a solid
Understand the concept
A cuboid is a three-dimensional solid with six rectangular faces, twelve edges, and eight vertices. The three perpendicular dimensions may be called length, width, and height. Opposite faces have equal dimensions and are parallel. A cube is a special cuboid with all edges equal, so its faces are squares. A rectangular sketch is a two-dimensional representation; identify which edges belong to the front, back, and depth before counting.
Fully worked examples and immediate checks
Read each example, predict how to begin, then follow its numerical explanation. The check below it is a fresh problem.
Worked example 1
A 13 cm by 14 cm by 4 cm cuboid has how many faces, edges, and vertices?
Count six rectangular faces, twelve meeting segments and eight corner points.
Check 1: try it yourself
A 12 cm by 15 cm by 5 cm cuboid has how many faces, edges, and vertices?
Show answer and reasoning
Count six rectangular faces, twelve meeting segments and eight corner points.
Worked example 2
A 9-cm tall cuboid has 6 faces. How many edges must it have, regardless of side lengths?
A cuboid always has twelve edges: four in each of three directions.
Check 2: try it yourself
A 11-cm tall cuboid has 6 faces. How many edges must it have, regardless of side lengths?
Show answer and reasoning
A cuboid always has twelve edges: four in each of three directions.
Worked example 3
Does a 11-cm long rectangular box have 8 or 12 vertices? Explain.
A vertex is a corner; there are four corners on each of two opposite rectangular ends.
Check 3: try it yourself
Does a 7-cm long rectangular box have 8 or 12 vertices? Explain.
Show answer and reasoning
A vertex is a corner; there are four corners on each of two opposite rectangular ends.
Guided practice
A 14 cm by 16 cm by 6 cm cuboid has how many faces, edges, and vertices?
Show answer and reasoning
Count six rectangular faces, twelve meeting segments and eight corner points.
If a box has three pairs of congruent rectangular faces and height 18 cm, how many faces are there altogether?
Show answer and reasoning
Two of each of three rectangle types gives six faces.
Does a 4-cm long rectangular box have 8 or 12 vertices? Explain.
Show answer and reasoning
A vertex is a corner; there are four corners on each of two opposite rectangular ends.
If a box has three pairs of congruent rectangular faces and height 17 cm, how many faces are there altogether?
Show answer and reasoning
Two of each of three rectangle types gives six faces.
Describe the number and sizes of faces of a 2-by-3-by-10 cuboid.
Show answer and reasoning
Each opposite-face pair is congruent, so the solid has six faces.
Independent practice
A 8 cm by 18 cm by 21 cm cuboid has how many faces, edges, and vertices?
Show answer and reasoning
Count six rectangular faces, twelve meeting segments and eight corner points.
A 5-cm tall cuboid has 6 faces. How many edges must it have, regardless of side lengths?
Show answer and reasoning
A cuboid always has twelve edges: four in each of three directions.
Describe the number and sizes of faces of a 6-by-8-by-6 cuboid.
Show answer and reasoning
Each opposite-face pair is congruent, so the solid has six faces.
If a box has three pairs of congruent rectangular faces and height 5 cm, how many faces are there altogether?
Show answer and reasoning
Two of each of three rectangle types gives six faces.
Describe the number and sizes of faces of a 7-by-7-by-9 cuboid.
Show answer and reasoning
Each opposite-face pair is congruent, so the solid has six faces.
A 12 cm by 13 cm by 14 cm cuboid has how many faces, edges, and vertices?
Show answer and reasoning
Count six rectangular faces, twelve meeting segments and eight corner points.
If a box has three pairs of congruent rectangular faces and height 12 cm, how many faces are there altogether?
Show answer and reasoning
Two of each of three rectangle types gives six faces.
Does a 3-cm long rectangular box have 8 or 12 vertices? Explain.
Show answer and reasoning
A vertex is a corner; there are four corners on each of two opposite rectangular ends.
Challenge and transfer
Use a diagram or verbal justification where appropriate. The goal is transfer rather than speed.
A 14 cm by 6 cm by 17 cm cuboid has how many faces, edges, and vertices? Explain or sketch the mathematical relationship.
Show answer and reasoning
Count six rectangular faces, twelve meeting segments and eight corner points. State which quantities or geometric properties justify the method.
A learner claims: “A cuboid has 12 corners and 8 edges.” Identify what is wrong or incomplete and give a mathematically correct explanation.
Show answer and reasoning
False: a cuboid has 8 vertices/corners and 12 edges. Explain the relevant quantities or properties, not merely the final answer.
Exit check
A 4 cm by 4 cm by 14 cm cuboid has how many faces, edges, and vertices?
Show answer and reasoning
Count six rectangular faces, twelve meeting segments and eight corner points.
A 12-cm tall cuboid has 6 faces. How many edges must it have, regardless of side lengths?
Show answer and reasoning
A cuboid always has twelve edges: four in each of three directions.
Does a 14-cm long rectangular box have 8 or 12 vertices? Explain.
Show answer and reasoning
A vertex is a corner; there are four corners on each of two opposite rectangular ends.
Mastery check: Solve a new instance without hints, explain why the method works, and return later to confirm retention.