Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Count three pairs of faces.
- Calculate capacity in cubic units.
- Remove the specified top face.
Before you start
Rectangular area and multiplying three numbers.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
An ideal rectangular box measures 4 cm long, 3 cm wide, and 2 cm high. Find its closed surface area, volume, and surface area without its top.
Why this math matters
Covering a box and filling a box require different quantities. A net reveals which faces exist, while stacked layers explain why volume multiplies three lengths.
Set up the model
A useful answer starts with clear assumptions:
- Faces are perpendicular rectangles with negligible thickness.
- The top and base both measure 4 cm by 3 cm.
- No overlap tabs, seams, or manufacturing allowance are included.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Separate a box's covering from its capacity
PausedQuestion: Start with the question. Paused.
Question
Start with the question
An ideal rectangular box measures 4 cm long, 3 cm wide, and 2 cm high. Find its closed surface area, volume, and surface area without its top.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Count paired faces
S = 2(4 × 3 + 4 × 2 + 3 × 2) = 52 cm²
The top matches the base, and opposite side faces match. Each of the three distinct areas occurs twice.
Stack the base area
V = (4 × 3) × 2 = 24 cm³
A 12-square-centimetre base extended through 2 centimetres gives a three-dimensional volume, not another surface area.
Remove one face
Sopen = 52 − 12 = 40 cm²
Only the top disappears. The base and all four walls remain in the open-container model.
The result
Closed surface area is 52 cm², volume is 24 cm³, and open surface area is 40 cm².
Removing a zero-thickness lid changes surface area without changing the dimensions of the available interior region.
Common mistakes to catch
- An open box loses one face, not both its top and base.
- Surface area uses square units; capacity uses cubic units.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Find surface area and volume of a closed 3 cm cube.
Show a hint
Use six equal faces.
Reveal answer and explanation
54 cm² and 27 cm³
Six faces of area 9 give 54; 3 × 3 × 3 gives 27.
Practice 2
A right prism has base area 15 cm² and volume 90 cm³. Find its height.
Show a hint
Divide volume by base area.
Reveal answer and explanation
6 cm
90 cm³ divided by 15 cm² leaves 6 cm.
Take the idea with you
Sketch a net whenever a packaging problem changes which faces are present.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Read a scale drawing without scaling area incorrectly
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