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
- Represent a cut-out as subtraction.
- Check an answer with a disjoint partition.
- Keep lengths and areas distinct.
Before you start
Rectangular area, multiplication, and subtraction.
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
A flat garden diagram begins as an 8 m by 6 m rectangle. A 3 m by 2 m rectangle is removed from its upper-right corner. What area remains?
Why this math matters
Rooms, drawings, and planting layouts often contain notches. You can measure their areas without inventing a new formula for each outline. A large bounding rectangle minus the missing part is often easier than adding many small strips.
Set up the model
A useful answer starts with clear assumptions:
- All corners are right angles and both removed dimensions are measured along the original edges.
- The removed corner is completely inside the original rectangle.
- Measurements describe a plane region; no material allowance is 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.
Break an L-shaped region into simple areas
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A flat garden diagram begins as an 8 m by 6 m rectangle. A 3 m by 2 m rectangle is removed from its upper-right corner. What area remains?
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.
Find the bounding area
Aouter = 8 × 6 = 48 m²
Temporarily fill the notch. The full rectangle contains every part of the garden plus the piece that will be removed.
Subtract the cut-out
Aremoved = 3 × 2 = 6 m²; A = 48 − 6 = 42 m²
The cut-out is an area, so both quantities use square metres. Removing two lengths from 48 would not represent this operation.
Check by adding two regions
8 × 4 + 5 × 2 = 32 + 10 = 42 m²
A bottom strip of height 4 and upper-left strip of width 5 cover the garden without overlapping. Their shared edge has no area.
The result
The remaining garden area is 42 m².
Two independent decompositions agree. The internal partition lines are bookkeeping devices, not additional boundaries to include in an outside-perimeter calculation. Their positions can change without changing the region's area.
Common mistakes to catch
- Adding the original rectangle and the two strips counts the region more than once.
- Subtracting 3 + 2 from 48 mixes lengths with an area.
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
Remove a 2 m by 3 m corner from a 7 m by 5 m rectangle. Find the remaining area.
Show a hint
Subtract the two rectangular areas.
Reveal answer and explanation
29 m²
The outer area is 35 and the cut-out is 6, giving 29 m².
Practice 2
An 8 m by 6 m rectangle loses 12 m² through a rectangular corner cut-out 3 m wide. Find the cut-out height.
Show a hint
Divide the missing area by its width.
Reveal answer and explanation
4 m
12 m² ÷ 3 m = 4 m, which fits within the 6 m outer height.
Take the idea with you
Draw a bounding shape and label holes before estimating the area of any irregular rectilinear plan.
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: Measure a circular fan's edge and surface
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