HSPT: Area: subtract the part that is cut out
Learn to find the area left over when one rectangle sits inside another, including frames and paths.
How do you find the area that is left when part of a rectangle is covered or cut out?
Short answer
Find the area of the outer rectangle and the area of the inner one, then subtract. Use area, not perimeter.
Know first: Area of a rectangle, Square units
Step 1 of 5
Find the big area, then take away the hole
When a shape has a piece cut out of it, such as a lawn with a pond or a wall with a window, the area that is left is the outer area minus the inner area.
Area counts square units, so work with length × width for each rectangle. The perimeter, the distance around, doesn't measure how much surface is covered and won't help here.
| Setup | Outer shape | Inner shape |
|---|---|---|
| Garden with a pond | The garden | The pond |
| Floor with a rug | The floor | The rug |
| Picture in a frame | The frame's outside edge | The picture |
| Pool with a path around it | Pool plus path | The pool |
In each case, the leftover area is outer area minus inner area.
Step 2 of 5
Subtract the inner rectangle
Worked example: A patio with a planter
A 15 m by 6 m patio has a 4 m by 3 m planter on it. How many square meters of patio are not covered by the planter?
- A78
- B102
- C90
- D28
- 1
Outer area
15 × 6 = 90 m².
- 2
Inner area
4 × 3 = 12 m².
- 3
Subtract
90 − 12 = 78 m².
- 4
See why the other choices are there
102 adds the areas instead of subtracting. 90 is the whole patio, with nothing taken away. 28 subtracts the perimeters, 42 − 14, which measure edges, not surface.
Answer
78 m²
Find the wrong step
A frame is 12 inches by 9 inches on the outside. The picture inside it is 9 inches by 6 inches. A student finds the area of the frame itself.
- 1
Subtracts the dimensions first: 12 − 9 = 3 and 9 − 6 = 3, then multiplies to get 9 square inches.
What went wrong
Subtracting lengths doesn't give the area of a border. The border wraps all the way around the picture, so it is much bigger than a 3 by 3 square.
- 2
Labels the answer in square inches.
- 3
Answers 9 square inches.
The fix
Outer area minus inner area: 12 × 9 = 108 and 9 × 6 = 54, so the frame covers 108 − 54 = 54 square inches.
Step 3 of 5
Outer area minus inner area
Check yourself · Question 1
A 9 ft by 7 ft rug lies on a 15 ft by 11 ft floor. How many square feet of floor are not covered?
Answer: B
Floor: 15 × 11 = 165 ft². Rug: 9 × 7 = 63 ft². Uncovered: 165 − 63 = 102 ft².
Trap answer: 20
Perimeter instead of area
- Why it looks right
- Perimeter is another way to measure a rectangle, and subtracting looks like the right move.
- Why it's wrong
- Perimeter measures the length of the edges. Floor space is area, in square feet.
- The right answer
- 102, because 165 − 63 = 102.
Check yourself · Question 2
A 12 m by 5 m pool has a path 1 m wide all the way around it. What is the area of the path?
Answer: B
The path adds 1 m on every side, so the outer rectangle is 12 + 2 = 14 m by 5 + 2 = 7 m, with area 98 m². The pool is 12 × 5 = 60 m². The path is 98 − 60 = 38 m².
Common mistake
"I subtract the lengths and widths, then multiply."
- Why it's tempting
- It looks like a shortcut that handles the outer and inner shapes at once.
- Do this instead
- Find each area on its own, then subtract the areas. Subtracting dimensions gives a much smaller, wrong region.
Step 4 of 5
What to remember
Remember
- Leftover area is outer area minus inner area.
- Multiply length by width for each rectangle before you subtract.
- A path or frame that goes all the way around adds twice its width to each dimension.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
What were the outer and inner areas in the practice question, and which wrong choice comes from using perimeter?