TACHS: Area of combined shapes: subtract or split
Find the area of shapes built from rectangles by subtracting a cut-out or splitting into pieces, including borders, joined tiles and missing side lengths.
How do you find the area of a shape that is a rectangle with a piece cut out?
Short answer
Take the area of the full rectangle and subtract the area of each piece removed, or split the shape into rectangles that do not overlap and add them. Use one method to check the other.
Know first: Area of a rectangle, Reading side lengths from a sketch
Step 1 of 6
Two ways to the same area
Picture a 14 m by 9 m yard with a 5 m by 2 m patio cut out of one corner. The grass is shaped like a wide letter L.
You can subtract: the full rectangle minus the cut-out. Or you can split: cut the L into two rectangles that do not overlap and add their areas. Both give the same answer, so one checks the other.
| Shape | Method |
|---|---|
| Rectangle with a corner removed | Full area − corner area |
| Two strips removed that do not overlap | Full area − each strip |
| Two tiles joined along an edge | Add the two areas |
| A path of width w inside every edge | Inner rectangle: each side shrinks by 2w |
| Leftover area given, corner side missing | Full − leftover = corner area, then divide |
Only overlapping pieces cause trouble. Make pieces meet along a line, never share space.
Step 2 of 6
Subtract, then split to check
Worked example: A yard with a patio
A yard is a 14 m by 9 m rectangle. A 5 m by 2 m patio is removed from one corner, with the 5 m side along the 14 m edge. What area of grass is left?
- A46
- B116
- C126
- D136
- 1
Subtract
Full yard: 14 × 9 = 126. Patio: 5 × 2 = 10. Grass: 126 − 10 = 116 square meters.
- 2
Split to check
Below the patio is a strip 14 m by 9 − 2 = 7 m: 98. Beside the patio is a strip 14 − 5 = 9 m by 2 m: 18. Total: 98 + 18 = 116.
- 3
See why the other choices are there
126 forgets to remove the patio. 136 adds the patio instead of removing it. 46 is the perimeter of the full yard, not an area.
Answer
116 square meters of grass are left.
Wrong: A 12 m by 9 m garden has a 2 m path inside all four edges. Planted center: (12 − 2) × (9 − 2) = 70 square meters.
Right: Planted center: (12 − 4) × (9 − 4) = 8 × 5 = 40 square meters.
The path runs along both ends of each side, so each side loses 2 m twice. Subtracting 2 once leaves the path on two edges inside the planted area.
Step 3 of 6
Work backward to a missing side
Some questions give the leftover area and ask for a side of the missing piece. Subtract to find the piece's area, then divide by the side you know.
A 9 m by 7 m rectangle has area 63. If a corner is removed and 51 square meters are left, the corner's area is 63 − 51 = 12. If the corner is 3 m wide, it is 12 ÷ 3 = 4 m long.
Step 4 of 6
Pick a method, then check
Check yourself · Question 1
A rectangular lawn measures 16 m by 11 m. A 5 m by 4 m corner is paved. What area of lawn remains, in square meters?
Answer: B
Full lawn: 16 × 11 = 176. Paved corner: 5 × 4 = 20. Lawn left: 176 − 20 = 156.
Trap answer: 196
Adding the removed piece
- Why it looks right
- Both areas are in the question, and combining them by adding feels natural.
- Why it's wrong
- The corner is part of the full rectangle already. Removing it means the lawn gets smaller, so the answer must be less than 176.
- The right answer
- 156 square meters, because 176 − 20 = 156.
Check yourself · Question 2
A 17 m by 11 m garden has a 2 m wide path running inside all four edges. What is the area of the planted center, in square meters?
Answer: A
Each side loses 2 m at both ends. Center: (17 − 4) × (11 − 4) = 13 × 7 = 91 square meters.
Check yourself · Question 3
A 12 m by 8 m rectangle has a rectangular corner removed, leaving 75 square meters. The missing corner is 3 m wide. How many meters long is it?
Answer: B
Full rectangle: 12 × 8 = 96. Corner area: 96 − 75 = 21. Length: 21 ÷ 3 = 7 m.
Common mistake
"The path is 2 m wide, so the inside is 2 m shorter each way."
- Why it's tempting
- The path width appears once in the question, so subtracting it once seems right.
- Do this instead
- A border has two ends on every side. Subtract twice the width from each side, then multiply.
Step 5 of 6
What to remember
Remember
- Subtract: full rectangle minus each removed piece. Split: add rectangles that do not overlap.
- A border of width w inside every edge shrinks each side by 2w.
- To find a missing side, subtract areas first, then divide by the known side.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, what is the area of the full rectangle, and which wrong choice forgets to remove the corner?