Subtract a cut-out and remove shared internal edges.
Core & Extended · Syllabus C5.2, C5.5, E5.2, E5.5
Notes and key methods
A compound plane figure can be split into nonoverlapping standard regions or calculated by removing a cut-out from a larger shape. Use the same units throughout. Areas add or subtract according to the occupied region.
For an external perimeter, walk around the complete outside boundary. Do not count an internal joining edge. Removing a rectangular corner from a rectangle changes the shape but leaves the perimeter unchanged when the two removed outer lengths are replaced by equal inner boundary lengths.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Compound areas and external perimeters
From the top-right corner of a 16 cm by 11 cm rectangle, remove a 5 cm wide by 2 cm high rectangular notch. The remaining figure is L-shaped.
Read the description and labels. Geometric sketches are schematic unless coordinate scales are shown.
Find the removed area in \(\displaystyle \mathrm{cm}^{2}\).
Find the remaining area in \(\displaystyle \mathrm{cm}^{2}\).
Find the length of the remaining top horizontal edge in cm.
Find the external perimeter of the L-shape in cm.
Show the reasoning
(a)
Multiply the two cut-out dimensions.
The cut-out rectangle has area \(\displaystyle 10 \mathrm{cm}^{2}\).
Answer: \(10\).
(b)
Subtract the nonoverlapping cut-out from the original rectangle.
Subtract 10 from 176 to obtain \(\displaystyle 166 \mathrm{cm}^{2}\).
Answer: \(166\).
(c)
Subtract the notch width from the original width.
The top edge loses width 5, leaving 11 cm.
Answer: \(11\).
(d)
Count each of the six boundary edges once, including the notch edges.
The two replacement notch edges equal the removed outer lengths, so the boundary remains 54 cm.
Answer: \(54\).
Example 2 · Compound areas and external perimeters
From the top-right corner of a 10 cm by 12 cm rectangle, remove a 2 cm wide by 3 cm high rectangular notch. The remaining figure is L-shaped.
Read the description and labels. Geometric sketches are schematic unless coordinate scales are shown.
Find the removed area in \(\displaystyle \mathrm{cm}^{2}\).
Find the remaining area in \(\displaystyle \mathrm{cm}^{2}\).
Find the length of the remaining top horizontal edge in cm.
Find the external perimeter of the L-shape in cm.
Show the reasoning
(a)
Multiply the two cut-out dimensions.
The cut-out rectangle has area \(\displaystyle 6 \mathrm{cm}^{2}\).
Answer: \(6\).
(b)
Subtract the nonoverlapping cut-out from the original rectangle.
Subtract 6 from 120 to obtain \(\displaystyle 114 \mathrm{cm}^{2}\).
Answer: \(114\).
(c)
Subtract the notch width from the original width.
The top edge loses width 2, leaving 8 cm.
Answer: \(8\).
(d)
Count each of the six boundary edges once, including the notch edges.
The two replacement notch edges equal the removed outer lengths, so the boundary remains 44 cm.
Answer: \(44\).
Example 3 · Compound areas and external perimeters
From the top-right corner of a 11 cm by 13 cm rectangle, remove a 3 cm wide by 4 cm high rectangular notch. The remaining figure is L-shaped.
Read the description and labels. Geometric sketches are schematic unless coordinate scales are shown.
Find the removed area in \(\displaystyle \mathrm{cm}^{2}\).
Find the remaining area in \(\displaystyle \mathrm{cm}^{2}\).
Find the length of the remaining top horizontal edge in cm.
Find the external perimeter of the L-shape in cm.
Show the reasoning
(a)
Multiply the two cut-out dimensions.
The cut-out rectangle has area \(\displaystyle 12 \mathrm{cm}^{2}\).
Answer: \(12\).
(b)
Subtract the nonoverlapping cut-out from the original rectangle.
Subtract 12 from 143 to obtain \(\displaystyle 131 \mathrm{cm}^{2}\).
Answer: \(131\).
(c)
Subtract the notch width from the original width.
The top edge loses width 3, leaving 8 cm.
Answer: \(8\).
(d)
Count each of the six boundary edges once, including the notch edges.
The two replacement notch edges equal the removed outer lengths, so the boundary remains 48 cm.
Answer: \(48\).
Common mistakes
Keep intermediate values exact until the final step.
