Perimeter — Part 2
Semester 4B · Chapter 11 · Lesson 5 · MATH-4B-C11-L05
Before you begin: prerequisite check
Recall: 11.04 and additive reasoning. Attempt these questions independently first.
A rectangle measures 12 cm by 7 cm. Find its perimeter.
Show answer and reasoning
Trace four outside sides: 12+7+12+7=38 cm.
A rectangle measures 9 cm by 7 cm. Find its perimeter.
Show answer and reasoning
Trace four outside sides: 9+7+9+7=32 cm.
A rectangle measures 10 cm by 3 cm. Find its perimeter.
Show answer and reasoning
Trace four outside sides: 10+3+10+3=26 cm.
Key vocabulary
- area
- region measured in square units
- perimeter
- distance around the outer boundary in linear units
- composite figure
- a shape assembled from simpler nonoverlapping regions
- dimension
- specified side length
Understand the concept
For a stepped or L-shaped figure, perimeter follows the outer boundary only. Do not include an interior line where two rectangles touch. Label every exposed horizontal and vertical side, then trace the perimeter in one direction to make sure each is counted once. Missing sides can be found by subtracting short lengths from a total width or height. For a rectangular corner cut out of a larger rectangle, consider which outer lengths disappear and which equal-length inner edges replace them before calculating.
Fully worked examples and immediate checks
Read each example, predict how to begin, then follow its numerical explanation. The check below it is a fresh problem.
Worked example 1
A 14-by-7 rectangle has a 4-by-1 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
The missing outer lengths 4 and 1 are replaced by inside-facing lengths of exactly 4 and 1. Total perimeter remains 2×(14+7)=42.
Check 1: try it yourself
A 12-by-6 rectangle has a 2-by-1 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 2 and 1 are replaced by inside-facing lengths of exactly 2 and 1. Total perimeter remains 2×(12+6)=36.
Worked example 2
A 9-by-8 rectangle has a 3-by-5 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
The missing outer lengths 3 and 5 are replaced by inside-facing lengths of exactly 3 and 5. Total perimeter remains 2×(9+8)=34.
Check 2: try it yourself
A 7-by-9 rectangle has a 1-by-6 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 1 and 6 are replaced by inside-facing lengths of exactly 1 and 6. Total perimeter remains 2×(7+9)=32.
Worked example 3
A 11-by-9 rectangle has a 6-by-3 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
The missing outer lengths 6 and 3 are replaced by inside-facing lengths of exactly 6 and 3. Total perimeter remains 2×(11+9)=40.
Check 3: try it yourself
A 12-by-10 rectangle has a 5-by-7 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 5 and 7 are replaced by inside-facing lengths of exactly 5 and 7. Total perimeter remains 2×(12+10)=44.
Guided practice
A 8-by-6 rectangle has a 3-by-1 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 3 and 1 are replaced by inside-facing lengths of exactly 3 and 1. Total perimeter remains 2×(8+6)=28.
A 9-by-7 rectangle has a 1-by-3 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 1 and 3 are replaced by inside-facing lengths of exactly 1 and 3. Total perimeter remains 2×(9+7)=32.
A 11-by-9 rectangle has a 8-by-1 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 8 and 1 are replaced by inside-facing lengths of exactly 8 and 1. Total perimeter remains 2×(11+9)=40.
A 11-by-7 rectangle has a 8-by-3 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 8 and 3 are replaced by inside-facing lengths of exactly 8 and 3. Total perimeter remains 2×(11+7)=36.
A 13-by-7 rectangle has a 1-by-2 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 1 and 2 are replaced by inside-facing lengths of exactly 1 and 2. Total perimeter remains 2×(13+7)=40.
Independent practice
A 7-by-5 rectangle has a 5-by-3 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 5 and 3 are replaced by inside-facing lengths of exactly 5 and 3. Total perimeter remains 2×(7+5)=24.
A 8-by-9 rectangle has a 3-by-3 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 3 and 3 are replaced by inside-facing lengths of exactly 3 and 3. Total perimeter remains 2×(8+9)=34.
A 13-by-8 rectangle has a 9-by-2 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 9 and 2 are replaced by inside-facing lengths of exactly 9 and 2. Total perimeter remains 2×(13+8)=42.
A 8-by-8 rectangle has a 4-by-2 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 4 and 2 are replaced by inside-facing lengths of exactly 4 and 2. Total perimeter remains 2×(8+8)=32.
A 9-by-7 rectangle has a 2-by-2 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 2 and 2 are replaced by inside-facing lengths of exactly 2 and 2. Total perimeter remains 2×(9+7)=32.
A 9-by-9 rectangle has a 2-by-5 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 2 and 5 are replaced by inside-facing lengths of exactly 2 and 5. Total perimeter remains 2×(9+9)=36.
A 8-by-6 rectangle has a 2-by-2 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 2 and 2 are replaced by inside-facing lengths of exactly 2 and 2. Total perimeter remains 2×(8+6)=28.
A 12-by-8 rectangle has a 3-by-2 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 3 and 2 are replaced by inside-facing lengths of exactly 3 and 2. Total perimeter remains 2×(12+8)=40.
Challenge and transfer
Use a diagram or verbal justification where appropriate. The goal is transfer rather than speed.
A 14-by-8 rectangle has a 5-by-1 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure. Explain why your units and method are appropriate.
Show answer and reasoning
The missing outer lengths 5 and 1 are replaced by inside-facing lengths of exactly 5 and 1. Total perimeter remains 2×(14+8)=44. For area count square units; for perimeter trace only the outside boundary.
A learner claims: “For an 8-by-6 rectangle with a 2-by-3 corner cut away, the perimeter must decrease by 5 units.” Identify what is wrong or incomplete and give a mathematically correct explanation.
Show answer and reasoning
False: the removed two outer edges are replaced by two inner edges of the same lengths; the perimeter stays 28 units. Explain the relevant quantities or properties, not merely the final answer.
Exit check
A 14-by-5 rectangle has a 10-by-1 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 10 and 1 are replaced by inside-facing lengths of exactly 10 and 1. Total perimeter remains 2×(14+5)=38.
A 12-by-5 rectangle has a 6-by-3 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 6 and 3 are replaced by inside-facing lengths of exactly 6 and 3. Total perimeter remains 2×(12+5)=34.
A 11-by-6 rectangle has a 2-by-2 corner notch removed (the notch touches the top-right corner). Find the perimeter of the L-shaped figure.
Show answer and reasoning
The missing outer lengths 2 and 2 are replaced by inside-facing lengths of exactly 2 and 2. Total perimeter remains 2×(11+6)=34.
Mastery check: Solve a new instance without hints, explain why the method works, and return later to confirm retention.