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Perimeter of L-Shapes: Find Missing Sides, Then Add

Find the perimeter of L-shapes and other composite figures: work out missing sides from opposite sides, add every edge once and compare perimeter with area.

Grades 3–5 · 30 practice questions · Original material
Learning objectiveFind unlabelled side lengths of composite shapes made of rectangles by using opposite sides, add every outside edge exactly once to find the perimeter, leave out inside cut lines, write perimeter in length units, and explain how a shape's perimeter and area can change differently.

The idea

Perimeter is the distance all the way around the outside of a shape. Imagine an ant walking along every edge once and coming back to where it started. In the last lesson we found the area of L-shapes by covering them with squares. Today we walk around their edges instead.

A rectangle has four sides, but a composite shape made of rectangles has more. An L-shape has six sides, a T-shape has eight and a U-shape has eight. Before adding, count the sides so you know how many numbers your total should include.

Drawings often leave some sides unlabelled. When every corner is a square corner, opposite sides balance: the edges that face up add up to the same total as the edges that face down, and the edges that face left add up to the edges that face right. In an L-shape, the top edge plus the step's flat edge equals the bottom edge, and the right edge plus the step's upright edge equals the left edge.

Only outside edges count. If you drew a cut line inside the shape to find its area, that line is not part of the perimeter, because the ant never walks across the middle. Counting the squares inside does not give the perimeter either; that is the area.

The same shape has two different measures. The 7 cm by 5 cm L-shape with a 3 cm by 2 cm corner missing has an area of 29 cm² but a perimeter of 24 cm. Perimeter is a length, so it uses cm or m, not cm² or m². Here is a surprise to test: cutting a corner off a rectangle made the area smaller, but the perimeter stayed the same as the full rectangle's. A notch cut into the middle of a side is different: it adds two extra sides.

Worked examples

Finding the perimeter of composite shapesThree panels. Panel 1: an L-shape 7 cm wide and 5 cm tall. The top edge is 4 cm and the right edge is 3 cm. The two unlabelled step edges are found from opposite sides: 7 minus 4 is 3 cm across, and 5 minus 3 is 2 cm down. Adding all six sides, 7 plus 5 plus 4 plus 2 plus 3 plus 3, gives 24 cm. Panel 2: the same L-shape with a dashed cut line inside it. The cut line is inside the shape, so it is not part of the perimeter. Sliding the two step edges outward makes the full 7 by 5 rectangle, whose perimeter is also 24 cm, while the area is 29 square centimetres. Panel 3: a U-shape 8 cm wide and 5 cm tall with a notch 2 cm wide and 3 cm deep cut down from the middle of the top. The notch adds two 3 cm sides, so the perimeter is 26 plus 3 plus 3, which is 32 cm.Perimeter: find the missing sides, then go all the way aroundBlue = a side you work out. Red dashed = an inside cut line, not part of the perimeter. Original schematic.1 · Find missing sides, add all 64 cm5 cm7 cm3 cm? = 7 − 4 = 3? = 5 − 3 = 22 · Cut lines are not edgescut line:not perimeter7 cm5 cmslide out3 · A notch adds length8 cm5 cm2 cm3 cm3 cm3 cmdeep7+5+4+2+3+3 = 24 cm7+5+7+5 = 24 cm26 + 3 + 3 = 32 cmSame L-shape, two measures: perimeter 24 cm (around), area 29 cm² (inside).Opposite sides balance: the top edges add up to the bottom edge, and the right-hand edges add up to the left edge.
Original schematic, drawn to the labelled centimetre lengths. Unlabelled sides are worked out from opposite sides, then every outside edge is added once. A cut line inside the shape is not part of the perimeter. Cutting a corner off a rectangle leaves the perimeter unchanged, but a notch in the middle of a side adds two extra sides.

Example 1 · Find the missing sides

An L-shape is 7 cm wide along the bottom and 5 cm tall on the left. Its top edge is 4 cm and its right edge is 3 cm. Two edges of the step are not labelled. How long are they?

  1. The flat edge of the step and the top edge together stretch as far as the bottom edge: 7 − 4 = 3 cm.
  2. The upright edge of the step and the right edge together are as tall as the left edge: 5 − 3 = 2 cm.
  3. Check: going across the top, 4 + 3 = 7, which matches the bottom. Going down the right, 2 + 3 = 5, which matches the left.

The step edges are 3 cm across and 2 cm down.

Example 2 · Add every side once

Find the perimeter of the same L-shape.

  1. An L-shape has 6 sides. List them going around: 4 cm, 2 cm, 3 cm, 3 cm, 7 cm, 5 cm.
  2. Tick each side as you add it so none is missed or counted twice.
  3. 4 + 2 + 3 + 3 + 7 + 5 = 24 cm.

The perimeter is 24 cm.

Example 3 · Why the L-shape matches the rectangle

Compare the L-shape with the full 7 cm by 5 cm rectangle around it.

  1. The rectangle's perimeter is 7 + 5 + 7 + 5 = 24 cm.
  2. Slide the step's 3 cm edge up to the top and its 2 cm edge out to the right. Together with the 4 cm and 3 cm edges they make the full rectangle's top and right side.
  3. So the L-shape and the rectangle have the same perimeter, 24 cm, even though the L-shape's area is smaller (29 cm² instead of 35 cm²).

Same perimeter (24 cm), different area. Cutting a corner off a rectangle does not change its perimeter.

Example 4 · A notch adds length

A U-shape fits in an 8 cm by 5 cm rectangle. A notch 2 cm wide and 3 cm deep is cut down from the middle of the top, leaving 3 cm of top edge on each side. Find the perimeter.

  1. Count the sides: the U-shape has 8.
  2. Go around: 3 (top left) + 3 (down into the notch) + 2 (notch bottom) + 3 (back up) + 3 (top right) + 5 (right) + 8 (bottom) + 5 (left).
  3. Total: 3 + 3 + 2 + 3 + 3 + 5 + 8 + 5 = 32 cm.
  4. Compare: the rectangle is 26 cm around. The notch adds its two 3 cm sides, 26 + 3 + 3 = 32 cm.

32 cm. A notch in the middle of a side makes the perimeter longer than the rectangle's.

Example 5 · Leave out the cut line

To find the area of the 7 cm by 5 cm L-shape, Kai cut straight across it with a 4 cm line. He then added that line to the perimeter: 24 + 4 = 28 cm. What went wrong?

  1. The cut line is inside the shape. Walking around the outside never crosses it.
  2. The cut helps with area (7 × 3 + 4 × 2 = 29 cm²), but it is not an edge of the shape.
  3. The perimeter is still 24 cm.

Inside cut lines are not part of the perimeter. The perimeter is 24 cm.

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Counting the squares inside the shape.

Why this fails: Counting squares measures area, the space inside. Perimeter adds the lengths of the outside edges.

Forgetting a side, especially an unlabelled one.

Why this fails: Count the sides first (an L-shape has 6) and tick each one as you add it. Unlabelled sides still need to be worked out and added.

Copying the wrong side for a missing length.

Why this fails: The step's upright edge is the left edge minus the right edge, not the same as the left edge. Check that the edges facing each way balance.

Adding a cut line that is inside the shape.

Why this fails: A cut line is a tool for finding area. The perimeter only includes edges on the outside boundary.

Mixing up area and perimeter, or their units.

Why this fails: Perimeter is a length (cm, m). Area counts squares (cm², m²). A shape can keep the same perimeter while its area changes.

Foundation

Foundation practice

Count the sides, find a missing side and add every edge once.

Rectangle

1. A rectangle on centimetre grid paper is 6 cm long and 4 cm wide. What is its perimeter?

  1. 24 cm
  2. 20 cm
  3. 10 cm
  4. 20 cm²
Count sides

2. How many sides does an L-shape made of two rectangles have?

  1. 4
  2. 5
  3. 6
  4. 8
Missing side

3. An L-shape is 8 m tall on the left. Its right edge is only 3 m tall. How long is the upright edge of the step?

  1. 5 m
  2. 11 m
  3. 3 m
  4. 8 m
Add the sides

4. The six sides of an L-shape are 9 cm, 6 cm, 5 cm, 2 cm, 4 cm and 4 cm. What is its perimeter?

  1. 26 cm
  2. 34 cm
  3. 25 cm
  4. 30 cm
Choose the measure

5. Which job needs the perimeter of an L-shaped garden?

  1. Covering the garden with soil
  2. Counting the grid squares inside the garden plan
  3. Spreading grass seed over the garden
  4. Putting a fence around the garden
Units

6. The same L-shape has a perimeter of 24 and an area of 29. Which pair has the correct units?

  1. 24 cm and 29 cm
  2. 24 cm² and 29 cm²
  3. 24 cm and 29 cm²
  4. 24 cm² and 29 cm
Cut line

7. To find the area of an L-shape, a student drew a dashed line across the inside to split it into two rectangles. Should the dashed line be added to the perimeter?

  1. Yes, every line in the drawing counts.
  2. No, because it is inside the shape, not on the outside edge.
  3. Yes, but only if it is the longest line.
  4. Only if the cut goes across, not up and down.
Missing side

8. The bottom edge of an L-shape is 12 m. The top edge, directly above it, is 7 m. How long is the flat edge of the step?

  1. 5 m
  2. 19 m
  3. 7 m
  4. 84 m

Practice

Practice practice

Work out the unlabelled sides first, then add all of them.

Find the perimeter

9. Find the perimeter of an L-shape that is 7 cm wide and 5 cm tall, with a top edge of 4 cm and a right edge of 3 cm. Two edges of the step are not labelled.

  1. 19 cm
  2. 22 cm
  3. 24 cm
  4. 29 cm
Find the perimeter

10. An L-shape is 12 m tall on the left and 10 m wide along the bottom. The top edge is 6 m and the right edge is 5 m tall. What is its perimeter?

  1. 44 m
  2. 33 m
  3. 92 m
  4. 40 m
Staircase

11. A staircase shape has three steps, each 2 cm tall. From bottom to top, the steps are 6 cm, 4 cm and 2 cm wide. What is its perimeter?

  1. 24 cm
  2. 36 cm
  3. 18 cm
  4. 20 cm
Grid shape

12. A shape on 1 cm grid paper has three rows of squares lined up on the left: 4 squares, then 4 squares, then 2 squares. What is its perimeter?

  1. 10 cm
  2. 14 cm
  3. 11 cm
  4. 20 cm
Fence with a gate

13. An L-shaped garden fits in a 10 m by 8 m rectangle with a 4 m by 3 m corner missing. A fence will go all the way around except for a 2 m gate. How much fence is needed?

  1. 36 m
  2. 68 m
  3. 34 m
  4. 38 m
Find the perimeter

14. An L-shape has a bottom edge of 8 cm and a left side of 6 cm. Its top edge is 5 cm and its right side is 4 cm. What is its perimeter?

  1. 23 cm
  2. 42 cm
  3. 25 cm
  4. 28 cm
Baseboard

15. A room is 6 m by 5 m with a 2 m by 2 m closet taking one corner, so the room is an L-shape. Baseboard goes along every wall of the room except a 1 m doorway. How much baseboard is needed?

  1. 21 m
  2. 22 m
  3. 26 m
  4. 17 m
T-shape

16. A T-shape has a top bar 9 cm wide and 2 cm tall. A stem 3 cm wide and 4 cm tall hangs from the middle, leaving 3 cm of the bar's underside on each side. What is its perimeter?

  1. 24 cm
  2. 30 cm
  3. 26 cm
  4. 30 cm²

Think

Think practice

Each question shows a tempting mistake. Explain why it fails.

Spot the error

17. Lena counts the 29 squares inside the 7 cm by 5 cm L-shape on centimetre grid paper and says the perimeter is 29 cm. What is wrong?

  1. She should count only the squares in the top row.
  2. Nothing; counting squares gives the perimeter.
  3. She found the area. The perimeter is the length around the edge, 24 cm.
  4. She should multiply 29 by 4.
Spot the error

18. Omar finds the perimeter of an L-shape that is 7 cm wide and 5 cm tall, with a top edge of 4 cm and a right edge of 3 cm by adding 7 + 5 + 4 + 3 = 19 cm. What did he forget?

  1. The two unlabelled step edges, 3 cm and 2 cm
  2. To multiply instead of adding
  3. To double his answer
  4. Nothing; 19 cm is correct
Spot the error

19. Kai cuts the 7 cm by 5 cm L-shape with a 4 cm line to find its area, then gives the perimeter as 24 + 4 = 28 cm. Why is that wrong?

  1. The cut line should be counted twice.
  2. He should have cut up and down instead.
  3. The perimeter should be 29 cm.
  4. The cut line is inside the shape, so it is not part of the perimeter.
Spot the error

20. An L-shape is 9 cm tall on the left, and its right edge is 4 cm tall. Ben says the upright edge of the step is 9 cm. What should it be?

  1. 4 cm
  2. 5 cm
  3. 13 cm
  4. 9 cm
True or false

21. True or false: an L-shape made by cutting a rectangular corner off a rectangle has the same perimeter as the full rectangle.

  1. True
  2. False
True or false

22. True or false: cutting a notch 3 cm deep out of the middle of a rectangle's top edge keeps the perimeter the same.

  1. True
  2. False
Compare

23. A 7 cm by 5 cm rectangle has a 3 cm by 2 cm corner cut off to make an L-shape. How do the L-shape's area and perimeter compare with the rectangle's?

  1. Both are smaller.
  2. The perimeter is smaller, but the area is the same.
  3. Both stay the same.
  4. The area is smaller, but the perimeter is the same.
Explain

24. Why do we not need a ruler to find the unlabelled sides of an L-shape?

  1. Unlabelled sides are always 1 cm.
  2. Opposite sides balance, so a missing side is a long side minus the shorter side beside it.
  3. The two missing sides are always equal.
  4. You can guess them from the picture.

Challenge

Challenge practice

Use perimeter in real situations, work backwards and compare with area.

Notch

25. A U-shape fits in an 8 cm by 5 cm rectangle, with a notch 2 cm wide and 3 cm deep cut down from the middle of the top. What is its perimeter?

  1. 32 cm
  2. 26 cm
  3. 34 cm
  4. 29 cm
Same perimeter?

26. Shape A is a 6 cm by 4 cm rectangle. Shape B is the same rectangle with a 2 cm by 2 cm corner cut off. Which statement is true?

  1. A has the larger perimeter and the larger area.
  2. B has the larger perimeter.
  3. They have the same perimeter, but A has the larger area.
  4. They have the same area, but A has the larger perimeter.
Work backwards

27. An L-shape has a perimeter of 30 cm. Five of its sides are 9 cm, 6 cm, 5 cm, 2 cm and 4 cm. How long is the sixth side?

  1. 26 cm
  2. 4 cm
  3. 56 cm
  4. 5 cm
Cost of fencing

28. An L-shaped dog run fits in a 10 m by 8 m rectangle with a 4 m by 3 m corner missing. Fencing costs $5 per metre. How much does it cost to fence all the way around?

  1. $400
  2. $340
  3. $36
  4. $180
Plus sign

29. A plus-sign shape is made of five 1 cm squares: one in the middle and one on each side of it. What is its perimeter?

  1. 5 cm
  2. 20 cm
  3. 12 cm
  4. 16 cm
Explain

30. Two L-shapes both fit in a 5 cm by 3 cm rectangle. One has a 1 cm by 1 cm corner missing; the other has a 2 cm by 2 cm corner missing. Find the perimeter and area of each, and explain what you notice.

Answer key

  1. 1. 20 cm
    6 + 4 + 6 + 4 = 20 cm. 24 cm is the area number (6 × 4), 10 cm adds only two of the four sides, and cm² is a unit for area, not for a distance around.
  2. 2. 6
    Going around an L-shape you turn six times, so it has 6 sides. 4 is a rectangle, 5 forgets one of the step edges, and 8 is a T-shape or U-shape.
  3. 3. 5 m
    The step edge and the right edge together are as tall as the left edge, so 8 − 3 = 5 m. 11 m adds instead of subtracting, and 3 m and 8 m just copy sides you were given.
  4. 4. 30 cm
    9 + 6 + 5 + 2 + 4 + 4 = 30 cm. 26 cm leaves out one of the 4 cm sides, 34 cm adds a 4 cm side twice, and 25 cm leaves out the 5 cm side.
  5. 5. Putting a fence around the garden
    A fence goes around the outside edge, so it needs the perimeter. Soil, grass seed and counting the squares inside are all about the space inside, which is area.
  6. 6. 24 cm and 29 cm²
    Perimeter is a length, so it is in cm. Area counts squares, so it is in cm². The other pairs give one or both measures the wrong kind of unit.
  7. 7. No, because it is inside the shape, not on the outside edge.
    Perimeter is only the outside boundary. A cut line is a helper for area. It is never part of the perimeter, whatever its length or direction.
  8. 8. 5 m
    The top edge and the step's flat edge together match the bottom edge: 12 − 7 = 5 m. 19 m adds the two edges, 7 m repeats the top edge, and 84 multiplies two lengths, which is not a length at all.
  9. 9. 24 cm
    The step edges are 7 − 4 = 3 cm and 5 − 3 = 2 cm. 7 + 5 + 4 + 3 + 3 + 2 = 24 cm. 19 cm adds only the four labelled sides, 22 cm forgets the 2 cm step edge, and 29 is the area in cm².
  10. 10. 44 m
    Missing edges: 10 − 6 = 4 m across and 12 − 5 = 7 m down. 12 + 10 + 6 + 5 + 4 + 7 = 44 m. 33 m adds only the labelled sides, 92 is the area in m², and 40 m forgets the 4 m edge.
  11. 11. 24 cm
    Going around: bottom 6, left side 2 + 2 + 2 = 6, top 2, then down and across the steps: 2, 2, 2, 2, 2. Total 6 + 6 + 2 + 2 + 2 + 2 + 2 + 2 = 24 cm, the same as a 6 cm by 6 cm square. 36 is that square's area, 18 adds only the widths and heights you were told (6 + 4 + 2 + 2 + 2 + 2), and 20 leaves out two of the 2 cm edges.
  12. 12. 14 cm
    The shape is 4 cm wide and 3 cm tall with a 2 cm by 1 cm corner missing. Going around: 4 + 2 + 2 + 1 + 2 + 3 = 14 cm. 10 counts the squares (the area), 11 cm leaves out the two step edges (2 cm and 1 cm), and 20 doubles the square count, which has nothing to do with the edges.
  13. 13. 34 m
    The sides are 6, 3, 4, 5, 10 and 8 m (10 − 4 = 6 and 8 − 3 = 5), so the perimeter is 36 m. The gate needs no fence: 36 − 2 = 34 m. 36 m forgets the gate, 68 is the area in m², and 38 m adds the gate instead of subtracting it.
  14. 14. 28 cm
    Missing edges: 8 − 5 = 3 cm and 6 − 4 = 2 cm. 8 + 6 + 5 + 4 + 3 + 2 = 28 cm. 23 cm adds only the four labelled sides, 42 is the area in cm² (48 − 6), and 25 cm forgets the 3 cm edge.
  15. 15. 21 m
    The room's sides are 4, 2, 2, 3, 6 and 5 m, so its perimeter is 22 m (the same as the full 6 by 5 rectangle). Take away the 1 m doorway: 21 m. 22 m forgets the doorway, 26 is the floor area in m², and 17 m leaves out the two 2 m closet walls as well as the doorway.
  16. 16. 30 cm
    The 8 sides are 9, 2, 3, 4, 3, 4, 3 and 2 cm, which add to 30 cm. 24 cm leaves out the two 3 cm edges under the bar, 26 cm leaves out the two 2 cm ends, and 30 cm² uses an area unit. (The area also happens to be 30 cm², but it is a different measure.)
  17. 17. She found the area. The perimeter is the length around the edge, 24 cm.
    Squares inside measure area (29 cm²). Adding the six edges gives 24 cm. Counting one row or multiplying by 4 does not measure the outside edge either.
  18. 18. The two unlabelled step edges, 3 cm and 2 cm
    An L-shape has 6 sides, and he added only 4. Adding 3 and 2 gives 24 cm. Doubling 19 gives 38, which is too much, and multiplying gives an area.
  19. 19. The cut line is inside the shape, so it is not part of the perimeter.
    Only outside edges count, so the perimeter stays 24 cm. The direction of the cut does not matter because no cut line is ever part of the perimeter, and 29 is the area.
  20. 20. 5 cm
    The step edge and the right edge stack up to match the left edge: 9 − 4 = 5 cm. Ben copied the left side. 4 cm copies the right side, and 13 cm adds them.
  21. 21. True
    The two step edges are exactly as long as the parts of the top and side that were removed, so they slide out to rebuild the rectangle's edges. The area gets smaller, but the perimeter does not change.
  22. 22. False
    The top edge still adds up to the same total, but the two 3 cm sides of the notch are new. The perimeter grows by 3 + 3 = 6 cm.
  23. 23. The area is smaller, but the perimeter is the same.
    Area drops from 35 cm² to 29 cm² because squares were removed. Perimeter stays 24 cm because the step edges replace the lengths that were cut away. Area and perimeter do not always change together.
  24. 24. Opposite sides balance, so a missing side is a long side minus the shorter side beside it.
    With square corners, the edges facing up add to the edges facing down, and the same for left and right. Missing sides can be any length and need not be equal, and a drawing is not always to scale, so guessing from the picture is not safe.
  25. 25. 32 cm
    The rectangle's 26 cm still counts, because the top edges plus the notch bottom add to 8 cm, and the notch adds two 3 cm sides: 26 + 3 + 3 = 32 cm. 26 cm ignores the notch sides, 34 is the area in cm², and 29 cm adds only one notch side.
  26. 26. They have the same perimeter, but A has the larger area.
    Both perimeters are 20 cm: B's step edges replace what was cut. A's area is 24 cm², B's is 24 − 4 = 20 cm². Cutting a corner removes area without changing perimeter.
  27. 27. 4 cm
    The five sides add to 9 + 6 + 5 + 2 + 4 = 26 cm, so the sixth is 30 − 26 = 4 cm. Check with opposite sides: the right side is 6 − 2 = 4 cm. 26 cm is the sum of the five, 56 cm adds instead of subtracting, and 5 cm copies a side.
  28. 28. $180
    The perimeter is 6 + 3 + 4 + 5 + 10 + 8 = 36 m, and 36 × 5 = $180. $400 multiplies the full rectangle's area (80) by 5, $340 multiplies the dog run's area (68) by 5, and $36 is the length in metres, not the cost.
  29. 29. 12 cm
    Each of the four arm squares shows 3 outside edges, and the middle square shows none: 4 × 3 = 12 cm. 5 counts the squares (the area), 20 counts all 4 edges of every square, including the ones hidden inside, and 16 counts the middle square's edges too.
  30. 30. Both perimeters are 16 cm, the same as the 5 cm by 3 cm rectangle. The areas are different: 15 − 1 = 14 cm² and 15 − 4 = 11 cm². Shapes with the same perimeter can have different areas.
    Cutting a corner off a rectangle never changes the perimeter, because the step edges replace the removed lengths, but it removes squares. A complete answer gives both measures with the right units.

Related practice

Area of L-shapes: split, add or subtractPerimeter vs areaClassify triangles and quadrilateralsMath hub