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Area of L-Shapes: Split, Add or Subtract

Find the area of L-shapes and other composite figures: split them into rectangles, subtract a missing corner, find missing sides and avoid double counting.

Grades 3–5 · 30 practice questions · Original material
Learning objectiveFind the area of composite shapes made of rectangles by splitting and adding or by subtracting a missing part, work out missing side lengths first, write answers in square units, and check that pieces do not overlap or leave gaps.

The idea

Area is the number of square units that cover a flat shape. For a rectangle, you can count rows of squares or multiply length × width. But many real spaces are not rectangles: an L-shaped room, a garden with a patio in one corner, or a T-shaped table. These are called composite shapes, because they are made of simpler shapes put together.

Method 1 is split and add. Cut the shape into rectangles, find the area of each one, then add. You can cut wherever is easiest, but the pieces must not overlap, and together they must cover the whole shape. Then every square is counted exactly once.

Method 2 is subtract. Imagine the full rectangle around the shape, find its area, then take away the part that is missing. This is often quickest when a small corner, notch or hole is missing.

Before you multiply, check that you know every side you need. Often a side length is not labelled. In an L-shape, the two short edges of the step can be found from the long edges: the top edge plus the step edge equals the bottom edge, and the same idea works up the sides.

Area is always in square units: cm² for square centimetres and m² for square metres. A good check is to solve the problem a second way. If split-and-add and subtract give different answers, one of them has a mistake.

Worked examples

Three ways to find the area of one L-shapeThe same L-shape is drawn three times on a centimetre grid. It is 7 cm wide and 5 cm tall, with a 3 cm by 2 cm corner missing at the top right. Panel 1 cuts across: a 7 by 3 bottom piece of 21 square centimetres plus a 4 by 2 top piece of 8 square centimetres makes 29. Panel 2 cuts up and down: a 4 by 5 left piece of 20 plus a 3 by 3 right piece of 9 makes 29. Panel 3 subtracts: the full 7 by 5 rectangle is 35, minus the 3 by 2 missing corner of 6, leaves 29. All three give 29 square centimetres.One L-shape, three ways to find its areaEach small square is 1 cm². Red dashed line = where we cut. Original schematic.7 cm5 cm4 cm3 cm2 cm3 cm7 × 3 = 214 × 2 = 87 cm5 cm4 cm3 cm2 cm3 cm4 × 5 = 203 × 3= 97 cm5 cm7 × 5 = 35− 3 × 2= 61 · Cut across2 · Cut up and down3 · Subtract21 + 8 = 29 cm²20 + 9 = 29 cm²35 − 6 = 29 cm²Pieces must not overlap or leave gaps. Every method gives the same area.Missing step sides: 7 − 4 = 3 cm across, 5 − 2 = 3 cm down.
Original schematic on a centimetre grid. The same L-shape is found three ways: cut across and add, cut up and down and add, or subtract the missing corner from the full rectangle. Any cut works if the pieces do not overlap and leave no gaps.

Example 1 · Split and add (cut across)

An L-shape is 7 cm wide and 5 cm tall, with a 3 cm by 2 cm corner missing at the top right. Find its area.

  1. Cut across where the step is. The bottom piece is 7 cm wide and 5 − 2 = 3 cm tall: 7 × 3 = 21 cm².
  2. The top piece is 7 − 3 = 4 cm wide and 2 cm tall: 4 × 2 = 8 cm².
  3. Add the pieces: 21 + 8 = 29 cm².

29 cm².

Example 2 · Split and add (cut up and down)

Find the area of the same L-shape by cutting up and down instead.

  1. The left piece is 4 cm wide and the full 5 cm tall: 4 × 5 = 20 cm².
  2. The right piece is 3 cm wide, but only 3 cm tall because the corner is missing: 3 × 3 = 9 cm².
  3. Add: 20 + 9 = 29 cm², the same answer, as it must be.

29 cm² again. Where you cut does not change the area.

Example 3 · Subtract the missing corner

Find the area of the same L-shape by subtracting.

  1. The full rectangle around the L-shape is 7 × 5 = 35 cm².
  2. The missing corner is 3 × 2 = 6 cm².
  3. 35 − 6 = 29 cm².

29 cm². Three methods, one answer.

Example 4 · Find the missing sides first

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. Find its area.

  1. The step’s horizontal edge is 10 − 6 = 4 m.
  2. The step’s vertical edge is 12 − 5 = 7 m.
  3. Split: left column 6 × 12 = 72 m², right part 4 × 5 = 20 m². Total 92 m².
  4. Check by subtracting: 10 × 12 = 120, missing corner 4 × 7 = 28, and 120 − 28 = 92 m².

92 m², checked two ways.

Example 5 · Spot the double count

A student uses the 7 cm by 3 cm bottom piece and the 4 cm by 5 cm left piece of the first L-shape and gets 21 + 20 = 41 cm². What happened?

  1. Shade both pieces: the bottom-left block, 4 cm by 3 cm, is inside both.
  2. That block, 4 × 3 = 12 cm², was counted twice.
  3. 41 − 12 = 29 cm², which matches the other methods.

The pieces overlapped. Pieces must not overlap.

Common mistakes

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

Multiplying the two longest outside edges of an L-shape.

Why this fails: That finds the area of the whole rectangle around it, including the missing corner. Subtract the missing part, or split the shape.

Using a full side length for a piece that is shorter.

Why this fails: After a cut, check each piece’s own length and width. In an L-shape, one piece is usually shorter than the whole shape.

Splitting into pieces that overlap.

Why this fails: Overlapping pieces count some squares twice, so the total is too big. Every square must be counted exactly once.

Adding side lengths instead of multiplying.

Why this fails: Adding the sides of a rectangle gives part of its perimeter, a length. Area needs length × width for each rectangle.

Writing cm instead of cm².

Why this fails: Area counts squares, so the unit is square centimetres (cm²) or square metres (m²). A plain cm or m is a length.

Foundation

Foundation practice

Count, multiply and use square units. Check that every piece is a rectangle.

Count and multiply

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

  1. 24 cm²
  2. 20 cm
  3. 10 cm²
  4. 24 cm
Split and add

2. An L-shape is split into two rectangles that do not overlap: one is 5 cm by 2 cm and the other is 3 cm by 4 cm. What is the area of the L-shape?

  1. 14 cm²
  2. 120 cm²
  3. 22 cm²
  4. 17 cm²
Units

3. A family measures an L-shaped living room floor in metres. Which unit should the area be written in?

  1. m
  2. cm
  3. m²
  4. km
Missing side

4. The bottom edge of an L-shape is 9 m long. The top edge, directly above it, is 4 m long. How long is the short horizontal edge of the step between them?

  1. 13 m
  2. 5 m
  3. 4 m
  4. 36 m
Subtract

5. A rectangle is 8 cm by 6 cm. A 2 cm by 3 cm piece is cut out of one corner. What area is left?

  1. 48 cm²
  2. 42 cm²
  3. 6 cm²
  4. 54 cm²
Split rule

6. Which rule must your pieces follow when you split a shape to find its area?

  1. The pieces must all be squares.
  2. The pieces must be the same size.
  3. You must always cut straight across the middle.
  4. The pieces must not overlap, and together they must cover the whole shape.
Count squares

7. A shape on a grid has three rows of squares: 5 squares in the top row, 5 in the middle row and 2 in the bottom row. What is its area in square units?

  1. 12 square units
  2. 10 square units
  3. 50 square units
  4. 15 square units

Practice

Practice practice

Split and add, or subtract the missing part. Find missing sides before you multiply.

Subtract

8. An L-shaped patio fits inside a 10 m by 7 m rectangle with a 4 m by 3 m corner missing. What is the patio’s area?

  1. 58 m²
  2. 70 m²
  3. 12 m²
  4. 82 m²
Split and add

9. An L-shape is split with an up-and-down cut. The left piece is 3 m wide and 8 m tall. The right piece is 5 m wide and 4 m tall. What is the total area?

  1. 40 m²
  2. 64 m²
  3. 20 m²
  4. 44 m²
Missing sides

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

  1. 120 cm²
  2. 92 cm²
  3. 28 cm²
  4. 102 cm²
Compare methods

11. For the 7 cm by 5 cm L-shape with a 3 cm by 2 cm corner missing, Noor splits it as 21 + 8 and Sam splits it as 20 + 9. Who is right?

  1. Only Noor
  2. Only Sam
  3. Both of them
  4. Neither of them
Tiles

12. A room is 6 m by 5 m, but a 2 m by 2 m closet in one corner will not be tiled. Each tile covers 1 m². Ignoring cuts and waste, how many tiles are needed?

  1. 30
  2. 22
  3. 34
  4. 26
T-shape

13. A T-shape has a top bar 9 cm wide and 2 cm tall, with a stem 3 cm wide and 4 cm tall below it. What is its area?

  1. 30 cm²
  2. 54 cm²
  3. 18 cm²
  4. 36 cm²
Subtract

14. A U-shape fits inside 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. What is the area of the U-shape?

  1. 40 cm²
  2. 34 cm²
  3. 37 cm²
  4. 30 cm²
Units

15. An L-shape is split into a 4 cm by 3 cm rectangle and a 2 cm by 2 cm square. Which answer is written correctly?

  1. 16 cm²
  2. 16 cm
  3. 11 cm²
  4. 16 square cm long

Think

Think practice

These questions target common mistakes. Explain why the tempting answer fails.

Spot the error

16. For a 7 cm by 5 cm L-shape with a 3 cm by 2 cm corner missing, Raj uses the bottom piece 7 × 3 = 21 and the left piece 4 × 5 = 20, and gets 41 cm². What went wrong?

  1. His two pieces overlap, so a 4 cm by 3 cm part was counted twice.
  2. He should have multiplied 21 × 20.
  3. His pieces leave a gap in the middle.
  4. Nothing; the area is 41 cm².
Spot the error

17. Ella finds the area of an L-shape by multiplying its two longest outside edges, 7 cm × 5 cm = 35 cm². Why is this not the area?

  1. She should add 7 + 5 instead.
  2. That is the area of the whole rectangle, but the L-shape has a corner missing that must be subtracted.
  3. 35 is the perimeter.
  4. It is correct for every L-shape.
Spot the error

18. An L-shape is 7 cm wide and 5 cm tall, with a 3 cm by 2 cm corner missing at the top right. Max cuts it up and down into a 4 cm-wide piece and a 3 cm-wide piece, then writes 4 × 5 + 3 × 5 = 35 cm². What is his mistake?

  1. He used the wrong width for the left piece.
  2. He should have cut across instead.
  3. The right piece is only 3 cm tall, not 5 cm, because the corner is missing.
  4. He forgot to add the pieces.
True or false

19. True or false: two different correct ways of splitting the same L-shape always give the same area.

  1. True
  2. False
True or false

20. True or false: if you turn an L-shape on its side, its area changes.

  1. True
  2. False
Missing side

21. An L-shape has a bottom edge of 11 cm and a top edge of 5 cm. How long is the short horizontal edge where the shape steps down?

  1. 16 cm
  2. 5 cm
  3. 11 cm
  4. 6 cm
Choose a method

22. A large rectangle has a small square missing from one corner. Which method is usually quickest?

  1. Count every square one by one.
  2. Find the area of the whole rectangle, then subtract the small missing square.
  3. Add up all the side lengths.
  4. Multiply every side length together.

Challenge

Challenge practice

Use composite area in real situations and work backwards. Show your reasoning.

Garden

23. A garden fits in a 12 m by 9 m rectangle, but a 5 m by 4 m patio fills one corner. Sod will cover the rest. How many square metres of sod are needed?

  1. 108 m²
  2. 20 m²
  3. 128 m²
  4. 88 m²
Shape with a hole

24. A 10 cm by 8 cm card has a 2 cm by 3 cm window cut out of the middle. What is the area of the card that is left?

  1. 74 cm²
  2. 80 cm²
  3. 75 cm²
  4. 86 cm²
Work backwards

25. An L-shape is made of a 6 cm by 4 cm rectangle and a square. The whole shape has an area of 33 cm². How long is each side of the square?

  1. 9 cm
  2. 3 cm
  3. 27 cm
  4. 4 cm
Compare

26. Shape A is a 6 cm by 6 cm square with a 2 cm by 2 cm corner cut off. Shape B is a 5 cm by 4 cm rectangle joined to a 3 cm by 4 cm rectangle. Which has the larger area?

  1. Shape A
  2. Shape B
  3. They have the same area.
  4. You cannot tell without counting.
Explain

27. Describe two different ways to find the area of a 7 cm by 5 cm L-shape that has a 3 cm by 2 cm corner missing, and give the area.

Round up

28. An L-shaped floor fits in an 8 m by 6 m rectangle with a 3 m by 2 m corner missing. One box of flooring covers 5 m². How many boxes must be bought?

  1. 8 boxes
  2. 9 boxes
  3. 10 boxes
  4. 42 boxes
Staircase

29. 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 the area?

  1. 36 cm²
  2. 12 cm²
  3. 24 cm²
  4. 20 cm²
Explain

30. Why can you not find the area of an L-shape by multiplying just two of its side lengths?

Answer key

  1. 1. 24 cm²
    There are 4 rows of 6 squares, and 4 × 6 = 24 square centimetres. 20 cm is the perimeter (the distance around), 10 comes from adding 6 + 4, and 24 cm uses a length unit for an area.
  2. 2. 22 cm²
    5 × 2 = 10 and 3 × 4 = 12, so 10 + 12 = 22 cm². 14 adds the four side lengths, 120 multiplies the two areas together, and 17 (10 + 7) adds the second rectangle’s sides instead of multiplying them.
  3. 3. m²
    Area counts square units, so metres become square metres (m²). Plain m measures a length such as a wall, cm is a smaller length unit, and km is a length unit for long distances.
  4. 4. 5 m
    The top edge and the step edge together stretch as far as the bottom edge, so the step is 9 − 4 = 5 m. 13 adds instead of subtracting, 4 repeats the top edge, and 36 multiplies two lengths.
  5. 5. 42 cm²
    Whole rectangle 8 × 6 = 48 cm², minus the cut-out 2 × 3 = 6 cm², leaves 42 cm². 48 forgets to remove the corner, 6 is only the missing piece, and 54 adds the piece instead of subtracting it.
  6. 6. The pieces must not overlap, and together they must cover the whole shape.
    Each square unit must be counted exactly once. Pieces can be rectangles of different sizes, and you can cut wherever is easiest; overlapping pieces count some squares twice, and gaps miss some.
  7. 7. 12 square units
    5 + 5 + 2 = 12 squares. 10 counts only the top two rows, 50 multiplies 5 × 5 × 2, and 15 assumes all three rows have 5 squares.
  8. 8. 58 m²
    10 × 7 = 70 m² for the whole rectangle, minus 4 × 3 = 12 m² for the missing corner, gives 58 m². 70 forgets the corner, 12 is just the corner, and 82 adds the corner.
  9. 9. 44 m²
    Left: 3 × 8 = 24 m². Right: 5 × 4 = 20 m². Total 24 + 20 = 44 m². 64 treats the whole shape as 8 by 8, 40 treats it as 8 wide and 5 tall, and 20 is only the right piece.
  10. 10. 92 cm²
    Split it: the left column is 6 × 12 = 72 cm², and the right part is 4 × 5 = 20 cm² (10 − 6 = 4 cm wide). 72 + 20 = 92 cm². Check by subtracting: 10 × 12 = 120, and the missing corner is 4 × 7 = 28 (12 − 5 = 7), so 120 − 28 = 92. 120 forgets the corner, and 102 subtracts a corner of the wrong size (18 instead of 28).
  11. 11. Both of them
    Noor cuts across: 7 × 3 = 21 and 4 × 2 = 8. Sam cuts up and down: 4 × 5 = 20 and 3 × 3 = 9. Both make 29 cm², because the area of a shape does not depend on where you cut it.
  12. 12. 26
    6 × 5 = 30 m² for the room, minus 2 × 2 = 4 m² for the closet: 26 m², so 26 tiles. 30 tiles the closet too, 34 adds the closet, and 22 subtracts 2 × 2 twice or treats the closet as 8 m².
  13. 13. 30 cm²
    Top bar 9 × 2 = 18 cm², stem 3 × 4 = 12 cm², total 30 cm². 54 uses the whole 9 by 6 rectangle, 18 forgets the stem, and 36 makes the stem 3 × 6, using the full height of the shape for the stem.
  14. 14. 34 cm²
    8 × 5 = 40 cm², minus the notch 2 × 3 = 6 cm², leaves 34 cm². 40 ignores the notch, 37 subtracts 2 + 3 instead of 2 × 3, and 30 subtracts too much.
  15. 15. 16 cm²
    4 × 3 = 12 and 2 × 2 = 4, so 16 square centimetres, written 16 cm². 16 cm is a length, 11 adds the side lengths, and “square cm long” mixes area and length words.
  16. 16. His two pieces overlap, so a 4 cm by 3 cm part was counted twice.
    The bottom-left 4 by 3 block is inside both pieces. 41 − 12 = 29 cm², the true area. Multiplying areas never gives an area, and there is no gap: the problem is double counting.
  17. 17. That is the area of the whole rectangle, but the L-shape has a corner missing that must be subtracted.
    7 × 5 covers the full rectangle around the L. Subtract the missing 3 × 2 = 6 cm² corner to get 29 cm². Adding gives 12, which is not an area, and 35 is not the perimeter (that is 24 cm).
  18. 18. The right piece is only 3 cm tall, not 5 cm, because the corner is missing.
    The right piece is 5 − 2 = 3 cm tall, so it is 3 × 3 = 9, and 20 + 9 = 29 cm². The left width 4 cm is correct, and cutting up and down is a fine choice.
  19. 19. True
    The shape covers the same squares however you cut it, as long as the pieces do not overlap and leave no gaps. If two methods disagree, one of them has a mistake.
  20. 20. False
    Turning moves the shape but does not add or remove any squares, so the area stays the same.
  21. 21. 6 cm
    The top edge and the step edge together equal the bottom edge, so 11 − 5 = 6 cm. 16 adds the edges, and 5 and 11 just repeat lengths you were given.
  22. 22. Find the area of the whole rectangle, then subtract the small missing square.
    Two multiplications and one subtraction are quick. Counting one by one works but is slow and easy to lose track of. Adding side lengths finds perimeter, and multiplying all the sides does not measure any real amount.
  23. 23. 88 m²
    12 × 9 = 108 m², minus the patio 5 × 4 = 20 m², gives 88 m². 108 sods the patio too, 20 is only the patio, and 128 adds the patio.
  24. 24. 74 cm²
    10 × 8 = 80 cm², minus the window 2 × 3 = 6 cm², leaves 74 cm². Subtracting works for a hole in the middle, not just a missing corner. 80 ignores the hole, 75 subtracts 2 + 3, and 86 adds the hole.
  25. 25. 3 cm
    The rectangle is 6 × 4 = 24 cm², so the square is 33 − 24 = 9 cm². A square with area 9 cm² has sides of 3 cm, because 3 × 3 = 9. 9 cm confuses the area with the side, and 27 and 4 do not make 9 when multiplied by themselves.
  26. 26. They have the same area.
    A: 36 − 4 = 32 cm². B: 20 + 12 = 32 cm². They look different but cover the same number of square centimetres. You can tell without counting by splitting and subtracting.
  27. 27. Split it: 7 × 3 + 4 × 2 = 21 + 8 = 29 cm² (or 4 × 5 + 3 × 3 = 20 + 9). Or subtract: 7 × 5 − 3 × 2 = 35 − 6 = 29 cm².
    Any non-overlapping split or the subtract method should reach 29 cm². If two methods disagree, look for an overlap or a missing side.
  28. 28. 9 boxes
    The floor is 48 − 6 = 42 m². 42 ÷ 5 = 8 R2, so 8 boxes cover only 40 m² and a ninth box is needed for the last 2 m². 10 boxes are more than needed, and 42 confuses square metres with boxes.
  29. 29. 24 cm²
    6 × 2 + 4 × 2 + 2 × 2 = 12 + 8 + 4 = 24 cm². 36 treats the shape as a full 6 by 6 square, 12 adds only the widths, and 20 forgets the top step (12 + 8).
  30. 30. Because an L-shape is not a rectangle. Multiplying two outside sides gives the whole rectangle around it, including the missing corner, so you must split it into rectangles or subtract the missing part.
    Length × width works only for rectangles. An L-shape needs two rectangles added, or one subtracted.

Related practice

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