KS3 · Maths

Area of composite rectilinear shapes

An L-shaped floor needs tiling. You can chop it into rectangles in one place, or somewhere completely different. Which cut gives the right area? Both. Here's why.

Maths · Area

One L-shape. Two ways to cut it.

Bottom edge 15 cm. Right edge 4 cm. Top edge 7 cm. Left edge 9 cm. Bottom: left of M 3.5 cm. Bottom: right of M 11.5 cm. Left: below N 2.0 cm. Left: above N 7.0 cm. Area of the whole L 95 cm². Classification: Two cuts, one area. Relationship: Cut along the horizontal inner line: 15 × 4 + 7 × 5 = 60 + 35 = 95. Cut along the vertical inner line: 7 × 9 + 8 × 4 = 63 + 32 = 95. Different rectangles, same total.

Bottom edge15 cmRight edge4 cmTop edge7 cmLeft edge9 cmBottom: left of M3.5 cmBottom: right of M11.5 cmLeft: below N2.0 cmLeft: above N7.0 cmArea of the whole L95 cm²Drag M and N to the inner lines

Two cuts, one area

Cut along the horizontal inner line: 15 × 4 + 7 × 5 = 60 + 35 = 95. Cut along the vertical inner line: 7 × 9 + 8 × 4 = 63 + 32 = 95. Different rectangles, same total.

Two sides of this L have no label, and the two inner lines show where you could cut it. Drag M along the bottom to the vertical inner line and read how the 15 cm splits. Then drag N up the left side to the horizontal inner line.

Watch out: 15 × 9 is not the area of this L-shape. It is the box around it, missing corner included. Every rectangle needs its own length and width.

Maths · Area

Which method for which shape?

Pick a shape, then pick the method you would reach for first.

Still to sort

Break into rectangles (0)

Cut the shape up, find each rectangle's area using its own length and width, then add them.

Where the line is: Suits a shape where the piece you would add in by completing is itself composite, and no part rearranges neatly.

Complete the rectangle (0)

Draw a line across the gap to make one whole rectangle, find its area, then subtract the area of the extra piece you added in.

Where the line is: Suits a shape with a gap in the middle, because the piece to subtract is one rectangle.

Rearrange into one rectangle (0)

Cut a rectangle off and slide it into the gap to make one whole rectangle, then multiply its length by its width. Moving a piece doesn't change the area.

Where the line is: Suits a shape where a rectangle can slide into the gap to make one whole rectangle.

5 of 5 still to sort.

Watch out: More than one method will usually work. You are choosing the one that needs the fewest lengths and the least fuss. Breaking into rectangles is not always the best choice, because sometimes there isn't enough information to use it.

Maths · Area

Why every method agrees

A shape is 9 cm tall all the way along. It is made of a 2 cm wide part and a 5 cm wide part. Fill in each part's area, then add them up.

Grid: each cell is its row times its column
×	2	5
9

Add the cells to get the total area:

Type x² as x^2 if you cannot type ². Spaces do not matter.

Worked example: a missing length

Problem

A shape is an 8 cm by 6 cm rectangle with a 2 cm wide slot cut down from its top edge. The floor of the slot is 2.7 cm above the bottom edge, and we aren't told how far along the slot sits. Find the area.

Maths · Algebra

Working backwards from the area

This time you know the area and a side is missing. Write the area as an expression, then solve it.

GoalA rectangle 5 cm wide and x cm tall has a 2 cm by 3 cm piece cut from one corner. The area is 54 cm². Find x.

5 × x − 2 × 3 = 54

write the area

The whole 5 by x rectangle, take away the 2 by 3 corner, equals the area we were given.

Step 1 of 5

The whole 5 by x rectangle, take away the 2 by 3 corner, equals the area we were given.

Watch out: Always check by working forwards: put your answer back into the area expression and see whether it gives the area you were told.

Maths · Area

Same perimeter, same area?

Shape A is an 8 cm by 10 cm rectangle. Shape B is the same rectangle with a 3 cm by 3 cm square cut out of one corner. Both have a perimeter of 36 cm.

Which is closest to what you think right now?

They have the same perimeter, so they must have the same area.B has had a piece taken away, so its perimeter must be smaller than A's.Area and perimeter are the same kind of measurement. Perimeter is just area for the outside.Perimeter is the distance round the edge and area is the surface inside, so two shapes can share a perimeter and still have different areas.

How sure are you?

I'm sureI think soI'm guessing

They have the same perimeter, so they must have the same area.

This is a common misconception

It's a tempting idea, because the distance round the edge really does match. But perimeter only measures that distance. Area is how much surface is inside, and B has lost a 3 by 3 square of it: 8 × 10 − 3 × 3 = 80 − 9 = 71, while A is 80. Same perimeter, different areas.

B has had a piece taken away, so its perimeter must be smaller than A's.

This is a common misconception

Look at where the piece went. Slide the two edges of the notch out to the sides of the 8 by 10 rectangle and you are back at A, the same distance round: 36. B's area dropped, but its perimeter didn't. That slide only works when the edges can move without overlapping. Where they would overlap, the perimeter is larger.

Area and perimeter are the same kind of measurement. Perimeter is just area for the outside.

This is a common misconception

They measure different things. Area is the size of the surface. Perimeter is the distance around the shape, which is a length. That's why 80 and 36 can both describe shape A without being the same measurement.

Perimeter is the distance round the edge and area is the surface inside, so two shapes can share a perimeter and still have different areas.

This matches the evidence

Exactly. A and B both have a perimeter of 36, but their areas are 80 and 71. Keep perimeter (distance around) and area (size of the surface) in separate boxes in your head.

Perimeter is the distance around the outside; area is the size of the surface. Matching perimeters don't guarantee matching areas.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Cut it, complete it or slide it: every honest way gives the same area.

What you need to know

A composite rectilinear shape is made from rectangles, so it can be broken back up into rectangles. The area of a rectangle is length × width (base × height).
Three methods: break the shape into rectangles and add, complete the rectangle and subtract the extra piece, or rearrange the shape into one rectangle.
Every honest method gives the same area, because the lines of working are equivalent calculations.
The best method depends on the shape's features. Breaking into rectangles is not always best, because there is not always enough information to use it.
If a side length isn't shown, deduce it from parallel sides that don't overlap, add it to the diagram, then choose your method.
You can compare areas without calculating: if two rectangles share one measurement, the one with the longer other measurement has the larger area, and a shape cut up and rearranged keeps its area.
Given the area, you can find a missing side by writing the area expression and solving it. Area is the size of the surface; perimeter is the distance around.

The big picture

A composite rectilinear shape is made from rectangles, so you can find its area by breaking it into rectangles, by completing the rectangle and subtracting the extra piece, or by rearranging it into one rectangle. Every honest method gives the same area, because the lines of working are equivalent calculations. You choose the method that suits the shape, deduce any missing sides from parallel, non-overlapping lengths, and you can even work backwards from a given area to a missing side.

Key points

1A composite rectilinear shape is just rectangles joined up, and a rectangle's area is length × width.

2Cut it, complete it or rearrange it: all three give the same area.

3Find missing sides first: parallel, non-overlapping lengths, one subtracted from the other.

4Choose the method that needs the fewest missing lengths.

5Same perimeter doesn't mean same area.

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Worked example

Problem

A rectangular sheet is 17 cm wide and 19 cm tall, with a rectangular hole cut out of the middle. Across the width the sheet shows 3 cm, then the hole, then 5 cm. Up the height it shows 2 cm, then the hole, then 11 cm. Find the area of sheet that is left.

⚠ Watch out

Multiplying any two lengths you can see. On an L-shape, 15 × 9 is the box around it, not the shape. Each rectangle uses its own length and width, and unlabelled sides are deduced from parallel, non-overlapping sides, not guessed.

🧠

Memory hook

Same shape, same area: 2 × 9 + 5 × 9, 14 × 9 − 7 × 9 and 7 × 9 are all seven lots of nine.

✓

Check yourself

A 10 cm by 7 cm rectangle loses a 4 cm by 2 cm corner. Which method suits, and what's the area? (Complete it: 10 × 7 − 4 × 2 = 62 cm².)

Flashcards

(15)

What is a composite rectilinear shape?

A shape made from rectangles, so it can be broken back up into rectangles.

How do you find the area of a rectangle?

Multiply length by width (base by height).

Two rectangles share one measurement. How can you tell which has the larger area without calculating?

The one whose other measurement is longer has the larger area.

A shape is cut up and rearranged into a different shape. What happens to its area?

It stays the same. It is the same surface, just moved.

Method 1: how do you break a shape into rectangles?

Cut it, work out each rectangle's area using that rectangle's own length and width, then add. The cut can go in different places and gives the same answer.

Method 2: how do you complete the rectangle?

Draw a line across the gap to make a whole rectangle, find its area, then subtract the area of the extra piece you added in.

Method 3: how do you rearrange a shape?

Cut a rectangle off and slide it into the gap to make one whole rectangle, then calculate its area.

Why do 2 × 9 + 5 × 9 and 14 × 9 − 7 × 9 both come to 63?

They are equivalent calculations. Each is seven lots of nine, and so is (2 + 5) × 9.

Which shape feature suggests completing the rectangle?

A gap in the middle: the piece to subtract is a single rectangle.

When is breaking into rectangles the better choice?

When the piece you'd add in would itself be composite and no part slots neatly into a gap. It isn't always best: sometimes there isn't enough information to use it.

A side length isn't shown. How do you find it?

Use parallel sides that don't overlap and subtract. A 15 cm side and a 7 cm side parallel to it leave 15 − 7 = 8 cm.

You have deduced the missing lengths. What next?

Add them to the diagram, then choose the approach that needs the fewest missing lengths.

You're given the area and one side is missing. What do you do?

Write the area expression, then solve it. For 5 × x − 2 × 3 = 54: 5x = 60, so x = 12. Then check by working back.

Do shapes with the same perimeter have the same area?

Not necessarily. An 8 by 10 rectangle (area 80) and the same rectangle with a 3 by 3 corner removed (area 71) both have perimeter 36.

What is the difference between area and perimeter?

Area is the size of the surface. Perimeter is the distance around a 2D shape, a special kind of length.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Which method for which shape?

Why every method agrees

Working backwards from the area

Same perimeter, same area?

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