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# Composite shapes

Area, volume, circles and Pythagoras · Area

<!-- Speaker note: Students cut composite shapes into rectangles and triangles, or take pieces away from a rectangle, and find the area and the perimeter (AC9M8M01, node 1). About 40 minutes (Predict and Run 8 min · Investigate 10 min · Modify 10 min · Practise and Make 12 min). -->

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## Today

**We are learning to** a composite shape is made of shapes you know. Cut it into pieces and add their areas, or see it as a big rectangle with pieces taken away; the area is the same either way. The perimeter is the distance round the outside.

- I can cut a composite shape into pieces I know and add their areas.
- I can find an area by taking pieces away from a big rectangle.
- I can work out a missing side and the perimeter.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need the area of rectangles and triangles (Area unit). The one idea: the area is the same however you cut it, as long as each piece uses its own lengths. The common mistake is giving a piece the whole shape's height (Ella's mistake). The board's three buttons cut across, cut up and down, or take away from the surrounding rectangle; it names each piece and gives its area. The perimeter needs every side, so missing sides (the ? sides) come first. Wider: floor plans of real rooms, shapes that need both adding and taking away. -->

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## Why it matters

> An L-shaped bedroom, a house front with a pointed roof, a photo frame: none is a single rectangle, but each is easy once you cut it up.

<!-- Speaker note: A **composite shape** is made of simple shapes joined together. To find its area: 1. **Cut** it into rectangles, triangles or other shapes you know, find each area and **add** them; or 2. see it as a **big rectangle** and **take away** the pieces that are missing. Read every length you need from the diagram: a missing side is often the difference of two others. The **perimeter** is the total length round the outside, in units of length (m), not area (m²). -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**An L-shaped room is 8 m wide and 6 m tall, with a 5 m by 3 m corner missing. What is its area?**

- 18 m²
- 28 m²
- 33 m²
- 48 m²

<!-- Speaker note: What happens: 33 m². The whole 8 by 6 rectangle is 48 m², and the missing corner is 15 m²: 48 − 15 = 33. (28 is the perimeter, in metres.) Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask a 48 student what is missing from the rectangle. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fmeasurement%2Fcomposite-shapes%2Fstep-2&code=widget%3A+composite%0Acorners%3A+0%2C0%3B+8%2C0%3B+8%2C3%3B+3%2C3%3B+3%2C6%3B+0%2C6%0Asplit%3A+on%0Aunits%3A+m%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/measurement/composite-shapes/?full=step-2#step-2)

<!-- Speaker note: Each button shows a different way to see the same shape. -->

<!-- Speaker note: Teacher: ask which way they would choose for this shape, and why. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 33 m². The whole 8 by 6 rectangle is 48 m², and the missing corner is 15 m²: 48 − 15 = 33. (28 is the perimeter, in metres.)

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Missing lengths

Here is the routine.

Now find the step.

$$\text{an L } 10 \text{ m by } 7 \text{ m, a } 6 \times 4 \text{ corner missing}$$

1) $\text{pieces: } 4 \times 7 \text{ and } 6 \times 3$ — Cut into shapes you know, with each piece's own lengths.
2) $28 + 18 = 46\ \text{m}^2$ — Add the pieces.
3) $10 \times 7 - 6 \times 4 = 70 - 24 = 46\ \text{m}^2$ — Or: the big rectangle, take away the missing corner.
4) $10 + 3 + 6 + 4 + 4 + 7 = 34\ \text{m}$ — The perimeter adds every side, in metres.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write an L 10 m by 7 m, a 6 × 4 corner missing and say "A different L, as an example." Line 1: write pieces: 4 × 7 and 6 × 3 and say "Cut into shapes you know, with each piece's own lengths." Line 2: write 28 + 18 = 46m^2 and say "Add the pieces." Line 3: write 10 × 7 - 6 × 4 = 70 - 24 = 46m^2 and say "Or: the big rectangle, take away the missing corner." Line 4: write 10 + 3 + 6 + 4 + 4 + 7 = 34m and say "The perimeter adds every side, in metres." -->

<!-- Speaker note: Teacher: ask how tall each of Ella's pieces is. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Missing lengths

**The L-shape is 8 m along the bottom and 3 m along the top. How long is the step across in the middle?**

* Answer: 5

<!-- Speaker note: The bottom (8 m) = the top (3 m) + the step: 8 − 3 = 5 m. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Ella finds the area of the L-shape. Click the line where her working goes wrong.**

```
 1  The L-shape is 8 m wide at the bottom and 6 m tall on the left.
 2  Ella cuts it up and down into a piece 3 m wide and a piece 5 m wide.
 3  Left piece: 3 × 6 = 18 m².
 4  Right piece: 5 × 6 = 30 m².
 5  Total: 18 + 30 = 48 m².
```

* Answer: line 4

<!-- Speaker note: Line 4: the right piece is only 3 m tall, not 6 m. 5 × 3 = 15, so the total is 18 + 15 = 33 m². -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Three ways

**Ali cuts the L across, Bea cuts it up and down, and Cal takes a corner away from a rectangle. Who gets the right area?**

- Only Cal
- All three, if each uses the right lengths
- None: you must use a formula for L-shapes
- Only Ali and Bea

* Answer: All three, if each uses the right lengths

<!-- Speaker note: The area does not depend on how you cut it, only on using each piece's own lengths. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Other shapes



- **Take away:** 
- **A house:** 
- **Missing sides:** 
- **Units:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/composite-shapes/?full=step-4#step-4)

<!-- Speaker note: What this step is for: A U-shape is quickest by taking away, a house needs a triangle, and the perimeter needs missing sides first; units separate area from length. Hints, in order: The rectangle around the U is 10 m by 8 m. / The gap is 4 m wide and 5 m deep. / Cut where the roof meets the walls. / The roof is a triangle: half of base times height. For students who need more support: Give the shape on grid paper so pieces can be counted as well as multiplied. For early finishers: Draw a floor plan of a room at home with at least six walls. Find its area two ways and its perimeter. The hard part, on purpose: Using each piece's own lengths. Ask students to write the two lengths on each piece before they multiply. When students get stuck: A student uses the whole height for every piece. Ask: How tall is this piece on its own? Fix: Write each piece's two lengths on it before you multiply. / A student cannot find a missing side. Ask: Which two sides together make the opposite side? Fix: A missing side is the difference of two others. / A student gives a perimeter in m². Ask: Are you counting squares or measuring a length? Fix: Perimeter is a length: m. Area counts squares: m². -->

<!-- Speaker note: Teacher: ask why taking away is quicker for the U. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Rectangle and triangle

**A shape is a rectangle 10 cm by 4 cm with a triangle on top (base 10 cm, height 3 cm). What is its area in cm²?**

* Answer: 55

<!-- Speaker note: 10 × 4 + ½ × 10 × 3 = 40 + 15 = 55 cm². -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A frame

**A photo frame is 30 cm by 20 cm on the outside, with a 24 cm by 14 cm hole for the photo. What is the frame's area in cm²?**

* Answer: 264

<!-- Speaker note: 30 × 20 − 24 × 14 = 600 − 336 = 264 cm². -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Perimeter shortcut

**An L-shape fits exactly inside an 8 m by 6 m rectangle (one corner cut out). What is its perimeter in metres?**

* Answer: 28

<!-- Speaker note: The steps in the L add up to the missing sides of the rectangle, so the perimeter is 8 + 6 + 8 + 6 = 28 m. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: triangles

**Review: what is the area of a triangle with base 6 cm and height 5 cm, in cm²?**

* Answer: 15

<!-- Speaker note: ½ × 6 × 5 = 15 cm². -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Another L

**An L-shape is 10 m wide along the bottom and 7 m tall on the left. Its left part is 4 m wide; its right part is 6 m wide and only 3 m tall. What is its area in m²?**

* Answer: 46

<!-- Speaker note: 4 × 7 + 6 × 3 = 28 + 18 = 46 m². -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: design a garden

Then find your garden's perimeter, and say how many metres of edging it needs.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/composite-shapes/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Choosing two lengths to hit a target area uses the decomposition in reverse. Hints, in order: The left part is always 3 × 6 = 18 m². / The right part must be 38 − 18 = 20 m². -->

<!-- Speaker note: Teacher: ask for a second pair of lengths that also works. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**A rectangle 12 m by 5 m has a 4 m by 2 m corner cut out. Find its area.**

* 12 × 5 − 4 × 2 = 60 − 8 = 52 m².

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: This starts AC9M8M01. Where it leads: irregular shapes (next lesson), prisms with composite bases (Volume), and circles in composite shapes. -->