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# Area of parallelograms

Area, volume, circles and Pythagoras · Area

<!-- Speaker note: Students cut a parallelogram along its height and move the triangle to make a rectangle, see that sliding the top keeps the area, and use A = bh with the perpendicular height (AC9M7M01, node 1). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** cut a triangle off one end of a parallelogram and move it to the other end: it becomes a rectangle with the same base and height. So the area of a parallelogram is base × perpendicular height.

- I can explain why a parallelogram has the same area as a rectangle with the same base and height.
- I can find the area of a parallelogram with base × perpendicular height.
- I can tell the perpendicular height from a slanted side.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need area of rectangles by counting rows of squares (Year 6) and the meaning of cm². The one idea: a parallelogram rearranges into a rectangle with the same base and height, so A = bh. The common mistake is using the slanted side (Ruby's mistake); the board's dashed height with its right-angle mark is the fix. -->

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## Why it matters

> Tiles, bricks in a wall, and the stripes of a pedestrian crossing seen from an angle are parallelograms. How much area does a slanted tile cover?

<!-- Speaker note: The **area** of a shape is how many unit squares it covers. A rectangle 6 cm long and 4 cm high has 4 rows of 6 squares: $6 \times 4 = 24$ square centimetres, written 24 cm². A **parallelogram** has two pairs of parallel sides, like a rectangle pushed over. Its **base** is one side, and its **perpendicular height** is the distance straight up from the base to the opposite side, at a right angle (marked ∟). The slanted side is longer than the height. -->

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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

**A rectangle and a parallelogram both have base 6 cm and perpendicular height 4 cm. Which covers more area?**

- They cover the same area
- The parallelogram: its slanted sides are longer
- The rectangle: it is not pushed over

<!-- Speaker note: What happens: The same: 24 cm² each. Cut a triangle off the parallelogram and move it, and it becomes the 6 by 4 rectangle. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose the parallelogram think longer sides mean more area. -->

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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%2Fparallelograms%2Fstep-2&code=widget%3A+area%0Ashape%3A+parallelogram%0Ab%3A+6%0Ah%3A+4%0As%3A+2%0Acut%3A+on%0Aunits%3A+cm%0A&fresh=1}}

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

<!-- Speaker note: The dashed orange line is the perpendicular height. The cut goes along it. -->

<!-- Speaker note: Teacher: ask where the cut went, and why there. -->

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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?

* The same: 24 cm² each. Cut a triangle off the parallelogram and move it, and it becomes the 6 by 4 rectangle.

<!-- 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>

## Cut, move, multiply

Here is why the formula works, then how to use it.

Now try one: base 8 cm, height 5 cm.

$$A = 6 \times 4 = 24 \text{ cm}^2$$

1) $\text{parallelogram} \to \text{rectangle}$ — Cut along the height and move the triangle: the same base and height.
2) $A = b \times h$ — So a parallelogram's area is base × perpendicular height.
3) $A = 8 \times 5 = 40 \text{ cm}^2$ — Use the height at a right angle to the base, never the slanted side.

<!-- 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 A = 6 × 4 = 24 cm^2 and say "A rectangle: 4 rows of 6 unit squares." Line 1: write parallelogram rectangle and say "Cut along the height and move the triangle: the same base and height." Line 2: write A = b × h and say "So a parallelogram's area is base × perpendicular height." Line 3: write A = 8 × 5 = 40 cm^2 and say "Use the height at a right angle to the base, never the slanted side." -->

<!-- Speaker note: Teacher: have students point to the height on the bug task's picture in their head: which length meets the base at a right angle? -->

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## Cut, move, multiply

**A parallelogram has base 8 cm and perpendicular height 5 cm. What is its area, in cm²?**

* Answer: 40

<!-- Speaker note: 8 × 5 = 40 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>

## Find the mistake

**Ruby finds the area of a parallelogram. Click the line where her working goes wrong.**

```
 1  A parallelogram has base 7 cm, slanted side 5 cm and height 4 cm.
 2  Area = base × height
 3  = 7 × 5
 4  = 35 cm²
```

* Answer: line 3

<!-- Speaker note: Line 3: 5 cm is the slanted side, not the height. The height is the perpendicular distance, 4 cm: 7 × 4 = 28 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>

## Why it works

**Why does a parallelogram have the same area as a rectangle with the same base and height?**

- The triangle cut off one end fits exactly into the gap at the other end
- Both shapes have four sides
- Its slanted sides are the same length as the rectangle's sides

* Answer: The triangle cut off one end fits exactly into the gap at the other end

<!-- Speaker note: Cut along the height and move the triangle: the parallelogram becomes the rectangle, with nothing added or lost. -->

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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>

## One change at a time



- **Straighten it:** Each task changes one thing.
- **Area 24:** 
- **A decimal height:** 
- **Backwards: the height:** 
- **Very slanted:** 

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

<!-- Speaker note: What this step is for: One change at a time: sliding the top to make a rectangle, building an area from a base, a decimal height, finding a missing height (backwards), then the edge case of a very slanted parallelogram. Hints, in order: Move the top until it sits straight above the left end of the base. / Which number times 8 makes 24? / Set b to 8 first. For students who need more support: Cut paper parallelograms and rearrange them; count squares on the board for small cases. For early finishers: Draw three different parallelograms on squared paper, each with area 24 cm², none of them a rectangle. The hard part, on purpose: Choosing the perpendicular height when a slanted side is also labelled. Ask students to trace the right-angle mark with a finger before they multiply, rather than telling them which number to use. When students get stuck: A student multiplies by the slanted side (Ruby's mistake). Ask: Which length meets the base at a right angle? Fix: Use the perpendicular height: the dashed line with the ∟. / A student adds base and height. Ask: How many rows of squares, and how many in each row? Fix: Area is rows × squares in a row: multiply. / A student writes cm for an area. Ask: Are we counting lengths or squares? Fix: Squares: cm². -->

<!-- Speaker note: Teacher: for the missing height, ask what operation undoes × 6. -->

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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>

## Base times height

**Base 9 cm, perpendicular height 7 cm. Area, in cm²?**

* Answer: 63

<!-- Speaker note: 9 × 7 = 63 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>

## Ignore the slant

**A parallelogram has base 10 cm, slanted side 6 cm and perpendicular height 4.5 cm. Area, in cm²?**

* Answer: 45

<!-- Speaker note: 10 × 4.5 = 45 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>

## Units

**A parallelogram's base is 5 m and its height 2 m. How is its area written?**

- 10 m²
- 10 m
- 10 m³
- 7 m²

* Answer: 10 m²

<!-- Speaker note: 5 × 2 = 10 square metres: 10 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>

## Review: decimals

**Review: work out $4.5 \times 6$.**

* Answer: 27

<!-- Speaker note: 4 × 6 = 24 and 0.5 × 6 = 3: 27. -->

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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: area 18, slanted

Then find a different base and height for the same area.

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

<!-- Speaker note: What this step is for: Building a slanted parallelogram with a given area means choosing a base and height that multiply to it, and seeing the slant does not matter. Hints, in order: Which pairs of numbers multiply to 18? / Then move the top so it is not a rectangle. -->

<!-- Speaker note: Teacher: collect different base and height pairs for 18. -->

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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 parallelogram has base 9 cm, slanted side 5 cm and perpendicular height 4 cm. Find its area.**

* 9 × 4 = 36 cm² (the 5 cm slanted side is not used).

<!-- 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: Next: Area of triangles (node 1), half of a parallelogram. Where it leads: area of trapeziums and composite shapes (Year 8), and volume of prisms (the base times the height again). -->