---
title: "Area of parallelograms"
subtitle: "Area, volume, circles and Pythagoras · Area · Lesson 1"
---

# Area of parallelograms

**Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}}

> **Key question:** 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?


:::teacher-note

**In one sentence:** 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).

**Students should be able to say:** 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.

**Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet.

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## Predict

Picture the parallelogram as the rectangle pushed over. Does anything get added or taken away?

:::keep-together

**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

My prediction, and why:

{{lines-3}}

:::


:::teacher-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.

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## Run

Try it on the board in the tutorial. Was your prediction right? What was different?

{{lines-2}}

## Base times height


:::keep-together

### Cut, move, multiply

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

1. $A = 6 \times 4 = 24 \text{ cm}^2$ — A rectangle: 4 rows of 6 unit squares.
2. $\text{parallelogram} \to \text{rectangle}$ — Cut along the height and move the triangle: the same base and height.
3. $A = b \times h$ — So a parallelogram's area is base × perpendicular height.
4. $A = 8 \times 5 = 40 \text{ cm}^2$ — Use the height at a right angle to the base, never the slanted side.

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

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

Answer: {{field-15}}

:::


:::teacher-note

**Answer:** 40

8 × 5 = 40 cm².

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:::keep-together

### 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
```

Line: {{field-15}}

Why is it wrong?

{{lines-2}}

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:::teacher-note

**Answer:** line 3

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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:::keep-together

### 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

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:::teacher-note

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

Cut along the height and move the triangle: the parallelogram becomes the rectangle, with nothing added or lost.

:::


:::typical

## One change at a time


**Straighten it.** Each task changes one thing.

**Area 24.** 

**A decimal height.** 

**Backwards: the height.** 

**Very slanted.** 

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

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## Parallelograms


:::keep-together

### Base times height

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

Answer: {{field-15}}

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:::teacher-note

**Answer:** 63

9 × 7 = 63 cm².

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:::keep-together

### Ignore the slant

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

Answer: {{field-15}}

:::


:::teacher-note

**Answer:** 45

10 × 4.5 = 45 cm².

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:::keep-together

### 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

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:::teacher-note

**Answer:** 10 m²

5 × 2 = 10 square metres: 10 m².

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:::keep-together

### Review: decimals

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

Answer: {{field-15}}

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:::teacher-note

**Answer:** 27

4 × 6 = 24 and 0.5 × 6 = 3: 27.

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## Make: area 18, slanted

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

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

## Exit ticket

:::keep-together

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

{{lines-3}}

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:::teacher-note

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

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