Year 7 Maths: Area Between Two Rectangles

Introduction

In Year 7 Maths, students learn how to calculate the area between two rectangles. This concept builds on basic area calculations and involves finding the area of two overlapping rectangles and determining the remaining space between them. This is useful in real-life situations, such as finding the area of a garden path or the space between two objects on a surface.

Key Concepts

1. Area of a Rectangle: The area of a rectangle is calculated by multiplying its length by its width:
   $$\text{Area} = \text{Length} \times \text{Width}$$
2. Area Between Two Rectangles: To find the area between two rectangles, you subtract the area of the smaller rectangle from the larger rectangle.
   $$\text{Area between rectangles} = \text{Area of large rectangle} – \text{Area of small rectangle}$$
3. Position of Rectangles: The two rectangles could be nested (one inside the other) or one could overlap the other partially.

Area Between Two Rectangles: Questions

Level 1: Easy

Find the area of the larger rectangle, the smaller rectangle, and the area between them:

1. Large rectangle: 10 cm by 8 cm, Small rectangle: 6 cm by 4 cm
2. Large rectangle: 12 cm by 10 cm, Small rectangle: 8 cm by 6 cm
3. Large rectangle: 15 cm by 7 cm, Small rectangle: 10 cm by 5 cm
4. Large rectangle: 9 cm by 5 cm, Small rectangle: 7 cm by 3 cm
5. Large rectangle: 20 cm by 15 cm, Small rectangle: 12 cm by 10 cm
6. Large rectangle: 14 cm by 8 cm, Small rectangle: 9 cm by 6 cm
7. Large rectangle: 18 cm by 12 cm, Small rectangle: 11 cm by 8 cm
8. Large rectangle: 16 cm by 14 cm, Small rectangle: 10 cm by 7 cm
9. Large rectangle: 22 cm by 11 cm, Small rectangle: 12 cm by 9 cm
10. Large rectangle: 25 cm by 20 cm, Small rectangle: 15 cm by 12 cm

Level 2: Medium

Find the area of the larger rectangle, the smaller rectangle, and the area between them. The large rectangle may overlap or partially cover the smaller rectangle:

1. Large rectangle: 30 cm by 20 cm, Small rectangle: 12 cm by 9 cm
2. Large rectangle: 35 cm by 18 cm, Small rectangle: 15 cm by 10 cm
3. Large rectangle: 28 cm by 14 cm, Small rectangle: 9 cm by 7 cm
4. Large rectangle: 32 cm by 25 cm, Small rectangle: 18 cm by 12 cm
5. Large rectangle: 40 cm by 30 cm, Small rectangle: 20 cm by 15 cm
6. Large rectangle: 24 cm by 16 cm, Small rectangle: 11 cm by 8 cm
7. Large rectangle: 50 cm by 40 cm, Small rectangle: 30 cm by 20 cm
8. Large rectangle: 45 cm by 35 cm, Small rectangle: 25 cm by 15 cm
9. Large rectangle: 38 cm by 22 cm, Small rectangle: 15 cm by 10 cm
10. Large rectangle: 55 cm by 45 cm, Small rectangle: 32 cm by 28 cm

Level 3: Hard

Find the area of the larger rectangle, the smaller rectangle, and the area between them. Some rectangles are irregularly placed and may only partially overlap:

1. Large rectangle: 60 cm by 45 cm, Small rectangle: 20 cm by 15 cm
2. Large rectangle: 70 cm by 50 cm, Small rectangle: 25 cm by 20 cm
3. Large rectangle: 80 cm by 60 cm, Small rectangle: 30 cm by 25 cm
4. Large rectangle: 85 cm by 55 cm, Small rectangle: 35 cm by 28 cm
5. Large rectangle: 90 cm by 70 cm, Small rectangle: 40 cm by 30 cm
6. Large rectangle: 100 cm by 80 cm, Small rectangle: 45 cm by 35 cm
7. Large rectangle: 95 cm by 65 cm, Small rectangle: 50 cm by 40 cm
8. Large rectangle: 110 cm by 90 cm, Small rectangle: 55 cm by 45 cm
9. Large rectangle: 120 cm by 100 cm, Small rectangle: 60 cm by 50 cm
10. Large rectangle: 150 cm by 130 cm, Small rectangle: 75 cm by 60 cm

Answers with Explanations

Level 1: Easy

Find the area of the larger rectangle, the smaller rectangle, and the area between them:

1. Large rectangle: $$10 \times 8 = 80 \, \text{cm}^2$$
   Small rectangle: $$6 \times 4 = 24 \, \text{cm}^2$$
   Area between them: $$80 – 24 = 56 \, \text{cm}^2$$
2. Large rectangle: $$12 \times 10 = 120 \, \text{cm}^2$$
   Small rectangle: $$8 \times 6 = 48 \, \text{cm}^2$$
   Area between them: $$120 – 48 = 72 \, \text{cm}^2$$
3. Large rectangle: $$15 \times 7 = 105 \, \text{cm}^2$$
   Small rectangle: $$10 \times 5 = 50 \, \text{cm}^2$$
   Area between them: $$105 – 50 = 55 \, \text{cm}^2$$
4. Large rectangle: $$9 \times 5 = 45 \, \text{cm}^2$$
   Small rectangle: $$7 \times 3 = 21 \, \text{cm}^2$$
   Area between them: $$45 – 21 = 24 \, \text{cm}^2$$
5. Large rectangle: $$20 \times 15 = 300 \, \text{cm}^2$$
   Small rectangle: $$12 \times 10 = 120 \, \text{cm}^2$$
   Area between them: $$300 – 120 = 180 \, \text{cm}^2$$
6. Large rectangle: $$14 \times 8 = 112 \, \text{cm}^2$$
   Small rectangle: $$9 \times 6 = 54 \, \text{cm}^2$$
   Area between them: $$112 – 54 = 58 \, \text{cm}^2$$
7. Large rectangle: $$18 \times 12 = 216 \, \text{cm}^2$$
   Small rectangle: $$11 \times 8 = 88 \, \text{cm}^2$$
   Area between them: $$216 – 88 = 128 \, \text{cm}^2$$
8. Large rectangle: $$16 \times 14 = 224 \, \text{cm}^2$$
   Small rectangle: $$10 \times 7 = 70 \, \text{cm}^2$$
   Area between them: $$224 – 70 = 154 \, \text{cm}^2$$
9. Large rectangle: $$22 \times 11 = 242 \, \text{cm}^2$$
   Small rectangle: $$12 \times 9 = 108 \, \text{cm}^2$$
   Area between them: $$242 – 108 = 134 \, \text{cm}^2$$
10. Large rectangle: $$25 \times 20 = 500 \, \text{cm}^2$$
    Small rectangle: $$15 \times 12 = 180 \, \text{cm}^2$$
    Area between them: $$500 – 180 = 320 \, \text{cm}^2$$

Level 2: Medium

Find the area of the larger rectangle, the smaller rectangle, and the area between them:

1. Large rectangle: $$30 \times 20 = 600 \, \text{cm}^2$$
   Small rectangle: $$12 \times 9 = 108 \, \text{cm}^2$$
   Area between them: $$600 – 108 = 492 \, \text{cm}^2$$
2. Large rectangle: $$35 \times 18 = 630 \, \text{cm}^2$$ Small rectangle: $$15 \times 10 = 150 \, \text{cm}^2$$
   Area between them: $$630 – 150 = 480 \, \text{cm}^2$$
3. Large rectangle: $$28 \times 14 = 392 \, \text{cm}^2$$
   Small rectangle: $$9 \times 7 = 63 \, \text{cm}^2$$
   Area between them: $$392 – 63 = 329 \, \text{cm}^2$$
4. Large rectangle: $$32 \times 25 = 800 \, \text{cm}^2$$
   Small rectangle: $$18 \times 12 = 216 \, \text{cm}^2$$
   Area between them: $$800 – 216 = 584 \, \text{cm}^2$$
5. Large rectangle: $$40 \times 30 = 1200 \, \text{cm}^2$$
   Small rectangle: $$20 \times 15 = 300 \, \text{cm}^2$$
   Area between them: $$1200 – 300 = 900 \, \text{cm}^2$$
6. Large rectangle: $$24 \times 16 = 384 \, \text{cm}^2$$
   Small rectangle: $$11 \times 8 = 88 \, \text{cm}^2$$
   Area between them: $$384 – 88 = 296 \, \text{cm}^2$$
7. Large rectangle: $$50 \times 40 = 2000 \, \text{cm}^2$$
   Small rectangle: $$30 \times 20 = 600 \, \text{cm}^2$$
   Area between them: $$2000 – 600 = 1400 \, \text{cm}^2$$
8. Large rectangle: $$45 \times 35 = 1575 \, \text{cm}^2$$
   Small rectangle: $$25 \times 15 = 375 \, \text{cm}^2$$
   Area between them: $$1575 – 375 = 1200 \, \text{cm}^2$$
9. Large rectangle: $$38 \times 22 = 836 \, \text{cm}^2$$
   Small rectangle: $$15 \times 10 = 150 \, \text{cm}^2$$
   Area between them: $$836 – 150 = 686 \, \text{cm}^2$$
10. Large rectangle: $$55 \times 45 = 2475 \, \text{cm}^2$$
    Small rectangle: $$32 \times 28 = 896 \, \text{cm}^2$$
    Area between them: $$2475 – 896 = 1579 \, \text{cm}^2$$

Level 3: Hard

Find the area of the larger rectangle, the smaller rectangle, and the area between them:

1. Large rectangle: $$60 \times 45 = 2700 \, \text{cm}^2$$
   Small rectangle: $$20 \times 15 = 300 \, \text{cm}^2$$
   Area between them: $$2700 – 300 = 2400 \, \text{cm}^2$$
2. Large rectangle: $$70 \times 50 = 3500 \, \text{cm}^2$$
   Small rectangle: $$25 \times 20 = 500 \, \text{cm}^2$$
   Area between them: $$3500 – 500 = 3000 \, \text{cm}^2$$
3. Large rectangle: $$80 \times 60 = 4800 \, \text{cm}^2$$
   Small rectangle: $$30 \times 25 = 750 \, \text{cm}^2$$
   Area between them: $$4800 – 750 = 4050 \, \text{cm}^2$$
4. Large rectangle: $$85 \times 55 = 4675 \, \text{cm}^2$$
   Small rectangle: $$35 \times 28 = 980 \, \text{cm}^2$$
   Area between them: $$4675 – 980 = 3695 \, \text{cm}^2$$
5. Large rectangle: $$90 \times 70 = 6300 \, \text{cm}^2$$
   Small rectangle: $$40 \times 30 = 1200 \, \text{cm}^2$$
   Area between them: $$6300 – 1200 = 5100 \, \text{cm}^2$$
6. Large rectangle: $$100 \times 80 = 8000 \, \text{cm}^2$$
   Small rectangle: $$45 \times 35 = 1575 \, \text{cm}^2$$
   Area between them: $$8000 – 1575 = 6425 \, \text{cm}^2$$
7. Large rectangle: $$95 \times 65 = 6175 \, \text{cm}^2$$
   Small rectangle: $$50 \times 40 = 2000 \, \text{cm}^2$$
   Area between them: $$6175 – 2000 = 4175 \, \text{cm}^2$$
8. Large rectangle: $$110 \times 90 = 9900 \, \text{cm}^2$$
   Small rectangle: $$55 \times 45 = 2475 \, \text{cm}^2$$
   Area between them: $$9900 – 2475 = 7425 \, \text{cm}^2$$
9. Large rectangle: $$120 \times 100 = 12000 \, \text{cm}^2$$
   Small rectangle: $$60 \times 50 = 3000 \, \text{cm}^2$$
   Area between them: $$12000 – 3000 = 9000 \, \text{cm}^2$$
10. Large rectangle: $$150 \times 130 = 19500 \, \text{cm}^2$$
    Small rectangle: $$75 \times 60 = 4500 \, \text{cm}^2$$
    Area between them: $$19500 – 4500 = 15000 \, \text{cm}^2$$

Explanation of Key Steps

• Step 1: Calculate the area of each rectangle by multiplying the length by the width.
• Step 2: Subtract the area of the smaller rectangle from the area of the larger rectangle to find the area between them.

This set of exercises is designed to progressively challenge students, from simple problems with nested rectangles to more complex problems involving larger dimensions and partial overlaps.