Year 7 Maths: Area between two triangles

Introduction

Understanding the concept of the area between two triangles is a vital skill in geometry. The area between two triangles generally refers to the difference in their areas when one triangle is subtracted from another. It can occur when triangles overlap or when one triangle is placed inside another. In Key Stage 3, students should know how to calculate the area of triangles using the formula:

$$\text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height}$$

When dealing with the area between two triangles, you simply calculate the area of each triangle separately and then subtract the smaller area from the larger one.

Key Points to Remember:

1. Formula for the area of a triangle:
   $$A = \frac{1}{2} \times \text{base} \times \text{height}$$
2. When the triangles are overlapping or when one triangle is subtracted from another, calculate the area of each and find the difference.
3. It’s essential to compare the base and height of each triangle accurately and ensure that both measurements are in the same units.

Question Set on Area Between Two Triangles

Easy Level

H1: Basic Calculation of Triangle Areas

Q1: Find the area of a triangle with a base of 5 cm and a height of 8 cm.

Q2: Calculate the area of a triangle with a base of 7 cm and a height of 10 cm.

Q3: What is the area of a triangle with a base of 6 cm and a height of 9 cm?

Q4: If a triangle has a base of 10 cm and a height of 5 cm, what is its area?

Q5: Find the area of a triangle with a base of 4 cm and a height of 7 cm.

H2: Comparing Two Triangles

Q6: Two triangles have areas of 20 cm² and 15 cm², respectively. What is the area between the two triangles?

Q7: Calculate the difference in areas between two triangles, one with a base of 6 cm and height of 9 cm, and the other with a base of 4 cm and height of 5 cm.

Q8: Triangle A has a base of 10 cm and height of 12 cm. Triangle B has a base of 5 cm and height of 6 cm. Find the difference in their areas.

Q9: Two triangles have areas of 35 cm² and 25 cm². What is the area between them?

Q10: Triangle A has an area of 50 cm², and Triangle B has an area of 30 cm². Find the difference in their areas.

H3: Identifying Overlapping Areas

Q11: Two triangles overlap such that Triangle A has an area of 40 cm², and Triangle B has an area of 30 cm². What is the area of the non-overlapping section?

Q12: Two triangles overlap, and the overlapping area is 5 cm². Triangle A has an area of 20 cm², and Triangle B has an area of 15 cm². Find the total area of the non-overlapping sections.

Q13: Triangle A has an area of 25 cm², and Triangle B has an area of 15 cm². If they overlap, covering an area of 10 cm², what is the non-overlapping area?

Q14: If two triangles overlap, covering an area of 8 cm², and their individual areas are 28 cm² and 20 cm², find the non-overlapping area.

Q15: The areas of two triangles are 18 cm² and 14 cm², and they overlap over an area of 6 cm². Find the non-overlapping area.

Medium Level

H4: Calculating the Area of Right-Angled Triangles

Q1: A right-angled triangle has a base of 8 cm and a height of 6 cm. Find its area.

Q2: Calculate the area of a right-angled triangle with a base of 9 cm and a height of 7 cm.

Q3: What is the area of a right-angled triangle with a base of 5 cm and height of 12 cm?

Q4: Find the area of a right-angled triangle with a base of 11 cm and a height of 10 cm.

Q5: Calculate the area of a right-angled triangle with a base of 13 cm and a height of 8 cm.

H5: Area Between Two Right-Angled Triangles

Q6: A right-angled triangle has a base of 10 cm and a height of 12 cm. Another right-angled triangle has a base of 6 cm and a height of 9 cm. Find the difference in their areas.

Q7: Two right-angled triangles have bases of 7 cm and 9 cm, and heights of 5 cm and 8 cm, respectively. Calculate the difference in their areas.

Q8: Find the area difference between two right-angled triangles, one with a base of 15 cm and height of 10 cm, and the other with a base of 10 cm and height of 8 cm.

Q9: Two right-angled triangles have areas of 36 cm² and 28 cm². What is the difference in their areas?

Q10: Calculate the difference in areas between two right-angled triangles, one with a base of 20 cm and height of 15 cm, and the other with a base of 10 cm and height of 10 cm.

H6: Composite Shapes with Triangles

Q11: A composite shape consists of two triangles. Triangle A has an area of 40 cm², and Triangle B has an area of 25 cm². What is the total area of the shape?

Q12: If a composite shape consists of two overlapping triangles with areas of 30 cm² and 20 cm², and the overlapping area is 10 cm², what is the total area?

Q13: Find the total area of a composite shape consisting of two triangles with areas of 50 cm² and 30 cm², with an overlapping area of 20 cm².

Q14: Calculate the total area of a composite shape formed by two triangles, one with an area of 45 cm² and another with an area of 35 cm², overlapping by 15 cm².

Q15: Two triangles form a composite shape, with areas of 60 cm² and 40 cm², overlapping by 25 cm². Find the total area of the shape.

Hard Level

H7: Advanced Calculations for Complex Triangles

Q1: A triangle has a base of 12 cm and height of 10 cm, and another triangle has a base of 14 cm and height of 9 cm. Find the area between them.

Q2: Two triangles have bases of 15 cm and 18 cm, with heights of 12 cm and 10 cm, respectively. Calculate the area between them.

Q3: A triangle has an area of 72 cm², and another has an area of 56 cm². Find the area between the two triangles.

Q4: Find the area between two triangles, one with a base of 25 cm and height of 15 cm, and the other with a base of 20 cm and height of 12 cm.

Q5: Calculate the area between two triangles with areas of 120 cm² and 80 cm².

H8: Overlapping Triangles and Subtractions

Q6: Two triangles overlap with areas of 100 cm² and 80 cm². If the overlapping area is 30 cm², what is the total non-overlapping area?

Q7: Calculate the non-overlapping area between two triangles with areas of 150 cm² and 120 cm², with an overlap of 40 cm².

Q8: Two triangles have areas of 90 cm² and 60 cm², and they overlap over 20 cm². Find the non-overlapping area.

Q9: Find the non-overlapping area between two triangles with areas of 110 cm² and 85 cm², overlapping by 45 cm².

Q10: Calculate the total non-overlapping area between two triangles with areas of 200 cm² and 150 cm², with an overlapping area of 50 cm².

H9: Advanced Applications with Right-Angled Triangles

Q11: A right-angled triangle has a base of 18 cm and a height of 10 cm, and another right-angled triangle has a base of 15 cm and a height of 12 cm. Find the area between them.

Q12: Calculate the area difference between two right-angled triangles, one with a base of 25 cm and height of 20 cm, and the other with a base of 15 cm and height of 10 cm.

Q13: Two right-angled triangles have bases of 30 cm and 20 cm, and heights of 15 cm and 12 cm, respectively. What is the area between them?

Q14: Find the difference in areas between two right-angled triangles, one with a base of 22 cm and height of 18 cm, and the other with a base of 16 cm and height of 14 cm.

Q15: Calculate the area between two right-angled triangles with areas of 200 cm² and 150 cm².

Answers and Explanations

Easy Level

Q1: 20 cm²
Explanation:
$$A = \frac{1}{2} \times 5 \times 8 = 20 \, \text{cm}^2$$

Q2: 35 cm²
Explanation:
$$A = \frac{1}{2} \times 7 \times 10 = 35 \, \text{cm}^2$$

Q3: 27 cm²
Explanation:
$$A = \frac{1}{2} \times 6 \times 9 = 27 \, \text{cm}^2$$

Q4: 25 cm²
Explanation:
$$A = \frac{1}{2} \times 10 \times 5 = 25 \, \text{cm}^2$$

Q5: 14 cm²
Explanation:
$$A = \frac{1}{2} \times 4 \times 7 = 14 \, \text{cm}^2$$

Q6: 5 cm²
Explanation: The difference in their areas is:
$$20 \, \text{cm}^2 – 15 \, \text{cm}^2 = 5 \, \text{cm}^2$$

Q7: 17 cm²
Explanation:

• $$A_1 = \frac{1}{2} \times 6 \times 9 = 27 \, \text{cm}^2$$
• $$A_2 = \frac{1}{2} \times 4 \times 5 = 10 \, \text{cm}^2$$
• Difference:
  $$27 \, \text{cm}^2 – 10 \, \text{cm}^2 = 17 \, \text{cm}^2$$

Q8: 45 cm²
Explanation:

• $$A_1 = \frac{1}{2} \times 10 \times 12 = 60 \, \text{cm}^2$$
• $$A_2 = \frac{1}{2} \times 5 \times 6 = 15 \, \text{cm}^2$$
• Difference:
  $$60 \, \text{cm}^2 – 15 \, \text{cm}^2 = 45 \, \text{cm}^2$$

Q9: 10 cm²
Explanation:
The difference between their areas is:
$$35 \, \text{cm}^2 – 25 \, \text{cm}^2 = 10 \, \text{cm}^2$$

Q10: 20 cm²
Explanation:
The difference between their areas is:
$$50 \, \text{cm}^2 – 30 \, \text{cm}^2 = 20 \, \text{cm}^2$$

Q11: 10 cm²
Explanation:
The difference between the areas is:
$$40 \, \text{cm}^2 – 30 \, \text{cm}^2 = 10 \, \text{cm}^2$$

Q12: 10 cm²
Explanation: The total area of the non-overlapping sections is:
$$20 \, \text{cm}^2 – 5 \, \text{cm}^2 + 15 \, \text{cm}^2 – 5 \, \text{cm}^2 = 10 \, \text{cm}^2$$

Q13: 20 cm²
Explanation:
The non-overlapping area is:
$$25 \, \text{cm}^2 – 10 \, \text{cm}^2 + 15 \, \text{cm}^2 – 10 \, \text{cm}^2 = 20 \, \text{cm}^2$$

Q14: 40 cm²
Explanation:
The non-overlapping area is:
$$28 \, \text{cm}^2 + 20 \, \text{cm}^2 – 8 \, \text{cm}^2 = 40 \, \text{cm}^2$$

Q15: 26 cm²
Explanation:
The non-overlapping area is:
$$18 \, \text{cm}^2 + 14 \, \text{cm}^2 – 6 \, \text{cm}^2 = 26 \, \text{cm}^2$$

Medium Level

Q1: 24 cm²
Explanation:
$$A = \frac{1}{2} \times 8 \times 6 = 24 \, \text{cm}^2$$

Q2: 31.5 cm²
Explanation:
$$A = \frac{1}{2} \times 9 \times 7 = 31.5 \, \text{cm}^2$$

Q3: 30 cm²
Explanation:
$$A = \frac{1}{2} \times 5 \times 12 = 30 \, \text{cm}^2$$

Q4: 55 cm²
Explanation:
$$A = \frac{1}{2} \times 11 \times 10 = 55 \, \text{cm}^2$$

Q5: 52 cm²
Explanation:
$$A = \frac{1}{2} \times 13 \times 8 = 52 \, \text{cm}^2$$

Q6: 24 cm²
Explanation:

• $$A_1 = \frac{1}{2} \times 10 \times 12 = 60 \, \text{cm}^2$$
• $$A_2 = \frac{1}{2} \times 6 \times 9 = 27 \, \text{cm}^2$$
• Difference:
  $$60 \, \text{cm}^2 – 27 \, \text{cm}^2 = 33 \, \text{cm}^2$$

Q7: 17 cm²
Explanation:

• $$A_1 = \frac{1}{2} \times 9 \times 8 = 36 \, \text{cm}^2$$
• $$A_2 = \frac{1}{2} \times 7 \times 5 = 17.5 \, \text{cm}^2$$
• Difference:
  $$36 \, \text{cm}^2 – 17.5 \, \text{cm}^2 = 18.5 \, \text{cm}^2$$

Q8: 45 cm²
Explanation:

• $$A_1 = \frac{1}{2} \times 15 \times 10 = 75 \, \text{cm}^2$$
• $$A_2 = \frac{1}{2} \times 10 \times 8 = 40 \, \text{cm}^2$$
• Difference:
  $$75 \, \text{cm}^2 – 40 \, \text{cm}^2 = 35 \, \text{cm}^2$$

Q9: 8 cm²
Explanation: The difference between their areas is:
$$36 \, \text{cm}^2 – 28 \, \text{cm}^2 = 8 \, \text{cm}^2$$

Q10: 75 cm²
Explanation:

• $$A_1 = \frac{1}{2} \times 20 \times 15 = 150 \, \text{cm}^2$$
• $$A_2 = \frac{1}{2} \times 10 \times 10 = 50 \, \text{cm}^2$$
• Difference:
  $$150 \, \text{cm}^2 – 50 \, \text{cm}^2 = 100 \, \text{cm}^2$$

Q11: 65 cm²
Explanation:
The total area of the composite shape is:
$$40 \, \text{cm}^2 + 25 \, \text{cm}^2 = 65 \, \text{cm}^2$$

Q12: 40 cm²
Explanation:
Total area:
$$30 \, \text{cm}^2 + 20 \, \text{cm}^2 – 10 \, \text{cm}^2 = 40 \, \text{cm}^2$$

Q13: 60 cm²
Explanation:
Total area:
$$50 \, \text{cm}^2 + 30 \, \text{cm}^2 – 20 \, \text{cm}^2 = 60 \, \text{cm}^2$$

Q14: 65 cm²
Explanation:
Total area:
$$45 \, \text{cm}^2 + 35 \, \text{cm

}^2 – 15 \, \text{cm}^2 = 65 \, \text{cm}^2$$

Q15: 75 cm²
Explanation:
Total area:
$$60 \, \text{cm}^2 + 40 \, \text{cm}^2 – 25 \, \text{cm}^2 = 75 \, \text{cm}^2$$

Hard Level

Q1: 9 cm²
Explanation:

• $$A_1 = \frac{1}{2} \times 12 \times 10 = 60 \, \text{cm}^2$$
• $$A_2 = \frac{1}{2} \times 14 \times 9 = 63 \, \text{cm}^2$$
• Difference:
  $$63 \, \text{cm}^2 – 60 \, \text{cm}^2 = 3 \, \text{cm}^2$$

Q2: 21 cm²
Explanation:

• $$A_1 = \frac{1}{2} \times 15 \times 12 = 90 \, \text{cm}^2$$
• $$A_2 = \frac{1}{2} \times 18 \times 10 = 90 \, \text{cm}^2$$
• Difference:
  $$90 \, \text{cm}^2 – 90 \, \text{cm}^2 = 0 \, \text{cm}^2$$

Q3: 16 cm²
Explanation: The difference between their areas is:
$$72 \, \text{cm}^2 – 56 \, \text{cm}^2 = 16 \, \text{cm}^2$$

Q4: 22.5 cm²
Explanation:

• $$A_1 = \frac{1}{2} \times 25 \times 15 = 187.5 \, \text{cm}^2$$
• $$A_2 = \frac{1}{2} \times 20 \times 12 = 120 \, \text{cm}^2$$
• Difference:
  $$187.5 \, \text{cm}^2 – 120 \, \text{cm}^2 = 67.5 \, \text{cm}^2$$

Q5: 40 cm²
Explanation: The difference between their areas is:
$$120 \, \text{cm}^2 – 80 \, \text{cm}^2 = 40 \, \text{cm}^2$$

H8: Overlapping Triangles and Subtractions

Q6: 50 cm²
Explanation: The non-overlapping area is:
$$100 \, \text{cm}^2 + 80 \, \text{cm}^2 – 30 \, \text{cm}^2 = 50 \, \text{cm}^2$$

Q7: 190 cm²
Explanation: The non-overlapping area is:
$$150 \, \text{cm}^2 + 120 \, \text{cm}^2 – 40 \, \text{cm}^2 = 190 \, \text{cm}^2$$

Q8: 130 cm²
Explanation: The non-overlapping area is:
$$90 \, \text{cm}^2 + 60 \, \text{cm}^2 – 20 \, \text{cm}^2 = 130 \, \text{cm}^2$$

Q9: 150 cm²
Explanation: The non-overlapping area is:
$$110 \, \text{cm}^2 + 85 \, \text{cm}^2 – 45 \, \text{cm}^2 = 150 \, \text{cm}^2$$

Q10: 300 cm²
Explanation: The non-overlapping area is:
$$200 \, \text{cm}^2 + 150 \, \text{cm}^2 – 50 \, \text{cm}^2 = 300 \, \text{cm}^2$$

H9: Advanced Applications with Right-Angled Triangles

Q11: 9 cm²
Explanation:

• $$A_1 = \frac{1}{2} \times 18 \times 10 = 90 \, \text{cm}^2$$
• $$A_2 = \frac{1}{2} \times 15 \times 12 = 90 \, \text{cm}^2$$
• Difference:
  $$90 \, \text{cm}^2 – 90 \, \text{cm}^2 = 0 \, \text{cm}^2$$

Q12: 25 cm²
Explanation:

• $$A_1 = \frac{1}{2} \times 25 \times 20 = 250 \, \text{cm}^2$$
• $$A_2 = \frac{1}{2} \times 15 \times 10 = 75 \, \text{cm}^2$$
• Difference:
  $$250 \, \text{cm}^2 – 75 \, \text{cm}^2 = 175 \, \text{cm}^2$$

Q13: 105 cm²
Explanation: The difference in their areas is:
$$180 \, \text{cm}^2 – 105 \, \text{cm}^2 = 75 \, \text{cm}^2$$