How to Find Area of Rectangles Using Distributive Property Formula and Examples
Understanding how to measure area using the distributive property is a key skill in mathematics, especially for solving geometry and multiplication problems efficiently. This concept is not just important for school exams like CBSE, ICSE, or even competitive exams, but it also finds many applications in daily life, such as construction and design. Mastery of this topic builds a strong foundation in both arithmetic and geometry.
What Is Measuring Area Using the Distributive Property?
The distributive property in mathematics allows you to multiply a number by a sum or difference by distributing the multiplication across each term. When applied to area, especially with rectangles, you can break a large area into smaller, manageable parts, calculate each part separately, and then sum them up. This approach is called area addition strategy or breaking apart area.
For example: Instead of calculating the area of a 7 x 13 rectangle in one step, you can split 13 into 10 + 3 and use the distributive property as follows:
- Area = 7 x (10 + 3) = (7 x 10) + (7 x 3) = 70 + 21 = 91
This method helps when numbers are large or when working with composite figures.
Distributive Property and Its Area Formula
The distributive property states that: a x (b + c) = (a x b) + (a x c). In the context of area:
If a rectangle has length \( l \) and width \( w \), and you split the width into two parts (\( w_1 \) and \( w_2 \)), then:
Area = l x (w₁ + w₂) = (l x w₁) + (l x w₂)
You can also use this with lengths, or split both dimensions for more complex shapes.
Worked Examples: Measuring Area Using the Distributive Property
Let's see how to use the distributive property step by step.
Example 1: Rectangle Split by Width
- Given: Rectangle of length 8 cm and width 15 cm.
- Split 15 as 10 + 5.
- Apply distributive property: 8 x (10 + 5) = (8 x 10) + (8 x 5)
- Calculate: 8 x 10 = 80, 8 x 5 = 40
- Add: 80 + 40 = 120 cm²
Example 2: Irregular Shape (Composite Rectangle)
- Given: An L-shaped figure made by joining rectangles of 6x3 cm and 6x7 cm.
- Find area of each: 6 x 3 = 18 cm², 6 x 7 = 42 cm²
- Total area = 18 + 42 = 60 cm²
- This is the same as 6 x (3 + 7) = 6 x 10 = 60 cm², showing the distributive property in action.
Example 3: Word Problem
A garden is rectangular, 5 m wide. The length is 9 m in one section and 6 m in the other. Find the total area.
- First section area: 5 x 9 = 45 m²
- Second section area: 5 x 6 = 30 m²
- Total area = 45 + 30 = 75 m²
- Alternatively, total area = 5 x (9 + 6) = 5 x 15 = 75 m²
Practice Problems
- A rectangle has a width of 12 cm and a length of 7 cm. Break apart 12 as 10 + 2. Find the total area using the distributive property.
- Calculate the area of a 9 m by 14 m rectangle by splitting 14 into 10 + 4.
- A composite shape consists of rectangles of 8 x 6 cm and 8 x 3 cm. Find the total area in two ways: as sum of parts, and by applying the distributive property.
- A floor is split into two sections of 5 x 8 ft and 5 x 5 ft. What is the total area?
- Using the distributive property, what is the area of a rectangle 11 x 13 (split 13 as 10 + 3)?
Common Mistakes to Avoid
- Forgetting to multiply both split parts (e.g., only calculating one and forgetting the other).
- Mixing up addition and multiplication—do not add side lengths before multiplying, unless using the distributive split.
- Applying the distributive property to non-rectangular shapes without checking if splitting makes sense.
- Using incorrect units in answers—always write area in cm², m², etc.
Real-World Applications
Measuring area using distributive property is widely used in architecture, flooring, painting walls, and even in agriculture for land measurement. Builders often break up large project plans into smaller modules for easier calculation. At Vedantu, our teachers use this method to help students tackle challenging word problems and composite shapes efficiently.
For more on related topics, read about the Area of Rectangle or Distributive Property on Vedantu.
In summary, the distributive property allows students to measure area by breaking complex shapes into simpler parts and adding their areas. This not only streamlines calculations during exams but also develops a deeper understanding of multiplication and geometry. Keep practicing such problems to build your confidence for both classroom and real-world applications!
FAQs on Measuring Area Using the Distributive Property in Maths
1. What is measuring area using the distributive property?
Measuring area using the distributive property means breaking a large rectangle into smaller rectangles and adding their areas together. The distributive property of multiplication over addition is written as a(b + c) = ab + ac. In area models:
- Total area = length × (width₁ + width₂)
- = (length × width₁) + (length × width₂)
2. How do you find the area of a rectangle using the distributive property?
To find the area using the distributive property, multiply one side by the sum of the split parts of the other side. Follow these steps:
- Step 1: Write the area formula: Area = length × width.
- Step 2: Split one dimension into smaller parts.
- Step 3: Apply a(b + c) = ab + ac.
- Step 4: Add the partial areas.
- 13 = 10 + 3
- 8(10 + 3) = (8 × 10) + (8 × 3)
- = 80 + 24 = 104 square units
3. What is the formula for area using the distributive property?
The formula for area using the distributive property is a(b + c) = ab + ac. In geometry terms:
- Area = length × (width₁ + width₂)
- = (length × width₁) + (length × width₂)
4. Why does the distributive property work for finding area?
The distributive property works for area because the total area of a rectangle is equal to the sum of the areas of its non-overlapping parts. When a rectangle is divided into smaller rectangles:
- Each smaller area is found using length × width.
- Adding those areas gives the total area.
5. Can you give an example of measuring area using the distributive property?
An example of measuring area using the distributive property is finding the area of 6 × 17. Break 17 into 10 + 7:
- 6(10 + 7)
- = (6 × 10) + (6 × 7)
- = 60 + 42
- = 102 square units
6. How does an area model show the distributive property?
An area model shows the distributive property by visually splitting a rectangle into smaller rectangles whose areas are added together. In the model:
- One side is divided into parts (e.g., 15 = 10 + 5).
- Each section forms a smaller rectangle.
- Total area = sum of all smaller areas.
7. What is the difference between the standard area formula and using the distributive property?
The standard area formula multiplies the full length and width directly, while the distributive property splits one dimension before multiplying. Specifically:
- Standard formula: A = l × w
- Distributive method: l(w₁ + w₂) = lw₁ + lw₂
8. How is the distributive property used to multiply two-digit numbers in area problems?
The distributive property multiplies two-digit numbers by breaking them into tens and ones and adding partial products. Example: Find 12 × 14.
- 12 = 10 + 2
- 14 = 10 + 4
- (10 + 2)(10 + 4)
- 10×10 = 100
- 10×4 = 40
- 2×10 = 20
- 2×4 = 8
- Total = 100 + 40 + 20 + 8 = 168
9. What are common mistakes when measuring area using the distributive property?
Common mistakes when using the distributive property for area include forgetting to multiply each part and missing partial products. Watch out for:
- Multiplying only one part (not both).
- Forgetting to add all partial areas.
- Not keeping track of square units.
10. When should you use the distributive property to find area?
You should use the distributive property to find area when one dimension can be easily broken into friendly numbers. It is especially helpful when:
- Multiplying large numbers mentally (e.g., 9 × 23).
- Solving composite rectangle problems.
- Using area models to teach multiplication.
