Step-by-Step Guide: Using the Distributive Property to Find Area
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 How to Measure Area Using the Distributive Property
1. How to use the distributive property to find area?
Using the distributive property to find area involves breaking down a complex shape into smaller, simpler shapes. Calculate the area of each smaller shape and then add the individual areas together to get the total area. This method is especially useful for irregular shapes or rectangles that are difficult to measure directly.
2. What is the distributive property of 4 x 8?
The distributive property states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding the products. For 4 x 8, you could break it down as 4 x (4 + 4) = (4 x 4) + (4 x 4) = 16 + 16 = 32. This concept is directly applicable to finding the area of rectangles by breaking them into smaller rectangles.
3. What is the distributive property of 5437 into 1001?
To apply the distributive property to 5437 x 1001, think of 1001 as (1000 + 1). Then, you can calculate (5437 x 1000) + (5437 x 1) = 5437000 + 5437 = 5442437. This demonstrates how the distributive property simplifies complex multiplication problems. It's a valuable technique for efficient area calculations.
4. What is the answer to the distributive property 7 x 5 + 3 x 12 + 7 x 5 x 5 x 12?
The expression 7 x 5 + 3 x 12 + 7 x 5 x 5 x 12 needs clarification. It appears to be a combination of addition and multiplication, not a direct application of the distributive property. The distributive property involves multiplying a number by a sum (or difference) in the form a(b + c) = ab + ac. To solve, calculate each term separately: 35 + 36 + 2100 = 2171.
5. What does the distributive property look like using area models?
In an area model, the distributive property is visually represented by breaking a larger rectangle into smaller rectangles. The area of the large rectangle equals the sum of the areas of the smaller rectangles. This is particularly helpful for understanding how multiplying binomials (e.g., (a + b)(c + d)) works. Each smaller rectangle represents a partial product.
6. How do I solve problems using the distributive property?
To solve problems using the distributive property, first identify a situation where a larger quantity can be broken into smaller parts. Then, break the larger quantity into smaller parts, multiply each part by the relevant factor, and finally add together the results. Remember, this approach efficiently handles calculations involving multiplication and addition.
7. What is the area formula using distributive property?
There isn't a single 'area formula' specifically for the distributive property. Instead, the distributive property is a *method* to simplify area calculations. For example, for a rectangle with sides (a + b) and c, the area is calculated as A = c(a + b) = ac + bc; this shows how the distributive property is applied to find the area by calculating the sum of areas of two smaller rectangles.
8. How can you use distributive property with complex areas?
The distributive property is particularly useful for complex areas. Break down irregular shapes into smaller, manageable rectangles, triangles, or other shapes whose areas you know how to calculate. Calculate the area of each smaller component separately and then add them to find the total area.
9. Why do we use the distributive property in area measurement?
The distributive property simplifies area calculations, especially for irregular or complex shapes. It breaks a large area into smaller, easier-to-calculate parts. Adding the areas of these smaller parts gives the total area of the complex shape more efficiently.
10. What is distributive property in area?
In area calculation, the distributive property lets you break down a larger shape into smaller shapes, find the area of each smaller shape, and then add those areas together to find the total area. This is particularly useful for complex or irregular shapes, making area calculation simpler.
