About Area Measurement Using Distributive Property
What is area?
The area of a shape is the amount of space taken up by the shape. All flat or two-dimensional surfaces have an area. Two-dimensional surfaces such as the screen of a smartphone, a piece of cloth, and a piece of land have a definite area that can be measured.
Standard units of area
The area of a shape can be measured using standard units of area. One way to measure the area of a shape is by counting the number of unit squares (squares of sides having unit length) that would fit inside the boundary of the shape. The length of the unit can be inches, centimeters, meters, and so on, depending on the area we are trying to measure. If the length of the side of the unit square is 1 inch, the area will be expressed in square inches.
In this figure, the area of the shaded region can be found by counting the number of unit squares present inside its boundary. Alternatively, we can calculate the area by multiplying the number of square units along its length by the number of square units along its width.
Area of rectangle = Number of square of square units along length\( \times \) Number of square units along width
Area of rectangle\( = 9\times 6\) = 54 square units
Measuring area using distributive property
Measuring the area of shapes using distributive property is an extension of the method that we have learned earlier. In this case, the shape is divided into two parts, and their individual areas are added up to get the total area of the original shape. Let’s take a look at an example.
In this case, the rectangle can be divided into two parts: A and B. Adding the individual areas of A and B gives us the total area of the rectangle given to us.
Area of the original rectangle = Area of A + Area of B
\(9\times 5 = (6+3)\times 5\)
\(9\times 5 = 6\times 5+3\times 5\)
45 = 30 + 15
45 square inches = 45 square inches
Therefore, we got the same result when we used the distributive property to find the area of the rectangle.
Examples on Finding Area Using Distributive Property
Example 1: Use the distributive property to find the area of the rectangle.
Solution:
We can split the provided rectangle into two parts.
Area of the original rectangle = Area of A + Area of B
\(4\times 6=3\times 6+1\times 6\)
\(4\times 6=(3+1)\times 6\)
\(24=4\times 6\)
Area of rectangle = 24 square inches
Example 2: Find the area of the rectangle using the distributive property.
Solution:
Let’s split this rectangle into two parts.
Area of the original rectangle = Area of A + Area of B
\(5\times 3=3\times 3+2\times 3\)
\(5\times 3=(3+2)\times 3\)
\(15=5\times 3\)
Area of rectangle = 15 square inches
Example 3: You cut a piece of paper into two rectangles. Suppose one rectangle is 7 inches wide and 6 inches long, and the other rectangle is 7 inches wide and 3 inches long. Find the dimensions and the area of the original piece of paper.
Solution:
Let’s place these pieces of paper on unit squares.
Area of the first piece of paper is represented by A, and the area of the second piece of paper is represented by B.
If we add the areas of A and B, we will get the area of the original piece of paper.
Area of A + Area of B
\(=6\times 7+3\times 7=(6+3)\times 7=9\times 7\)
Area of A + Area of B = 63 square inches
Length of the original rectangle = Length of A + Length of B = 6 + 3 = 9 inches
Width of the original rectangle = 7 inches
Yes, the distributive property can be used to find the area of all closed two-dimensional shapes.
The method of finding the area of a shape using distributive property is just an extension of the standard unit method. We can’t find the area of a shape using the distributive property without using the standard square units.