The area of 2-dimensional figures, such as rectangles, is describing the amount of surface those shapes cover. We measure areas in square units of any fixed size.
Some examples of square units of measure are square centimeters, square miles, or square inches.
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When we need to find the area of a polygon, for example, we count how many squares of a specific size are covering the inside region of the polygon.
Let’s take a look at a 4 x 4 (four by four) square:
You can easily count that we have 16 squares, so our area is 16 square units. Counting out the 16 squares doesn’t take that long, but what if we need to find the area if it is a larger square or if our units are smaller? Well, that could take a much longer time to count.
Fortunately, we can use multiplication. Because we have 4 (four) rows of 4 (four) squares, we can multiply 4 • 4 (four times four) to get 16 squares! And we can generalize this to one formula for finding areas of squares with any length (s).
Area = s • s = s2.
We can write “in2“ for square inches and “ft2“ for square feet.
To help us find the areas of different categories of polygons, mathematicians developed formulas. Those formulas are helping us to find those measurements more quickly than if we would simply count them. These formulas we’re going to look at are all created from the understanding that we’re counting all of the numbers of square units inside the given polygon. Let’s take a look at a rectangle:
We can count all of the squares individually, but it would be a lot easier if we would multiply 3 times 5 to come up with the number more quickly. And generally, the area of a rectangle can be easily found if we multiply length times width.
The problem: we have a rectangle with a length of 8 cm. and a width of 3 cm. Find its area.
Area (A) = l • w Begin with the formula for a rectangle’s area, which is multiplying length times width.
Area (A) = 8 • 3 Now substitute 8 for length and 3 for width.
The answer is: Area (A) = 24 cm2
We must include the units, so in this case, square centimeters (cm).
So we would need 24 squares that each measure 1 (one) cm on a side, to cover our rectangle.
Now, the formula to measure the area of a parallelogram (remember that a rectangle is just a type of parallelogram) is exactly the same as for any rectangle: Area = l • w.
Note well that in a rectangle, the width and the length are perpendiculars. This counts as well for all parallelograms.
Base (b) for the base length and height (h) for the perpendicular to the baseline’s height are often used. So our formula for parallelograms is generally written as A = b • h.
The problem: Find the area of this parallelogram:
Area (A ) is b • h. We begin with our formula for a parallelogram’s area: Area = base • height.
Area = 4 • 2 Substitute our values into the formula.
A = 8 Then, multiply.
The answer is: The area of our parallelogram is 8 ft2.
Now, what is the area of a parallelogram that has a height of 12 ft. and a base of 9 ft?
A) 21 ft2
B) 54 ft2
C) 42 ft
D) 108 ft2
A) 21 ft2
Incorrect. It seems like you’ve added the dimensions. Remember that when we want to find the area, we multiply its base by its height. Our correct answer would be 108 ft2
B) 54 ft2
Incorrect. It seems like you’ve multiplied the parallelogram’s base by its height and then divided it by 2. For finding the area of a parallelogram, we multiply its base by its height. Correct answer: 108 ft2.
C) 42 ft
Incorrect again. It seems like you’ve added 12 + 12 + 9 + 9. This gives you the perimeter (P) of a 12 by 9 rectangle. To find the area (A) of any parallelogram, we need to multiply its base by its height. Our correct answer would be 108 ft2.
D) 108 ft2
This correct. The height of our parallelogram is 12, while the base of our parallelogram is 9. Area (A) is 12 times 9, which gives us 108 ft2.