GED Mathematical Reasoning: Area

 

It is important to know that the measure of area is stated using SQUARE units. For example, one square foot is a square that has a length of one foot and a width of one foot.

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 1 ft^2

 

Area of a rectangle

Finding the area of a rectangle entails multiplying the length by the width.

Area (A) of a rectangle = length (L) \times width (W)

A = LW

 

Area of a square

Like a rectangle, a square is also a four sided figure containing four right angles where opposite sides are parallel. However, in a square, all four sides have the same length.

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The area of a square is equal to the length times the width, but since those measures are the same for a square – we’ll call it “s” – the formula for area is:

Area (A) of a square = length \times width

A = s \times s
A = s^2

 

Area of a parallelogram

These formulas use two measures we haven’t yet discussed – base and height.

The “base”, represented by the letter b, is the measure of one side of the figure.

The “height” of a figure, denoted by the letter h, is the length from the vertex to the base, forming a right angle.

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The formula for finding the area of a parallelogram is:

Area of a parallelogram = base \times height

 A = bh

 

Area of a triangle

To find the area of a triangle, we’ll need the formula: A equals one-half times base times height.

meas67Area of a triangle = \frac{1}{2} \times base \times height

A = \frac{1}{2}bh

 

Area of a trapezoid 

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Area of a trapezoid =\frac{1}{2} \times height \times (base_1 + base_2)

A = \frac{1}{2}h(b_1 + b_2)

When finding the area of a trapezoid, it is very important to simplify according to the order of operations. In other words, be sure to add the bases before multiplying.

 

Area of a circle

To find the area of a circle, multiply pi times the radius and then times the radius AGAIN. The formula used to represent this calculation is:

A = \pi r^2

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Example 1

Marie would like to re-floor her home office, which is a rectangular room with a length measuring 14 feet and a width measuring 9 feet. How much flooring will she need to purchase?

 

Let’s begin example one by sketching a picture of Marie’s home office. It is a rectangular room with a length of 14 feet and a width of 9 feet.

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To calculate the amount of flooring needed, we will compute the AREA of the room. This is because, when we install flooring, we lay it on the entire surface that the room covers.

Finding the area of a rectangle entails multiplying the length by the width.

Area (A) of a rectangle = length (L) \times width (W)

A = LW

So to find the area of Marie’s room, we’ll multiply the length of 14 feet by the width of 9 feet.

Area = A = 14 \times 9 = 126

To state our final answer, it is important to know that the measure of area is stated using SQUARE units, since a rectangle has TWO dimensions – length and width. The shorthand way of writing “square feet” is using the abbreviation for feet, which is lowercase “f-t,” along with an exponent of 2.

So we will write the final area as:

Area = 126 ft^2

Relating this back to the example, Marie’s room covers a surface equal to 126 square feet, so the amount of flooring she will need is 126 square feet.

Now that you know HOW to calculate area, let’s talk about what it means. As I mentioned, we measure area in square units. ONE square unit, in this case, is a square that has a length of one foot and a width of one foot.

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So the fact that the floor in Marie’s home office measures 126 square feet means that it covers a surface the same size as 126 of those square units.

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Example 2

Find the area of figure ABC.

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To find the area of a triangle, we’ll need the formula: A equals one-half times base times height.

Using the diagram provided, we see that the base is 10 centimeters and the height is 6 centimeters.

A = \frac{1}{2}(10)(6)
A = 30cm^2

 

Example 3

Find the area of a circle with a diameter that measures 8 meters.

 

We’re given the diameter of 8 meters, but to find the area of a circle we need the radius for the formula. Since the diameter is given to be 8 meters, we need to divide the diameter in half, by 2, to get the radius. 8 meters divided by 2 is 4 meters.

r = 8m \div 2 = 4m

Now we have all the information we need to calculate the area of the circle.

A = \pi r^2 = 3.14{(4)}^2 = 3.14(16) = 50.24m^2

So the area of the circle is: 50.24 m^2

ONE square unit, in this case, is a square that has a length of one meter and a width of one meter.

It may seem strange that we measure a circular area using square units, but this means our circle – with an area of 50.24 m^2  – covers a surface the same size as 50.24 of those square units.

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