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ELM: CSU Math Study Guide16 chapters | 140 lessons

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

Area is the size of a two-dimensional surface. This lesson will define area, give some of the most common formulas, and give examples of those formulas. A quiz at the end of the lesson will allow you to work out some area problems on your own.

The mathematical term **'area'** can be defined as the amount of two-dimensional space taken up by an object. The use of area has many practical applications in building, farming, architecture, science, and even deciding how much paint you need to paint your bedroom. The area of a shape can be determined by placing the shape over a grid and counting the number of squares that the shape covers, like in this image:

The area of many common shapes can be determined using certain accepted formulas. Let's take a look at the most common formulas for finding area.

To find the area of a rectangle, you use this formula:

*Area = length * width*

The area of a square is found with this formula:

*Area = s2, where s = side*

The formula for the area of a triangle is:

*Area = (1/2) b * h, where b = base and h = height*

To find the area of a circle, use this formula:

*Area = pi * r2, where r = radius*

The area of a parallelogram is found using this formula:

*Area = b * h, where b = base and h = vertical height*

The formula for the area of a trapezoid is:

*Area = (1/2) * (a + b) * h, where a =base 1, b = base 2, and h = vertical height*

An ellipse's area is found this way:

*Area = pi * a * b, where a = radius of major axis and b = area of minor axis*

Finding the area of a shape always requires the multiplication of two lengths. In a square, it's side multiplied by side. In a circle, it's the radius squared. For an ellipse, it's the radius of the major axis multiplied by the radius of the minor axis. Due to this, the units given to area will always be squared (feet squared, inches squared, etc.). Anything multiplied to itself is squared, whether it is a number or not.

Let's practice finding the area with some example problems.

*What is the area of a square with side length of 5 inches?*

Remember, the formula for finding the area of a square is *A* = *s*2. The sides of this particular square are 5 inches. Plug that into the formula to get *A* = 52 = 25 in2.

*What is the area of this parallelogram?*

Remember, the formula is *A* = *b* * *h*. So, for this example, the area would be *A* = 3 * 12 = 36 mm2.

If you are asked to find the area of an uncommon shape, it can be done by breaking the shape into more common shapes, finding the area of those shapes, and then adding the areas together. Let's look at some examples:

*Find the area of this shape:*

The first step to solving this problem is to divide the shape into shapes we can find the area of easily. This shape can be divided into a triangle and a square.

You can use the information given to determine the lengths you need to calculate the area. Since you know the height from the point of the triangle to the bottom of the square is 10 cm and the height of the square is 8 cm, the height of the triangle must be 2 cm. The base of the triangle is equal to a side of the square, which is 8 cm. You can use these numbers to determine the area. *A* of square = *s*2 = 82 = 64 cm2. *A* of triangle = (1/2) * *b* * *h* = (1/2)* 8 * 2 = 8 cm2. Then you just add the areas together to get the total area of the figure. *A* = 64 + 8 = 72 cm2.

*Find the area of the figure shaded in red, given that the dimensions of the rectangle are 11 inches by 7 inches*

This example is a bit different, since you only want the area of a small portion of the figure. This figure can be broken down into a rectangle and a circle only, this time, the area of the circle needs to be subtracted from the area of the rectangle to get the remaining area. *A* of rectangle = *l* * *w* = 11 * 7 = 77 in2. *A* of circle = pi * *r*2 = pi * (3.52) = 38.47 in2.

The radius of the circle is determined from the diameter of the circle, which is equal to the width of the rectangle because the circle is as wide as the rectangle. The radius is half of the diameter (1.2 * 7 = 3.5). *A* = 77 - 38.47 = 38.53 in2.

The **area** of a two-dimensional figure is a calculation of the space taken up by the figure. Figures such as squares, triangles, circles, and others have specific formulas that can be used to find their area. The area of other figures can be determined by breaking the figure into parts whose area can be easily determined.

As you watch the video lesson, your increasing knowledge could prepare you to:

- State the definition of area and recognize its applications
- Express the units for area
- Identify and apply the formulas for finding the area of common shapes
- Find the area of uncommon shapes

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ELM: CSU Math Study Guide16 chapters | 140 lessons

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