# Geometry

## Area


### Learning Objectives

After completing this section, you should be able to:
1. Calculate the area of triangles.
2. Calculate the area of quadrilaterals.
3. Calculate the area of other polygons.
4. Calculate the area of circles.
Some areas carry more importance than other areas. Did you know that in a baseball game, when the player hits the ball and runs to first base that he must run within a 6-foot wide path? If he veers off slightly to the right, he is out. In other words, a few inches can be the difference in winning or losing a game. Another example is real estate. On Manhattan Island, one square foot of real estate is worth far more than real estate in practically any other area of the country. In other words, we place a value on area. As the context changes, so does the value.
Area refers to a region measured in square units, like a square mile or a square foot. For example, to purchase tile for a kitchen floor, you would need to know how many square feet are needed because tile is sold by the square foot. Carpeting is sold by the square yard. As opposed to linear measurements like perimeter, which in in linear units. For example, fencing is sold in linear units, a linear foot or yard. Linear dimensions refer to an outline or a boundary. Square units refer to the area within that boundary. Different items may have different units, but either way, you must know the linear dimensions to calculate the area.
Many geometric shapes have formulas for calculating areas, such as triangles, regular polygons, and circles. To calculate areas for many irregular curves or shapes, we need calculus. However, in this section, we will only look at geometric shapes that have known area formulas. The notation for area, as mentioned, is in square units and we write sq in or sq cm, or use an exponent, such as  or  Note that linear measurements have no exponent above the units or we can say that the exponent is 1.

### Area of Triangles

The formula for the area of a triangle is given as follows.
For example, consider the triangle in .
The base measures 4 cm and the height measures 5 cm. Using the formula, we can calculate the area:
    
In , the triangle has a base equal to 7 cm and a height equal to 3.5 cm. Notice that we can only find the height by dropping a perpendicular to the base. The area is then
    

### Area of Quadrilaterals

A quadrilateral is a four-sided polygon with four vertices. Some quadrilaterals have either one or two sets of parallel sides. The set of quadrilaterals include the square, the rectangle, the parallelogram, the trapezoid, and the rhombus. The most common quadrilaterals are the square and the rectangle.

### Square

In , a  grid is represented with twelve  squares across each row, and twelve  squares down each column. If you count the little squares, the sum equals 144 squares. Of course, you do not have to count little squares to find area—we have a formula. Thus, the formula for the area of a square, where , is . The area of the square in  is 

### Rectangle

Similarly, the area for a rectangle is found by multiplying length times width. The rectangle in  has width equal to 5 in and length equal to 12 in. The area is 
Many everyday applications require the use of the perimeter and area formulas. Suppose you are remodeling your home and you want to replace all the flooring. You need to know how to calculate the area of the floor to purchase the correct amount of tile, or hardwood, or carpet. If you want to paint the rooms, you need to calculate the area of the walls and the ceiling to know how much paint to buy, and the list goes on. Can you think of other situations where you might need to calculate area?

### Parallelogram

The area of a parallelogram can be found using the formula for the area of a triangle. Notice in , if we cut a diagonal across the parallelogram from one vertex to the opposite vertex, we have two triangles. If we multiply the area of a triangle by 2, we have the area of a parallelogram:
    
For example, if we have a parallelogram with the base be equal to 10 inches and the height equal to 5 inches, the area will be 

### Trapezoid

Another quadrilateral is the trapezoid. A trapezoid has one set of parallel sides or bases. The formula for the area of a trapezoid with parallel bases  and  and height  is given here.
For example, find the area of the trapezoid in  that has base  equal to 8 cm, base  equal to 6 cm, and height equal to 6 cm.
The area is .

### Rhombus

The rhombus has two sets of parallel sides. To find the area of a rhombus, there are two formulas we can use. One involves determining the measurement of the diagonals.
For our purposes here, we will use the formula that uses diagonals. For example, if the area of a rhombus is  and the measure of  find the measure of  To solve this problem, we input the known values into the formula and solve for the unknown. See .
We have that
    

### Area of Polygons

To find the area of a regular polygon, we need to learn about a few more elements. First, the apothem  of a regular polygon is a line segment that starts at the center and is perpendicular to a side. The radius  of a regular polygon is also a line segment that starts at the center but extends to a vertex. See .
For example, consider the regular hexagon shown in  with a side length of 4 cm, and the apothem measures 
We have the perimeter,  We have the apothem as  Then, the area is:
    

### Changing Units

Often, we have the need to change the units of one or more items in a problem to find a solution. For example, suppose you are purchasing new carpet for a room measured in feet, but carpeting is sold in terms of yards. You will have to convert feet to yards to purchase the correct amount of carpeting. Or, you may need to convert centimeters to inches, or feet to meters. In each case, it is essential to use the correct equivalency.

### Area of Circles

Just as the circumference of a circle includes the number  so does the formula for the area of a circle. Recall that  is a non-terminating, non-repeating decimal number: . It represents the ratio of the circumference to the diameter, so it is a critical number in the calculation of circumference and area.
For example, to find the of the circle with radius equal to 3 cm, as shown in , is found using the formula 
We have
    

### Area within Area

Suppose you want to install a round hot tub on your backyard patio. How would you calculate the space needed for the hot tub? Or, let’s say that you want to purchase a new dining room table, but you are not sure if you have enough space for it. These are common issues people face every day. So, let’s take a look at how we solve these problems.

### Check Your Understanding


### Key Terms

1. triangle
2. square
3. rectangle
4. rhombus
5. apothem
6. radius
7. circle

### Key Concepts

1. The area  of a triangle is found with the formula  where  is the base and  is the height.
2. The area of a parallelogram is found using the formula  where  is the base and  is the height.
3. The area of a rectangle is found using the formula  where  is the length and  is the width.
4. The area of a trapezoid is found using the formula  where  is the height,  is the length of one base, and  is the length of the other base.
5. The area of a rhombus is found using the formula  where  is the length of one diagonal and  is the length of the other diagonal.
6. The area of a regular polygon is found using the formula  where  is the apothem and  is the perimeter.
7. The area of a circle is found using the formula  where  is the radius.

### Formulas

The area of a triangle is given as  where  represents the base and  represents the height.
The formula for the area of a square is  or 
The area of a rectangle is given as 
The area of a parallelogram is 
The formula for the area of a trapezoid is given as 
The area of a rhombus is found using one of these formulas:
    
    
The area of a regular polygon is found with the formula  where  is the apothem and  is the perimeter.
The area of a circle is given as  where  is the radius.