After completing this section, you should be able to:
- Calculate the area of triangles.
- Calculate the area of quadrilaterals.
- Calculate the area of other polygons.
- 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.
The area of a triangle is given as where represents the base and represents the height.
For example, consider the triangle in Figure 10.133.
The base measures 4 cm and the height measures 5 cm. Using the formula, we can calculate the area:
In Figure 10.134, 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
Finding the Area of a Triangle
Find the area of this triangle that has a base of 4 cm and the height is 6 cm (Figure 10.135).
Using the formula, we have
Your Turn 10.41
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.
In Figure 10.137, 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 Figure 10.137 is
The formula for the area of a square is or
Similarly, the area for a rectangle is found by multiplying length times width. The rectangle in Figure 10.138 has width equal to 5 in and length equal to 12 in. The area is
The area of a rectangle is given as
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?
Finding the Area of a Rectangle
You have a garden with an area of 196 square feet. How wide can the garden be if the length is 28 feet?
The area of a rectangular region is Letting the width equal :
Your Turn 10.42
Determining the Cost of Floor Tile
Jennifer is planning to install vinyl floor tile in her rectangular-shaped basement, which measures 29 ft by 16 ft. How many boxes of floor tile should she buy and how much will it cost if one box costs $50 and covers
The area of the basement floor is We will divide this area by Thus, Therefore, Jennifer will have to buy 24 boxes of tile at a cost $1,200.
Your Turn 10.43
The area of a parallelogram can be found using the formula for the area of a triangle. Notice in Figure 10.139, 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:
The area of a parallelogram is
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
Finding the Area of a Parallelogram
In the parallelogram (Figure 10.140), if find the exact area of the parallelogram.
Using the formula of we have
Your Turn 10.44
Finding the Area of a Parallelogram Park
The boundaries of a city park form a parallelogram (Figure 10.142). The park takes up one city block, which is contained by two sets of parallel streets. Each street measures 55 yd long. The perpendicular distance between streets is 39 yd. How much sod, sold by the square foot, should the city purchase to cover the entire park and how much will it cost? The sod is sold for $0.50 per square foot, installation is $1.50 per square foot, and the cost of the equipment for the day is $100.
Step 1: As sod is sold by the square foot, the first thing we have to do is translate the measurements of the park from yards to feet. There are 3 ft to a yard, so 55 yd is equal to 165 ft, and 39 yd is equal to 117 ft.
Step 2: The park has the shape of a parallelogram, and the formula for the area is :
Step 3: The city needs to purchase of sod. The cost will be $0.50 per square foot for the sod and $1.50 per square foot for installation, plus $100 for equipment:
Your Turn 10.45
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.
The formula for the area of a trapezoid is given as
For example, find the area of the trapezoid in Figure 10.143 that has base equal to 8 cm, base equal to 6 cm, and height equal to 6 cm.
The area is .
Finding the Area of a Trapezoid
(Figure 10.144) is a regular trapezoid with Find the exact perimeter of , and then find the area.
The perimeter is the measure of the boundary of the shape, so we just add up the lengths of the sides. We have Then, the area of the trapezoid using the formula is .
Your Turn 10.46
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.
The area of a rhombus is found using one of these formulas:
- where and are the diagonals.
- where is the base and is the height.
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 Figure 10.146.
We have that
Finding the Area of a Rhombus
Find the measurement of the diagonal if the area of the rhombus is and the measure of
Use the formula with the known values:
Your Turn 10.47
Finding the Area of a Rhombus
You notice a child flying a rhombus-shaped kite on the beach. When it falls to the ground, it falls on a beach towel measuring by You notice that one of the diagonals of the kite is the same length as the width of the towel. The second diagonal appears to be 2 in longer. What is the area of the kite (Figure 10.147)?
Using the formula, we have:
Your Turn 10.48
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 Figure 10.148.
The area of a regular polygon is found with the formula where is the apothem and is the perimeter.
For example, consider the regular hexagon shown in Figure 10.149 with a side length of 4 cm, and the apothem measures
We have the perimeter, We have the apothem as Then, the area is:
Finding the Area of a Regular Octagon
Find the area of a regular octagon with the apothem equal to 18 cm and a side length equal to 13 cm (Figure 10.150).
Using the formula, we have the perimeter Then, the area is
Your Turn 10.49
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.
Carpeting comes in units of square yards. Your living room measures 21 ft wide by 24 ft long. How much carpeting do you buy?
We must convert feet to yards. As there are 3 ft in 1 yd, we have and Then,
Your Turn 10.50
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.
The area of a circle is given as where is the radius.
For example, to find the of the circle with radius equal to 3 cm, as shown in Figure 10.152, is found using the formula
Finding the Area of a Circle
Find the area of a circle with diameter of 16 cm.
The formula for the area of a circle is given in terms of the radius, so we cut the diameter in half. Then, the area is
Your Turn 10.51
Determining the Better Value for Pizza
You decide to order a pizza to share with your friend for dinner. The price for an 8-inch diameter pizza is $7.99. The price for 16-inch diameter pizza is $13.99. Which one do you think is the better value?
The area of the 8-inch diameter pizza is The area of the 16-inch diameter pizza is Next, we divide the cost of each pizza by its area in square inches. Thus, per square inch and per square inch. So clearly, the 16-inch pizza is the better value.
Your Turn 10.52
Applying Area to the Real World
You want to purchase a tinted film, sold by the square foot, for the window in Figure 10.153. (This problem should look familiar as we saw it earlier when calculating circumference.) The bottom part of the window is a rectangle, and the top part is a semicircle. Find the area and calculate the amount of film to purchase.
First, the rectangular portion has . For the top part, we have a semicircle with a diameter of 5 ft, so the radius is 2.5 ft. We want one half of the area of a circle with radius 2.5 ft, so the area of the top semicircle part is Add the area of the rectangle to the area of the semicircle. Then, the total area to be covered with the window film is
Your Turn 10.53
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.
Finding the Area within an Area
The patio in your backyard measures 20 ft by 10 ft (Figure 10.155). On one-half of the patio, you have a 4-foot diameter table with six chairs taking up an area of approximately 36 sq feet. On the other half of the patio, you want to install a hot tub measuring 6 ft in diameter. How much room will the table with six chairs and the hot tub take up? How much area is left over?
The hot tub has a radius of 3 ft. That area is then The total square feet taken up with the table and chairs and the hot tub is The area left over is equal to the total area of the patio, minus the area for the table and chairs and the hot tub. Thus, the area left over is
Your Turn 10.54
Finding the Cost of Fertilizing an Area
A sod farmer wants to fertilize a rectangular plot of land 150 ft by 240 ft. A bag of fertilizer covers and costs $200. How much will it cost to fertilize the entire plot of land?
The plot of land is It will take 7.2 bags of fertilizer to cover the land area. Therefore, the farmer will have to purchase 8 bags of fertilizer at $200 a bag, which comes to $1,600.
Your Turn 10.55
People in Mathematics
Heron of Alexandria
Heron of Alexandria, born around 20 A.D., was an inventor, a scientist, and an engineer. He is credited with the invention of the Aeolipile, one of the first steam engines centuries before the industrial revolution. Heron was the father of the vending machine. He talked about the idea of inserting a coin into a machine for it to perform a specific action in his book, Mechanics and Optics. His contribution to the field of mathematics was enormous. Metrica, a series of three books, was devoted to methods of finding the area and volume of three-dimensional objects like pyramids, cylinders, cones, and prisms. He also discovered and developed the procedures for finding square roots and cubic roots. However, he is probably best known for Heron’s formula, which is used for finding the area of a triangle based on the lengths of its sides. Given a triangle (Figure 10.159),
Heron’s formula is where is the semi-perimeter calculated as