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Formulas for Area

Why might we need many different formulas for calculating area? 

To answer this question, consider the following polygon. We cannot find the area of this polygon using rectangles and squares alone. 

Knowing how to calculate the areas of other shapes — like triangles, parallelograms, and trapezoids — will help us here.

In general, we realize that the more formulas we have, the more options we have for decomposing a complex shape into simpler shapes.

Lesson Goals

• Review the definition of area.
• Develop and apply the formulas for finding the areas of parallelograms, triangles, and trapezoids. 

Now you may already know some of these. But it's always good to remind ourselves.

Try This!

Consider the following parallelogram and trapezoid with side lengths labelled.

If the area of the parallelogram is \(33\) cm\(^2\), can you find the area of the trapezoid without using a formula?

Think about this problem, then move on to the next part of the lesson.

Area of a Rectangle

Units of Area

Recall: Area is a measure of the space enclosed by a 2D shape.

The basic unit of measurement is a square. And keep in mind that any square can be used. Common units in the metric system include the square millimetre, square centimetre, and square metre:

From a unit square, we can build rectangles.

Knowing this, how can we use the size lengths of a rectangle to find its area?

Area of a Rectangle

Consider the following rectangle with length \(5\) cm and width \(4\) cm. Recall that a square centimetre has side lengths that are \(1\) cm in length. So notice that this rectangle is formed from \(4\) rows of \(5\) square centimetres.

As before, you can count the number of square centimeters in the rectangle to determine that the area is \(20\) cm\(^2\). But, you can also calculate the area by multiplying the length by the width of the rectangle. 

\(\begin{align*} A &= \text{ length } \times \text{ width } \\ & \; \class{timed in3}{= 5 \times 4} \\ & \; \class{timed in4}{= 20 } \end{align*}\)

Notice that to make our calculations simpler, we have omitted the units and we will put those back in in the concluding statement. From our calculation, we conclude that the area of the rectangle is \(20\) cm\(^2\).

 

Example 1

Solution

We can calculate the area of this rectangle by multiplying the length by the width. So start with the formula \(A=l\times w\) and substitute the information we are given.

\[\begin{align*} A &= l \times w \\ & \; \class{timed in3}{= 5 \times 2.5} \\ & \; \class{timed in4}{= 12.5} \end{align*}\]

Therefore, the area of the rectangle is \(12.5\) cm\(^2\).

Check Your Understanding 1

Question

Calculate the area, in mm\(^2\), of the rectangle shown below.

Answer

\(273\) mm\(^2\)

Feedback

Use the formula for the area of a rectangle,

\(\begin{align*} A & = l \times w \\  & = 15 \times 18.2 \\ & = 273\end{align*}\)

Therefore, the area of the rectangle is \(273\) mm\(^2\).

Example 2

Solution

 I want you to first notice that the width is given in centimetres and the length in metres. Before we can calculate the area, we need to convert all of our measurements to the same unit. 

Step 1: Convert all measurements to the same unit

We can rewrite \(80\) cm as \(0.8\) m.

Step 2: Calculate the area

\(\begin{align*} A &= l \times w \\ & \; = 1.5 \times 0.8 \\ & \; =1.2 \end{align*}\)

Therefore, the area of the rectangle is \(1.2\) m\(^2\). 

Check Your Understanding 2

Question

Calculate the area, in mm\(^2\), of the rectangle shown below.

Answer

\(135\) mm\(^2\)

Feedback

Step 1: Convert all measurements to the same unit

We can rewrite \(1.5\) cm as \(15\) mm.

Step 2: Calculate the area

\(\begin{align*}A & = l \times w \\ & = 15 \times 9 \\ & = 135\end{align*}\)

Therefore, the area of the rectangle is \(135\) mm\(^2\).

Area of a Parallelogram

What Is a Parallelogram?

To determine the formula for the area of a parallelogram, we first recall that a parallelogram is a quadrilateral in which pairs of opposite sides are parallel and equal in length. 

Base

The length of any side of the parallelogram.

Height

The perpendicular distance from the base to the opposite side.

Given this definition, every parallelogram can have two base/height combinations. 

Rearranging the Area of a Parallelogram

Notice that we can cut any parallelogram into two pieces by cutting along the perpendicular distance from the base to the opposite side. The sum of the areas of these ‌\(2\) pieces is equal to the area of the parallelogram. Thus, if we rearrange these two pieces, the resulting shape will always have the same area as the original parallelogram.

It looks like the rearranged pieces form a rectangle. We already know how to calculate the area of a rectangle!

That would be really nice if they did because we already know how to calculate the area of a rectangle. But we have to be careful because appearances can sometimes be deceptive. How do we convince ourselves that this new polygons is, in fact, a rectangle?

Explore This

Description

Convince yourself that a parallelogram can be transformed into a rectangle.

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/x6VYxpHp

Online Version

https://ggbm.at/x6VYxpHp

Formula for Area of a Parallelogram

From the investigation, you should have convinced yourself that when you rearrange the two pieces of a parallelogram as shown, the resulting polygon has the properties of a rectangle. That is that it has all \(90^\circ\) angles and the opposite sides are equal in length.

Next, we notice that the base and height of the rectangle are the same as the base and height of the original parallelogram. We know the formula for the area of a rectangle. In this case, it's base times height. So, the area of the parallelogram has to be the same.

Example 3

Solution

To calculate the area, we start with the formula for the area of a parallelogram, \(A=b\times h\). But what are the base and height in this diagram? We need to identify the important information.

Remember that the base and the height must be perpendicular to each other. That means that the base is \(b =3\) cm and the height is \(h =4 \) cm. 

Putting this information into the formula, we have

\[\begin{align*} A &= b \times h \\ & \; \class{timed in2}{= 3 \times 4} \\ & \; \class{timed in3}{= 12} \end{align*}\]

The area of the given parallelogram is \(12\) cm\(^2\). 

Check Your Understanding 3

Question

Calculate the area, in cm\(^2\), of the parallelogram shown below.

Answer

\(123.6\) cm\(^2\)

Feedback

Using the formula for the area of a parallelogram with a base of \(12\) cm and a height of \(10.3\) cm, we have:

\(\begin{align*} A & = b\times h \\ & = 12\times 10.3 \\ & = 123.6\end{align*}\)

Therefore, the area of the parallelogram is \(123.6\) cm\(^2\).

Example 4

Solution

Notice that the height \(h\) is perpendicular to the side that is \(7\) m in length. Therefore, the \(7\) meter side is what we need to use as our base.

We have that the area of this parallelogram is equal to \(A=b\times h\).

\(\begin{align*} A &= b \times h \\ 21 & \; = 7 \times h \\ 3 & \; = h \end{align*}\)

The height of the given parallelogram is \(3\) m. 

Check Your Understanding 4

Question

The area of a parallelogram is \(112\) m\(^2\). If the height of the parallelogram is \(16\) m, what is the base?

Answer

\(7\) m

Feedback

Using the formula for the area of a parallelogram, we have:

\(\begin{align*} A & = b\times h \\ 112 & = b\times 16 \\  7 & = b\end{align*} \)

Therefore, the base of the parallelogram is \(7\) m.

Area of a Triangle

Decomposing a Parallelogram

Knowing how to find the area of a parallelogram can help us to find the area of a triangle. Notice that if you take any parallelogram and cut along one of its diagonals, two identical triangles are created.

The area of one triangleh is \(\dfrac{1}{2}\) the area of the corresponding parallelogram. Since the area of the parallelogram is base times height, the area of the triangle is \(\dfrac{1}{2}(b\times h)\). 

Example 5

Solution

We must use the base and the height of the triangle that are perpendicular to each other to calculate the area. So we start with the formula \(A = \dfrac{1}{2}(b\times h) \) and substitute \(b =10\) and \(h =4\) into the equation.

\(\begin{align*} A &= \dfrac{1}{2}(b\times h) \\ & \; \class{timed in3}{= \dfrac{1}{2}(10 \times 4)} \\ & \; \class{timed in4}{= \dfrac{1}{2}(40)} \\ & \; \class{timed in5}{= 20} \end{align*}\)

Therefore, the area of the triangle is \(20\) mm\(^2\).

Check Your Understanding 5

Question

Calculate the area, in cm\(^2\), of the triangle shown below.

Answer

\(49.7\) cm\(^2\)

Feedback

Notice the base of the triangle is \(14\) cm and height is \(7.1\) cm. Using the formula for the area of a triangle, we have:

\(\begin{align*} A & = \dfrac{1}{2} (b\times h ) \\ & = \dfrac{1}{2} (14\times 7.1) \\ & = \dfrac{1}{2} (99.4)\\ & = 49.7\end{align*}\)

Therefore, the area of the triangle is \(49.7\) cm\(^2\).

Example 6

Strategy

Let's start by coming up with a strategy for how we might solve this problem.

To solve for the length of \(\overline{RS}\), need to be able to look at triangle \(PQR\) from two different perspectives. If we look at this diagram as it is drawn, then we can say that the triangle has a base measuring \(3\) cm and a height measuring \(5\) cm.

Alternatively, if we rotate this triangle and look at it from a different perspective, then we can say that this triangle has a base measuring \(6\) cm and the height is the length of the line segment \(\overline{RS}\). 

What is important to recognize is that no matter how we choose the base and the height, the area of the triangle will always be the same.

So we can use the first base height pair to find the area of the triangle. Then we can use the area to find the length of \(\overline{RS}\). Using this strategy, take a moment to try and find the length of \(\overline{RA}\) for yourself before I go through the solution.

Let's now take the strategy that we started with and put the ideas together into a solution. 

Solution

Area of a Trapezoid

Try This Problem Revisited

Solution

First, we now know how the area of the parallelogram, which is \(33\) cm\(^2\), was calculated. We did so by multiplying the base, which is \(11\) cm, by the height, which is \(3\) cm. 

Now to get back to the question, you may have noticed that the trapezoid has a lot in common with the left side of the parallelogram. So much so that it makes sense that we might try superimposing the two images as shown.

When we do this, we notice that the part of the parallelogram that's left unshaded is actually a second copy of the given trapezoid, just upside down.

The area of the given trapezoid is \(\dfrac{1}{2}\) the area of the parallelogram.

The area of the parallelogram is \(33\) cm\(^2\).

Therefore, the area of the trapezoid is

\(\dfrac{1}{2} \text{ of } 33 = 16.5 \text{ cm}^2\)

Formula for the Area of a Trapezoid

To determine the formula for the area of any trapezoid, we generalize the observation from the Try This problem. To determine the area of a trapezoid, we must know the perpendicular height, labelled \(h\), and the lengths of the two parallel sides labelled \(a\) and \(b\).

When we place two identical trapezoids together as shown, we get a parallelogram.

The area of the trapezoid is \(\dfrac{1}{2}\) the area of the corresponding parallelogram.

To calculate the area of this parallelogram, we would multiply the length of the base, which is \(a+b\), by the height \(h\).

Area of the parallelogram \(= (a+b)\times h\) 

Example 7

Solution

The two parallel sides are \(6\) m and \(12\) m in length. The perpendicular distance between them is \(4\) m.

\(\begin{align*} A &= \dfrac{1}{2}(a+b)h \\ &\; = \dfrac{1}{2}(6+12)(4) \\ & \; = \dfrac{1}{2}(18)(4) \\ & \; = (9)(4) \\ & \; = 36 \end{align*}\)

Therefore, the area of the trapezoid is \(36\) m\(^2\).

Check Your Understanding 6

Question

Calculate the area, in mm\(^2\), of the trapezoid shown below.

Answer

\(128\) mm\(^2\)

Feedback

Notice the two parallel sides are \(8\) mm and \(12\) mm in length, and the height is \(12.8\) mm. Using the formula for the area of a trapezoid, we have:

\(\begin{align*} A & = \dfrac{1}{2} (a+b)h \\ & = \dfrac{1}{2} (8+12)(12.8) \\ & = \dfrac{1}{2} (20)(12.8) \\ & = (10)(12.8) \\ & = 128\end{align*}\)

Therefore, the area of the trapezoid is \(128\) mm\(^2\).

Summary of Area Formulas

Let's now take a moment to recap all of the formulas that we saw in this lesson. We begin by reviewing the formula for the area of a rectangle.

Rectangle

\(A=l\times w\)

Then we showed that the area of a parallelogram can be calculated by multiplying its base by its height.

Parallelogram

\(A = b\times h\)

Triangle

\(A= \dfrac{1}{2}(b\times h)\)

We also learned how to find the area of a trapezoid if we know the lengths of the parallel sides and the height.

Trapezoid

\(A = \dfrac{1}{2}(a+b)\times h\)

          

In the following Check Your Understanding, practice using all of these formulas to solve area problems.

Check Your Understanding 7

Question

Calculate the area, in m\(^2\), of the triangle shown below.

Answer

\(105.3\) m\(^2\)

Feedback

Notice the base of the triangle is \(13\) m and the height is \(16.2\) m. Using the formula for the area of a triangle, we have:

\(\begin{align*} A & = \dfrac{1}{2} (b\times h) \\ & = \dfrac{1}{2} (13\times 16.2) \\ & = \dfrac{1}{2} (210.6) \\ & = 105.3 \end{align*}\)

Therefore, the area of the triangle is \(105.3\) m\(^2\).

Take It With You

The following pair of parallel lines are drawn so that they are \(4\) cm apart.

A rectangle, triangle, parallelogram, and trapezoid are drawn so that every vertex is on a parallel line.

If they are all equal in area, what do you know about the dimensions of each polygon?