Basic Ideas of Area
Measuring area is just like measuring anything; before we can do it, we need to agree on standard units. People need to say, “These are the basic units of area.” This is a matter of history. Let’s re-create some of the thinking that went into decisions about standard units of area.
Example 1
What is the area of the rectangle below?
What should we use for a basic unit of area?
As one possibility, suppose we decided to use the space inside this circle as the unit of area.
To find the area, you need to count how many of these circles fit into the rectangle, including parts of circles.
So far you can see that the rectangle’s space is made up of whole circles. Determining the fractional parts of circles that would cover the remaining white space inside the rectangle would be no easy job! And this is just for a very simple rectangle. The challenge is even more difficult for more complex shapes.
Instead of filling space with circles, people long ago realized that it is much simpler to use a square shape for a unit of area. Squares fit together nicely and fill space with no gaps. The square below measures on each side, and it is called .
Now it’s an easy job to find the area of our rectangle.
The area is , because is the number of units of area (square feet) that will exactly fill, or cover, the rectangle.
The principle we used in Example 1 is more general.
The area of a two-dimensional figure is the number of square units that will fill, or cover, the figure.
Two Area Postulates
Congruent Areas
If two figures are congruent, they have the same area.
This is obvious because congruent figures have the same amount of space inside them. However, two figures with the same area are not necessarily congruent.
Area of Whole is Sum of Parts
If a figure is composed of two or more parts that do not overlap each other, then the area of the figure is the sum of the areas of the parts.
This is the familiar idea that a whole is the sum of its parts. In practical problems you may find it helpful to break a figure down into parts.
Example 2
Find the area of the figure below.
Luckily, you don’t have to learn a special formula for an irregular pentagon, which this figure is. Instead, you can break the figure down into a trapezoid and a triangle, and use the area formulas for those figures.
Basic Area Formulas
Look back at Example 1 and the way it was filled with unit area squares.
Notice that the dimensions are:
base (or length)
height (or width)
But notice, too, that the base is the number of feet in one row of unit squares, and the height is the number of rows. A counting principle tells us that the total number of square feet is the number in one row multiplied by the number of rows.
Area of a Rectangle
If a rectangle has base units and height units, then the area, , is square units.
Example 3
What is the area of the figure shown below?
Break the figure down into two rectangles.
Now we can build on the rectangle formula to find areas of other shapes.
Parallelogram
Example 4
How could we find the area of this parallelogram?
Make it into a rectangle
The rectangle is made of the same parts as the parallelogram, so their areas are the same. The area of the rectangle is , so the area of the parallelogram is also .
Warning: Notice that the height of the parallelogram is the perpendicular distance between two parallel sides of the parallelogram, not a side of the parallelogram (unless the parallelogram is also a rectangle, of course).
Area of a Parallelogram
If a parallelogram has base units and height units, then the area, , is square units.
Triangle
Example 5
How could we find the area of this triangle?
Make it into a parallelogram. This can be done by making a copy of the original triangle and putting the copy together with the original.
The area of the parallelogram is , so the area of the triangle is or
Warning: Notice that the height (also often called the altitude) of the triangle is the perpendicular distance between a vertex and the opposite side of the triangle.
Area of a Triangle
If a triangle has base units and altitude units, then the area, , is or square units.
or
Lesson Summary
Once we understood the meaning of measures of space in two dimensions—in other words, area—we saw the advantage of using square units. With square units established, the formula for the area of a rectangle is simply a matter of common sense. From that point forward, the formula for the area of each new figure builds on the previous figure. For a parallelogram, convert it to a rectangle. For a triangle, double it to make a parallelogram.
Points to Consider
As we study other figures, we will frequently return to the basics of this lesson—the benefit of square units, and the fundamental formula for the area of a rectangle.
It might be interesting to note that the word geometry is derived from ancient Greek roots that mean Earth (geo-) measure (-metry). In ancient times geometry was very similar to today’s surveying of land. You can see that land surveying became easily possible once knowledge of how to find the area of plane figures was developed.
Review Questions
Complete the chart. Base and height are given in units; area is in square units.
Base Height 'Area
1a. ?
1b. ?
1c. ?
1d. ?
1e. ?
1f. ?
The carpet for a by room cost . The same kind of carpet cost for a room with a square floor. What are the dimensions of the room?
Explain how an altitude of a triangle can be outside the triangle.
Line and line are parallel.
Explain how you know that , and all have the same area.
Lin bought a tract of land for a new apartment complex. The drawing below shows the measurements of the sides of the tract. Approximately how many acres of land did Lin buy?
A hexagon is drawn on a coordinate grid. The vertices of the hexagon are and What is the area of
Review Answers
1a. 1b.
1c.
1d.
1e.
1f.
by
This happens in a triangle with an obtuse angle. Each altitude to a side of the obtuse angle is outside the triangle.
All of the triangles have the same base and altitude, so in each triangle is the same as in each of the other triangles.
Trapezoids, Rhombi, and Kites
Learning Objectives
Understand the relationships between the areas of two categories of quadrilaterals: basic quadrilaterals (rectangles and parallelograms), and special quadrilaterals (trapezoids, rhombi, and kites).
Derive area formulas for trapezoids, rhombi, and kites.
Apply the area formulas for these special quadrilaterals.
Introduction
We’ll use the area formulas for basic shapes to work up to the formulas for special quadrilaterals. It’s an easy job to convert a trapezoid to a parallelogram. It’s also easy to take apart a rhombus or kite and rebuild it as a rectangle. Once we do this, we can derive new formulas from the old ones.
We’ll also need to review basic facts about the trapezoid, rhombus, and kite.
Area of a Trapezoid
Recall that a trapezoid is a quadrilateral with one pair of parallel sides. The lengths of the parallel sides are the bases. The perpendicular distance between the parallel sides is the height, or altitude, of the trapezoid.
To find the area of the trapezoid, turn the problem into one about a parallelogram. Why? Because you already know how to
compute the area of a parallelogram.
Make a copy of the trapezoid.
Rotate the copy .
Put the two trapezoids together to form a parallelogram.
Two things to notice:
The parallelogram has a base that is equal to .
The altitude of the parallelogram is the same as the altitude of the trapezoid.
Now to find the area of the trapezoid:
The area of the parallelogram is .
The parallelogram is made up of two congruent trapezoids, so the area of each trapezoid is one-half the area of the parallelogram.
The area of the trapezoid is one-half of .
Area of Trapezoid with Bases and and Altitude
Trapezoid with bases and and altitude
or
Notice that the formula for the area of a trapezoid could also be written as the "Average of the bases time the height." This may be a convenient shortcut for memorizing this formula.
Example 1
What is the area of the trapezoid below?
The bases of the trapezoid are and . The altitude is .
Area of a Rhombus or Kite
First let’s start with a review of some of the properties of rhombi and kites.
Kite Rhombus
Congruent sides Pairs All
Opposite angles congruent Pair yes. Pair maybe Both pairs yes
Perpendicular diagonals Yes Yes
Diagonals bisected Yes. maybe Both yes
Now you’re ready to develop area formulas. We’ll follow the command: “Frame it in a rectangle.” Here’s how you can frame a rhombus in a rectangle.
Notice that:
The base and height of the rectangle are the same as the lengths of the two diagonals of the rhombus.
The rectangle is divided into congruent triangles; of the triangles fill the rhombus, so the area of the rhombus is one-half the area of the rectangle.
Area of a Rhombus with Diagonals and
We can go right ahead with the kite. We’ll follow the same command again: “Frame it in a rectangle.” Here’s how you can frame a kite in a rectangle.
Notice that:
The base and height of the rectangle are the same as the lengths of the two diagonals of the kite.
The rectangle is divided into triangles; of the triangles fill the kite. For every triangle inside the kite, there is a congruent triangle outside the kite so the area of the kite is one-half the area of the rectangle.
Area of a Kite with Diagonals and
Lesson Summary
We see the principle of “no need to reinvent the wheel” in developing the area formulas in this section. If we wanted to find the area of a trapezoid, we saw how the formula for a parallelogram gave us what we needed. In the same way, the formula for a rectangle was easy to modify to give us a formula for rhombi and kites. One of the striking results is that the same formula works for both rhombi and kites.
Points to Consider
You’ll use area concepts and formulas later in this course, as well as in real life.
Surface area of solid figures: the amount of outside surface.
Geometric probability: chances of throwing a dart and landing in a given part of a figure.
Carpet for floors, paint for walls, fertilizer for a lawn, and more: areas needed.
Tech Note - Geometry Software
You saw earlier that the area of a rhombus or kite depends on the lengths of the diagonals.
This means that all rhombi and kites with the same diagonal lengths have the same area.
Try using geometry software to experiment as follows.
Construct two perpendicular segments.
Adjust the segments so that one or both of the segments are bisected.
Draw a quadrilateral that the segments are the diagonals of. In other words, draw a quadrilateral for which the endpoints of the segments are the vertices.
Repeat with the same perpendicular, bisected segments, but making a different rhombus or kite. Repeat for several different rhombi and kites.
Regardless of the specific shape of the rhombus or kite, the areas are all the same.
The same activity can be done on a geoboard. Place two perpendicular rubber bands so that one or both are bisected. Then place another rubber band to form a quadrilateral with its vertices at the endpoints of the two segments. A number of different rhombi and kites can be made with the same fixed diagonals, and therefore the same area.
Review Questions
Quadrilateral has vertices and in a coordinate plane.
Show that is a trapezoid.
What is the area of ?
Prove that the area of a trapezoid is equal to the area of a rectangle with height the same as the height of the trapezoid and base equal to the length of the median of the trapezoid.
Show that the trapezoid formula can be used to find the area of a parallelogram.
Sasha drew this plan for a wood inlay he is making.
is the length of the slanted side. is the length of the horizontal line segment. Each shaded section is a rhombus.
The shaded sections are rhombi. Based on the drawing, what is the total area of the shaded sections?
Plot on a coordinate plane. The points are the vertices of a rhombus.
The area of the rhombus is .
Tyra designed the logo for a new company. She used three congruent kites.
What is the area of the entire logo?
In the figure below: is a square
What is the area of ?
In the figure below:
is a square
What is the area of ?
The area of is what fractional part of the area of ?
Review Answers
Slope of , slope of
are parallel.
slope of
and are the bases, is an altitude.
.
For a parallelogram, = (the “bases” are two of the parallel sides), so by the trapezoid formula the area is
Length of long diagonal of one rhombus is . Length of other diagonal is (each rhombus is made of right triangles).
Total area is
Many rhombi work, as long as the product of the lengths of the diagonals is .
Areas of Similar Polygons
Learning Objectives
Understand the relationship between the scale factor of similar polygons and their areas.
Apply scale factors to solve problems about areas of similar polygons.
Use scale models or scale drawings.
Introduction
We’ll begin with a quick review of some important features of similar polygons. You remember that we studied similar figures rather extensively in Chapter 7. There you learned about scale factors and perimeters of similar polygons. In this section we’ll take similar figures one step farther. We’ll see that the areas of similar figures have a very specific relationship to the scale factor—but it’s just a bit tricky! We wrap up the section with some thoughts on why living things are the “right” size, and what geometry has to do with that!
Review - Scale Factors and Perimeter
Example 1
The diagram below shows two rhombi.
a. Are the rhombi similar? How do you know?
Yes.
The sides are parallel, so the corresponding angles are congruent.
Using the Pythagorean Theorem, we can see that each side of the smaller rhombus has a length of , and each side of the larger rhombus has a length of .
So the lengths of the sides are proportional.
Polygons with congruent corresponding angles and proportional sides are similar.
b. What is the scale factor relating the rhombi?
The scale factor relating the smaller rhombus to the larger one is
c. What is the perimeter of each rhombus?
Answer
d. What is the ratio of the perimeters?
e. What is the area of each rhombus?
What do you notice in this
example? The perimeters have the same ratio as the scale factor.
But what about the areas? The ratio of the areas is certainly not the same as the scale factor. If it were, the area of the larger rhombus would be , but the area of the larger rhombus is actually
What IS the ratio of the areas?
The ratio of the areas is Notice that or in decimal, .
So at least in this case we see that the ratio of the areas is the square of the scale factor.
Scale Factors and Areas
What happened in Example 1 is no accident. In fact, this is the basic relationship for the areas of similar polygons.
Areas of Similar Polygons
If the scale factor relating the sides of two similar polygons is , then the area of the larger polygon is times the area of the smaller polygon. In symbols let the area of the smaller polygon be and the area of the larger polygon be . Then:
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