Estimate and Find the Area of Polygons
MA69123
Activity Introduction
Math News, top story: Estimating the area of a polygon! For more on our story, let’s go
to our friend of poly-cotton blends… Sallllllllly Newsworth!
Direct Instruction
That’s right, Rupert! More specifically, right now you’re going to look at triangles: their
measurements and the formula you’d use to find the area within them. Who better to
take us on the journey than
Commander Christy Powers?
Model 1
Polygons are by far my favorite among shapes. No offense to trapezoids, pentagons or
hexagons, but triangles are my absolute favorite of the bunch.
(calls after him) But you’re my favorite quadrilateral, Zilch!
Today you’re going to find the area of (whispers) my favorite shape, (regular voice) the
triangle.
To do that, you need two measurements: the base and the height. You’ll use those
measurements in the formula to find the area of a triangle: one-half times base, times
height.
As you can see, the base of the triangle is the bottom side of the triangle.
The height of the triangle is the perpendicular distance from the base to the vertex. A
vertex, by the way, can be any of the points of a triangle. So you should be looking at
the vertex that is across from, or opposite the base.
Now let’s plug these measurements into the formula to find the area of this triangle.
Here we go…
The base (pause) is ten inches. The height (pause) is six inches. Your equation now
looks like this.
And it’s computation time! Ten times six is sixty, making your new equation: one-half
times sixty. That’s the same as sixty divided by two…
Which, by my calculations… is thirty. The eight inches and the seven inches are extra,
not necessary for the formula. Okay, I’ve gotta run… Zilch?
Zilch is a sensitive robot. I hope he doesn’t rust. Speaking of which… are you feeling
rusty about anything you just heard? If so, maybe a review’s in order! But if you think
you’ve got it, feel free to journey on!
Model 2
So… what if you don’t exactly know the base or height of a triangle, but you still have to
find the area? Hmmm… Commander Christy’s got a technique for that! Let’s take a
look!
Zilch makes the best grids in the whole galaxy.
We’ll need that… Not for chess, Zilch. We’re going to talk about a technique for
estimating the area of a triangle when-He’s still a bit sensitive about being a quadrilateral. (sighs)
Where were we? Right… There’s a technique for estimating the area of a triangle when
you aren’t given the base and height, but the triangle is over… a grid.
First, count the complete squares that are within the triangle. There are twenty complete
squares total. Next, count the number of partial squares. There are twenty-six partial
squares, total.
Now, add the number of complete squares plus the number of partial squares divided
by two. After computation, you see that the total area of the triangle is approximately
thirty-three square units.
Zilch? You make a great triangle.
And let’s pause here for a sec. How ya doin’? If ya wanna see that one more time for
good measure, lemme know. Otherwise, we can keep this ship in orbit.
Model 3
Commander Christy tells me that the formula for finding the area of a trapezoid is only
slightly different than the formula for the area of a triangle. Check it out…
Trapezoids are interesting shapes because they have two bases and a height. That’s
why the formula for finding the area of a trapezoid also shows two bases.
Notice that the other parts of the area formula for a trapezoid, are the same as the area
formula for a triangle
.
Watch how it works. The bases and the height of the trapezoid are given here. Plug in
base one, (pause) base two, (pause) and the height (pause) into the formula.
Now, add the two bases, and multiply the sum by the height: six. Then, multiply onehundred-ninety-two by one-half, which is actually the same as dividing one-hundredninety-two by two. So, now you know… the area of this trapezoid is ninety-six square
inches.
You can also estimate the area of a trapezoid on a grid using the same technique used
for a triangle.
Here’s a quick reminder: Once you’ve counted the complete squares, add that to the
number of partial squares divided by two.
After careful computation, you’ll find that the approximate, or estimated, area of this
trapezoid is twenty-one inches.
Feelin’ confident? If so, let’s move on to the next thing. If not, no sweat. We can go
over that again… if you think it’ll help.
Direct Instruction
And we’re back! Cruising the earth’s perimeter among constellations and planets with
Commander Christy. What’s a perimeter, you say? Let’s find out!
Model 1
When you say you want to find the perimeter of an object, you’re saying that you want
to find the distance around that object.
I don’t think your tail counts as an object, Zilch. (sighs)
Anyway, when you say you want to find the perimeter of any figure, even an irregular
figure, like this one, you’re looking for the measure of the distance around the object.
So, what you’d do is simply add all sides. Fifteen plus twenty, plus twenty, plus twentyfive, plus ten equals ninety. The sum is the perimeter. In this case, the perimeter is
ninety feet.
If any of that left you feeling dizzy and confused like our friend Zilch, let’s go back to
review it again. But if you think you’ve got it, journey on, my friend.
Model 2
Are you getting the hang of finding the areas and measuring perimeters for regular
shapes? (intrigued) Well, here’s where it gets tricky. Take a look…
You’ve learned how to find the area and perimeter of regular figures, like triangles and
my new favorite… quadrilaterals.
Now, you’ll learn how to find the area of irregular figures. Irregular figures are basically
regular figures that touch or overlap.
Take this irregular figure, for example. If you draw a line here, you can see that there is
merely a triangle on top of a rectangle.
If you add the areas of these two shapes, you’ll find the area of the original irregular
figure.
So let’s do it! Let’s find the area of this irregular shape… which looks curiously like the
omicron trec-tangular galaxy in the nine-hundred-eighty-eighth quadrant. Anyway…
First, the rectangle. The length is eight inches and the width is five inches. So the length
times the width gives you forty-square-inches.
Now… the triangle. The base of the triangle is also the width of the rectangle. So it is
also five inches. The height is seven inches.
The area of a triangle is one-half of the base times the height. The area of the triangle is
seventeen-and-one-half square inches. The rectangle area plus the triangle area equals
a total area of fifty-seven-and-one-half.
And guess what? Your new knowledge of estimation will come in handy here. Know
why? Because, estimating the area of an irregular shape on a grid, is exactly the same
as estimating the area of a triangle or a trapezoid on a grid.
So here, again, you add the number of complete squares, eighteen, with half of the
partial squares, twenty-six. Then the estimate is thirty-one square units.
By my estimation, we should pause here. I’ll wait for you to tell me what you wanna do
next.
Model 3
Now that you’ve learned how to find the area of shapes, I think the last thing you need
to know is how to find the area of shapes with missing parts.
Notice anything? There’s a square inside this circle. And how do you think you’d find
the area of the shaded part of the circle? Any ideas?
Well, the answer is as simple as subtraction! The area of the circle minus the area of
the square. Let’s take a look…
In this case, if the side of the square is four inches, and all sides of a square are equal,
you know the area of the square is sixteen-square-inches.
Now, if the radius of the circle is three inches, then pi-r-squared, the formula for the area
of a circle, gives you an area of twenty-eight-and-twenty-six-hundredths square inches.
Cosmic comet tails! You can now subtract: the area of the circle--twenty-eight-andtwenty-six-hundredths; minus the area of the square—sixteen.
The area of the shaded part of the circle is twelve-and-twenty-six-hundredths square
inches.
Let’s try it with a rectangle that has a triangle in it. Keep in mind that you’re trying to find
the area of the shaded part of the figure.
First, find the area of both shapes. The rectangle’s area equals the length times the
width. Notice that the height of the triangle is the same as the width of the rectangle.
The triangle’s area equals one-half the base times the height. Subtract the area of the
triangle, seventeen-and-five-tenths-square-inches, from the area of the rectangle, onehundred-five-square-inches.
Now you know that the area of the shaded part of the rectangle is eighty-seven-andfive-tenths-square-inches.
Have I told you lately that you… are … awesome! Before we jump into practice
problems, I wanna make sure you got all of that. Go back and see that again? Or
move it along? What do ya think?
End of Activity Review
Well, we covered a lot of ground today, huh? Let's go over it again.
Today, you learned to find and estimate the areas of triangles, (pause) trapezoids,
(pause) and irregular figures using known and unknown measurements.
Short and sweet! That’s how I like it! Or… Was it too short? If you wanna hear it
again, I’d be glad to go back over it. Lemme know if you’ve got this by hitting the
button.