Professor Dave here, let’s learn to calculate area.
Once we know how to calculate one-dimensional lengths like perimeters and diameters, it
is time to learn how to calculate two-dimensional areas.
This is the amount of surface that a figure covers.
We must understand that while side lengths and perimeters can be measured in any unit
of distance, like meters, area will be expressed in square units, like square meters, which
are meters squared.
To see why, let’s look at something simple, like a square.
Say this square has a side length of five meters.
How much surface does it cover?
To answer that, we can take this line segment that represents one side of the square, and
realize that to span the shape, we are extending this line segment across the shape.
So it’s a five meter length that is being pushed across the five meter length of the
adjacent side.
Five meters times five meters gives us twenty-five meters squared.
We must remember that units will be manipulated just like algebraic variables, so meters times
meters is meters squared just the way that X times X is X squared.
So with that, we understand the formula for the area of a square.
It is a side times a side, and by definition the sides of a square are the same, so it’s
side squared.
If we are looking at some other rectangle that is not a square, the area will be length
times width, which will be different numbers.
Here is a rectangle that is four meters by six meters, so the area is twenty-four meters
squared.
This approach works for any parallelogram as well, but instead of multiplying the sides,
we multiply the base times the height.
The height of a parallelogram is essentially its width, which we get if we draw a line
between the base and the opposite side that is perpendicular to both.
It becomes very clear why this works if we draw the height right here, chop off this
triangular segment, and move it to the other side to make a perfect rectangle.
Now we can see that base times height on the parallelogram is the same as length times
width on the corresponding rectangle.
Now that we are back to rectangles, let’s draw the diagonal, thus splitting it into
two triangles.
Each triangle has half the area of the rectangle.
That’s how we get the area of a triangle, we multiply the base times the height, which
would give us the area of the rectangle this triangle fits into, but then we cut it in
half.
One half base times height.
So this triangle, with a base of six meters and a height of four meters, will have an
area of twelve square meters.
This works whether the triangle is a right triangle or not.
Even though it’s a little harder to see how this triangle is half of this rectangle,
because it splits the rectangle up into more than two pieces, we can still use one half
base times height.
Trapezoids are a little different.
For these, we do one half the height times the sum of the two bases.
So with bases of six and ten meters and a height of four meters, we plug these all in
to get an area of thirty-two square meters.
Sometimes we will want to find the area of a shape that is quite irregular.
To do this, we might break these figures up into simpler shapes that are more familiar.
Take this shape.
Given all of these side lengths, we can split this up into two rectangles.
Then we just get the areas of the two rectangles and add them together.
Same with this figure, which can be split up into a rectangle and a triangle.
We just get all the side lengths, and then plug values into their respective equations
for area.
With circles, things are a bit different.
The area for a circle is given by pi R squared, or pi times the radius squared.
So if a circle has a radius of three meters, the area will be nine pi square meters.
Knowing this, we can calculate the area of even more irregular figures, as there are
ones that will have rectangular pieces and semicircular pieces.
We can split these up just like we did before, and the rectangular part will be easy, while
for the semicircle we figure out a way to get the radius of the circle, and then we
calculate pi R squared and just cut it in half, since it’s half a circle.
And that’s really all we need to know to calculate the area of simple shapes and some
irregular ones, so it’s a good idea to memorize this handful of formulas for the areas of
a square, rectangle, parallelogram, triangle, trapezoid, and circle.
Let’s check comprehension.