Welcome to a lesson on geometric formulas.
In this lesson we will determine the circumference
of a circle, the volume of a cone,
and also determine the length a missing side
of a right triangle using the Pythagorean Theorem.
In example one, we're told the circumference of a circle
with radius r is given by the formula C equals 2 pi r.
The circumference of a circle is the distance around
the circle, so we can also think of the circumference
as the perimeter of the circle.
Radius of a circle is the distance from the center
to the circle, this distance here
which was normally label with the variable r.
We're asked to determine the circumference of a circle
with radius 32 centimeters,
and we're asked to write the answer in exact form
in terms of pi
and in approximate form rounded to the nearest hundredth,
meaning two decimal places.
So start with our formula, circumference equals two pi r,
we'll substitute 32 for r
because the radius equals 32 centimeters.
So we'll have the circumference C is equal to
two times pi times 32.
So for the exact circumference,
we want the circumference in terms of pi
so we'll have a pi in our circumference
and because two times 32 equals 64,
the exact circumference would be 64 pi
and because the radius is in centimeters,
the circumference is also in centimeters.
So this is the exact circumference.
And now to approximate the circumference,
we'll actually do this two ways.
A common approximation for pi
is 3.14,
but most calculators also have a pi key,
which would give a more accurate approximation for pi.
So we'll determine the approximate circumference
using both methods.
So we're going to the calculator,
let's first use the pi key on the calculator,
which would give us a more accurate approximation.
So enter 64 and then to access the pi key,
we press second and then the exponent key here
and then enter.
To the hundredths place value,
because we have a one and a thousandths place value,
we would round down,
we'd approximate the circumference to be
201.06.
Or 102 and six hundredths.
Let's go ahead and record this.
Again this would be centimeters.
And this is the approximation
using pi on the calculator.
Now let's make another approximation
using pi as approximately 3.14.
So enter 64
times 3.14.
Notice how it does give us a different value.
This approximation is not as accurate
as this approximation here.
So our second approximation is going to be
200.96.
Or two hundred and 96 hundredths centimeters.
And we'll say this is the approximation
using
pi as approximately 3.14.
So it is important when doing the homework
to read the directions carefully.
My suggestion would be to use the pi key on the calculator
unless it indicates to use a different approximation for pi.
In example two, we're told the formula
for the volume of a cone of base radius r and height h
is given by V equals one-third pi r squared h.
A cone is pictured below,
this length here would be the height.
And this length here
would be the base radius.
We're asked to determine the volume of a cone
with a base radius of five inches
and a height of 12 inches.
So because we're determining volume,
we measure volume in cubic units
so our volume is going to be in cubic inches.
We're asked to write the exact form in terms of pi
and also in approximate form to the nearest hundredth.
So beginning with the volume formula,
volume V equals one-third
pi r squared h.
We'll substitute five for r
and 12 for h.
So the volume is equal to one-third pi
times five squared,
times 12.
Now following the order of operations,
we would simplify the exponents first,
five squared is equal to five times five or 25.
So we'd have volume V equals one-third pi
times 25 times 12.
Now before we multiply though,
because we have a fraction here,
notice how if we write 25 times 12 as a fraction
with the denominator of one,
we should recognize we are going to have a common factor
of three between three and 12.
There's one three in three
and four threes in 12.
Now we can multiply and notice how the denominator is one.
Which means the product is going to be
pi times 25 times four.
Which equals 100 pi.
So the exact volume
is equal to 100 pi,
and again the units would be cubic inches.
Now let's make our approximations for the volume.
Let's first use the pi key on the calculator.
So enter 100 and then using the pi key
press second and then the exponent key, enter.
Notice here we have a nine in the thousandths place value
so we round up to 314.16
or 314 and 16 hundredths.
Again this is cubic inches
and this is the approximation using pi on the calculator.
And now let's make another approximation using 3.14.
In this one we shouldn't need the calculator.
3.14 times 100 we should recognize will be 314.
Let's go ahead and show it,
100 times 3.14
is 314.
Again this is the approximation using pi
as approximately 3.14.
Before we look at example three,
let's look at the Pythagorean Theorem.
The Pythagorean Theorem states that
given any right triangle with legs a and b
and hypotenuse c,
as below,
notice how the two legs form the right angle.
The hypotenuse is opposite the right angle,
it's also the longest side of the triangle.
The following relationship is always true.
A squared plus b squared equals c squared.
Now we can use this theorem to find the length of one side,
if the length of the two other sides is known.
These three equations here are derived from
a squared plus b squared equals c squared,
solving for a, b, and c.
We're asked to find the length of the leg x
of the right triangle shown below
and then write the answer in exact form
and in approximate form rounded to the nearest thousandth.
Let's first make sure we label
the three sides correctly.
Remember a and b must be the legs.
Let's let this leg be a
and this leg be b,
and the hypotenuse must be c.
Because we're determining the length
of this leg here which we labeled a,
we'll go ahead and use this equation here
that's already solved for a.
So we have a equals
the square root of c squared minus b squared.
So in our case we'll have a equals the square root.
C squared will be 14 squared
minus b squared which will be eight squared.
Simplifying we have a equals the square root
of 14 squared is equal to 196
minus eight squared is equal to 64.
So we have a equals the square root
of 196 minus 64 is equal to 132.
So the exact value of a is
the square root of 132.
And because the lengths are measured in centimeters,
the length of side a is exactly
the square root of 132 centimeters.
Depending on what class you're in,
you may be able to stop here.
But I also wanted to show how
the square root of 132 can be simplified.
You look at the prime factorization on 132,
132
is equal to two times 66.
Two is prime,
66 is equal to two times 33.
Two is prime,
33 is equal to three times 11,
both are prime.
So the square root of 132
is equal to the square root of two,
times two times three times 11.
Two times two which equals four
is a perfect square,
and we know this because we have two equal factors here.
We can simplify the square root of 132
to two square root 33.
The two outside the square root came from
the square root of two times two,
of the square root of four,
which equals two.
So let's also express the exact length
as a
equals two times the square root of 33 centimeters.
So here are two ways to express the exact length.
Let's also show the approximate form.
So we'll go back to the calculator.
And we can enter either of these two forms,
let's check to make sure they're the same.
So second x squared brings up the square root,
then we enter 132, enter.
Let's also make sure that two
square root 33 gives us the same value.
Notice it does.
And we're asked to round to the nearest thousandths,
which would be to three decimal places.
So because there's a one
in the 10 thousandths place value,
or fourth decimal place,
we're going to round down.
The length is approximately
11.489 centimeters.
And again this is the approximate length.
I hope you found this helpful.