>> Welcome to a video on the area of a triangle using Heron's Formula.
The goal of the video is to determine the area of a triangle
from the length of the three sides.
So if we have any triangle with sides of length a, b, and c,
then we let s equal the semi-perimeter or half of the perimeter, than the area is going
to equal the square root of s times the quantity s minus a times the quantity s minus b times the
quantity s minus c. Now, this formula can be proven using the Pythagorean Theorem,
but in this video, we're just going to look at some examples on how to apply this formula.
Let's go and take a look at our first example.
Here, we want to determine the area of a triangle with sides of length three centimeters,
four centimeters, and five centimeters.
So the area is going to equal the square root
of s times s minus a times s minus b times s minus c. So let's go ahead
and let a equal 3, b equal 4, and c equal 5.
Remember, s is the semi-perimeter or half of the perimeter.
So let's go ahead and find the value of s. It's going
to be one-half times the quantity 3 plus 4 plus 5.
That's going to give us one-half times 12.
So s is equal to 6.
So now we can go ahead and apply for the formula.
Replace s with 6, a with 3, b with 4, and c with 5.
So we're going to have the square root of 6 times 6 minus 3 times 6 minus 4 times 6 minus 5.
Let's go ahead and simplify this.
We're going to have 6 times 3 times 2 times 1.
Well, this is going to be 6 times 3 is 18 times 2 is 36, the square root of 36.
So the area of this triangle with sides length three, four,
and five centimeters is equal to 6 centimeters squared.
Now I'm not sure if you recognize this, but a triangle with sides length three, four,
and five is a right triangle where the shortest leg would be three,
the longest side would be five, the hypotenuse, and in this case, the height would be four.
So using a traditional area formula for a triangle,
area equals one-half base times height, we could verify the area
that we found using Heron's formula.
It'd be one-half 3 times 4, which, of course, is also 6.
Now remember to use this formula here, we do have to find the height of the triangle.
We lucked out here because it was a right triangle, but what's special
about Heron's formula is we don't have to have the height,
and it doesn't have to be a right triangle.
We just have to know the length of the three sides.
Let's go and take a look at one more example.
Here, we have a triangle with sides of length 8, 14, and 16 inches.
Just to keep things organized, let's go ahead and call this a, we'll call this b,
and we'll call this c. Let's go and write out Heron's Formula.
The area is going to equal the square root of s times s minus a times s minus b times s minus c.
So let's go ahead and determine the value of s, which is half of the perimeter.
So we're going to have one-half times the quantity of 8 plus 14 plus 16.
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It's going to give us one-half times, this looks like we'd have, this would be 30 plus 8.
So we have one-half times 38.
So our semi-perimeter is equal to 19.
Let's go ahead and apply our formula.
The area of this triangle is going to be equal to the square root
of 19 times 19 minus a, which is 8.
So 19 minus 8 times 19, minus b. So 19 minus 14, and then,
lastly, we have 19 minus c, and c is 16.
Let's go ahead and simplify this.
We have 19 times 11 times 5 times 3.
This product is 3,135.
So we have the square root of 3,135.
Now depending on your class, you may need to write this and simplify radical form,
or just get a decimal approximation.
To simplify this, we want to determine if this has any perfect score or factors.
So let's go ahead and make a factor tree and see what we have.
It ends in 5.
So it must be divisible by 5.
That would be 5 times 627.
Now the sum of these digits, 6 plus 2 plus 7, that would be 15.
Remember, if the sum of the digits is divisible by 3, so is this number.
So this breaks down into 3 times 209.
Now 209 is kind of a tough one to break down,
but I know that 11 times 20 would be 220, and this is 11 less than that.
So it's going to be 11 times 19.
Now, notice 3,135 does not have any perfect score or factors.
Therefore, this does not simplify.
So the best we can do here is get a decimal approximation.
Let's go ahead and do that.
So second x squared, 3,135.
Well, it looks like it's approximately 55.99 square inches.
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OK. That's going to do it for this video.
I hope you found it helpful. ------------------------------7e71a525af4d--