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## 6th grade (Illustrative Mathematics)

### Unit 1: Lesson 5

Lesson 9: Formula for area of a triangle# Area of a triangle

CCSS.Math:

Understand why the formula for the area of a triangle is one half base times height, which is half of the area of a parallelogram.

## Want to join the conversation?

- is there another formula(76 votes)
- Interesting question!

Given the length of any base and the height (altitude) perpendicular to the side that is chosen as the base, the area formula of one half base times height is about as simple as it gets.

If instead the lengths of the three sides are given (but no heights are given), there is a much more complex formula for the area of the triangle, called Heron's formula. Let a, b, and c represent the lengths of the sides, and let S = (a+b+c)/2, that is, S represents half the perimeter.

Then the area is given by A = squareroot[S(S - a)(S - b)(S - c)].(89 votes)

- Why is math important?(28 votes)
- Math helps us think analytically and have better reasoning abilities. Analytical thinking refers to the ability to think critically about the world around us. ... Analytical and reasoning skills are essential because they help us solve problems and look for solutions(23 votes)

- Is the answer still units squared or square units?(13 votes)
- Yes, the answer will be in units squared (even though you are measuring a △ and not a □)!(2 votes)

- i really don't get this concept can any one discripe it in a better form or discription(6 votes)
- So if you know how to find area of a rectangle or square this should make sense. Use the formula Base x Height divided by 2. Hope that helped!(13 votes)

- What if the tringle has 1 number and you have to find the area?(4 votes)
- To calculate the area of a triangle given one side and two angles, solve for another side using the Law of Sines, then find the area with the formula: area = 1/2 × b × c × sin(A) video link is https://youtu.be/wTkH288r84s.

also i need 25 upvotes on this answer plz

Ty(15 votes)

- how do you find the base if you know the area and the height?(5 votes)
- we know that Area = (base * height)/2 (formula for area of a triangle). if we know the area, suppose it is 4 for this example, and the height is 2 we get

4=(x*2)/2.

multiply by 2 on both sides to get

8=x*2, divide both sides by 2 to get

4=x, or 4=the base

use this method for the actual numbers(5 votes)

- I still don't get it I am bad at math can someone explain this to me?(3 votes)
- Ok, so let's get started with right triangles.

Visualise a right triangle as a half of a rectangle. The hypotenuse is the diagonal of the rectangle. The hypotenuse is the longest side of a triangle. The legs of the triangles are the 2 adjacent sides of the rectangle. Adjacent sides are sides that share a common point. In other words, adjacent sides are side-by-side. The area of a rectangle is length times the breadth, or*lb*. Now we know our right triangle is half of our rectangle. So the area will be half of that of the rectangle. Therefore, the area is*lb*/2.

Watch this video where Sal describes the proof of Triangles.

https://www.khanacademy.org/math/geometry-home/geometry-area-perimeter/geometry-area-triangle/v/triangle-area-proofs(7 votes)

- Given the length of any base and the height (altitude) perpendicular to the side that is chosen as the base, the area formula of one half base times height is about as simple as it gets.

If instead the lengths of the three sides are given (but no heights are given), there is a much more complex formula for the area of the triangle, called Heron's formula. Let a, b, and c represent the lengths of the sides, and let S = (a+b+c)/2, that is, S represents half the perimeter.

Then the area is given by A = squareroot[S(S - a)(S - b)(S - c)].(4 votes) - does it matter what formula you use by the shape of the triangle(2 votes)
- The area of
*any*triangle can be found using the formula Area = base * height / 2(5 votes)

- Is there another formula?(4 votes)
- Nice question!

There is Heron’s formula for the area of a triangle, if the side lengths a, b, and c are given.

Let s represent the semiperimeter, which equals (a+b+c)/2.

Then the area A is given by the formula

A = square root[s(s-a)(s-b)(s-c)].(0 votes)

## Video transcript

- [Voiceover] We know that we can find the area of a rectangle by multiplying the base times the height. The area of a rectangle is
equal to base times height. In another video, we saw that, if we're looking at the
area of a parallelogram, and we also know the length of a base, and we know its height, then the area is still going to be base times height. Now, it's not as obvious when you look at the parallelogram, but in that video, we did a little manipulation of the area. We said, "Hey, let's take this "little section right over here." So we took that little
section right over there, and then we move it over
to the right-hand side, and just like that, you see that, as long as the base and
the height is the same, as this rectangle here,
I'm able to construct the same rectangle by
moving that area over, and that's why the area
of this parallelogram is base times height. I didn't add or take away
area, I just shifted area from the left-hand side
to the right-hand side to show you that the area
of that parallelogram was the same as this
area of the rectangle. It's still going to be base times height. So hopefully that convinces you that the area of a parallelogram
is base times height, because we're now going to use that to get the intuition for
the area of a triangle. So let's look at some triangles here. So that is a triangle, and we're given the base and the height, and we're gonna try to think about what's the area of this
triangle going to be, and you can imagine it's going to be dependent on base and height. Well, to think about that, let me copy and paste this triangle. So let me copy, and then let me paste it, and what I'm gonna do is, so now I have two of the triangles, so this is now going to be twice the area, and I'm gonna rotate it around, I'm gonna rotate it around like that, and then add it to the
original area, and you see something very interesting is happening. I have now constructed a parallelogram. I have now constructed a parallelogram that has twice the area
of our original triangle, 'cause I have two of
our original triangles right over here, you saw me do it, I copied and pasted it,
and then I flipped it over and I constructed the parallelogram. Now why is this interesting? Well, the area of the
entire parallelogram, the area of the entire parallelogram is going to be the length of
this base times this height. You also have height written with the "h" upside down over here. It's going to be base times height. That's going to be for the
parallelogram, for the entire-- let me draw a parallelogram
right over here. That's going to be the area
of the entire parallelogram. So what would be the area
of our original triangle? What would be the area
of our original triangle? Well, we already saw that this
area of the parallelogram, it's twice the area of
our original triangle. So our original triangle is just going to have half the area. So this area right over
here is going to be one half the area of the parallelogram. One half base-- let me
do those same colors. One half base times height. One half base times height. And you might say, "OK, maybe
it worked for this triangle, "but I wanna see it work
for more triangles." And so, to help you there, I've added another
triangle right over here, you could do this as an obtuse triangle, this angle right over here
is greater than 90 degrees, but I'm gonna do the same trick. We have the base, and
then we have the height. Here, you can think of, if you start at this
point right over here, and if you drop a ball, the length that the ball goes,
if you had a string here, to kind of get to the ground level, you could view this as the ground level right over there, that that's
going to be the height, it's not sitting in the
triangle like we saw last time, but it's still the height of the triangle. If this was a building of some kind, you'd say, "Well, this is the height." How far off the ground is it? Well, what's the area of this going to be? Well, you can imagine, it's going to be one
half base times height. How do we feel good about that? Well, let's do the same magic here. So let me copy and paste this, so I'm gonna copy and then paste it. Whoops, that didn't work. Let me copy, and then paste it. And so, I have two of these triangles now, but I'm gonna flip this one over, so that I can construct a parallelogram. So I'm gonna flip it over,
and move it over here, I'm gonna have to rotate
it a little bit more. So, I think you get the general idea. So now I have constructed a parallelogram that has twice the area
of our original triangle. It has twice the area of
our original triangle. And so, if I talked about the area of the entire parallelogram, it would be base times the
height of the parallelogram. Base times the height
of the parallelogram. But if we're only talking
about the area of -- If we're only talking about this area right over here, which
is our original triangle, it's going to be half the
area of the parallelogram, so it's gonna be one half of that. So our area of our original triangle is one half base times height. So hopefully that makes
you feel pretty good about this formula that
you will see in geometry, that area of a triangle is
one half base times height, while the area of a
rectangle or a paralleogram is going to be base times height.