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Area of a triangle

The area of a rectangle and a parallelogram is found by multiplying the base by the height. For a triangle, the area is half of a parallelogram's, so it's calculated by multiplying the base by the height and then dividing by 2.

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  • leaf blue style avatar for user jordan.martinson
    is there another formula
    (91 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      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)].
      (112 votes)
  • duskpin tree style avatar for user Midnight the wolf
    Why is math important?
    (39 votes)
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    ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣷⣦⣤⣀⣀⣀⣴⣿⣿⣿⣿⣿⡿⠻⠆⠸⣎⣧⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
    ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣠⡄⠀⣿⢹⡇⢸⡀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿
    (42 votes)
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  • orange juice squid orange style avatar for user Link#JesusIsKing ✝
    What if the tringle has 1 number and you have to find the area?
    (10 votes)
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  • male robot johnny style avatar for user Evan
    Is the answer still units squared or square units?
    (19 votes)
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  • aqualine ultimate style avatar for user erin husky🐺
    i really don't get this concept can any one discripe it in a better form or discription
    (9 votes)
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  • blobby green style avatar for user Evelyn Pastrana Gutierrez
    I still don't get it I am bad at math can someone explain this to me?
    (10 votes)
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  • starky seedling style avatar for user academic_girl
    This is so much better than other videos and teacheRs explaining
    (15 votes)
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  • blobby green style avatar for user leonidt668
    This video has broken down my brain. I was studying in school for 10 years and hadn`t understood why is it 1/2bh, but this video... oh my god
    (14 votes)
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  • duskpin tree style avatar for user mr aizawa
    1.has anybody seen my will to live
    2.when would we use this in life
    3. can I have some sleep
    (11 votes)
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Video transcript

- [Instructor] 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, that 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 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, well what's, what's this area? 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, 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, base times this height. You also have the height written on the h upside down over here was going to be base times height. That's gonna be for the parallelogram, for the entire, lemme draw a parallelogram right over here. That's gonna 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, lemme do those same colors. One half base times height, one half base times height. And you might say, "Okay, 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 view 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. Or if you had a string here to kinda 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 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 gonna 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, so 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've 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, 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, if we're only talking about this area right over here, which is our original triangle, it's gonna 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 parallelogram is going to be base times height.