I need to find out exactly WHY a trapezoid's area formula is "1/2 (base one + base two) times height."
And not because "that's what the math book says" or something lol, but I need the actual REASON why it's formula is 1/2(base 1 + base 2)height ... for example I know the reason a triangle's formula is "1/2 base times height." Because two triangles make one square, and a square's area is "base times height" but since a triangle only occupies half of that square, it is HALF base times height for a triangle.
Now what about for a trapezoid? Thanks!
And not because "that's what the math book says" or something lol, but I need the actual REASON why it's formula is 1/2(base 1 + base 2)height ... for example I know the reason a triangle's formula is "1/2 base times height." Because two triangles make one square, and a square's area is "base times height" but since a triangle only occupies half of that square, it is HALF base times height for a triangle.
Now what about for a trapezoid? Thanks!
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If you draw a diagonal from two opposite corners on the trapezoid, it splits it into two triangles. The area of one triangle will be 1/2 base one x height. THe area of the other triangle will be 1/2 base two x height.
Add the two triangle areas to get trapezoid area formula.
Add the two triangle areas to get trapezoid area formula.
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Robert Larue gave an excellent answer.
Another approach is to draw a parallel line midway between the bases, joining the two sides. The length of that line is the the average of the bases, namely (1/2)(base one + base two). At each end of that mid-line draw a line segment perpendicular to the bases and note that if you cut with scissors the little triangle that sticks out it could be moved to the vacant place on the other side of the mid-line which is a congruent "missing triangle". In effect, doing this you cut the trapezoid into pieces that reassemble into a rectangle. So that rectangle must have the same area as the trapezoid. But the base of the rectangle is (1/2)(base one + base two) and its height is the height of the trapezoid. Since area of a rectangle is base times height, you are done.
Another approach is to draw a parallel line midway between the bases, joining the two sides. The length of that line is the the average of the bases, namely (1/2)(base one + base two). At each end of that mid-line draw a line segment perpendicular to the bases and note that if you cut with scissors the little triangle that sticks out it could be moved to the vacant place on the other side of the mid-line which is a congruent "missing triangle". In effect, doing this you cut the trapezoid into pieces that reassemble into a rectangle. So that rectangle must have the same area as the trapezoid. But the base of the rectangle is (1/2)(base one + base two) and its height is the height of the trapezoid. Since area of a rectangle is base times height, you are done.
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well its like finding the area of a square almost. you have to do base times height. since your height is always the same you leave it like it is, but since the top and bottom (which are the bases) are different, doing 1/2(base1 plus base2) is just finding the average of the bases since they are different. you don't need to do that on a square because the top and bottom lengths will always be the same. hope you get it.