Given an isosceles trapezoid with known height,h, bottom base, b, and top base, a<b, how to find the width any any height < h?

Thanks.

## Isosceles trapezoid

### Re: Isosceles trapezoid

It's a linear function. If x is the height where you want to measure, and y is the width at that height, the function is a line that goes through:joe wrote:Given an isosceles trapezoid with known height,h, bottom base, b, and top base, a<b, how to find the width (at) any height < h?

Thanks.

(0,b) -- because the bottom width is b

(h,a) -- because the top width is a

Can you find the equation for the line that goes through those two points?

Isosceles doesn't matter. This technique works for any trapezoid.

### Re: Isosceles trapezoid

Thank you very much for the response. I was having trouble locating a discussion of this online. I can work out the equation for the line.

I guess my next question is, how do you know it's a linear function?

I guess my next question is, how do you know it's a linear function?

### Re: Isosceles trapezoid

Hmmm, I guess I know that because I skipped some steps, which I'll explain now.

Set the trapezoid on the plane such that the bottom base is along the x-axis from (0,0) to (b,0). The top base will go from ((b-a)/2,h) to (a+(b-a)/2,h). The equations for the top and bottom sides are y=0 and y=h. For the left and right sides you can work it out.

Now for any height y1 (note that I have changed the variables from my previous post), consider the line y=y1. It intersects the left and right sides at points (x1,y1) and (x2,y1). The distance between those points is x2-x1. At the bottom (y1=0), you have x1=0 and x2=b. At the top, their difference is going to be a. You can work out a formula for any point in between, and it will be linear because it's the difference of two linear functions.

Set the trapezoid on the plane such that the bottom base is along the x-axis from (0,0) to (b,0). The top base will go from ((b-a)/2,h) to (a+(b-a)/2,h). The equations for the top and bottom sides are y=0 and y=h. For the left and right sides you can work it out.

Now for any height y1 (note that I have changed the variables from my previous post), consider the line y=y1. It intersects the left and right sides at points (x1,y1) and (x2,y1). The distance between those points is x2-x1. At the bottom (y1=0), you have x1=0 and x2=b. At the top, their difference is going to be a. You can work out a formula for any point in between, and it will be linear because it's the difference of two linear functions.

### Re: Isosceles trapezoid

Thanks for the additional explanation. That helps a lot.