Area of rectangle knowing diagonal and angle between diagonal and edgeI found on the web...

John Landry

John Landry

Answered

2022-07-21

Area of rectangle knowing diagonal and angle between diagonal and edge
I found on the web that the area of a rectangle with the diagonal of length d, and inner angle (between the diagonal and edge) θ is d 2 cos ( θ ) sin ( θ ). However, I wasn't able to deduce it myself. I tried applying law of sines or generalised Pythagorean theorem but I couldn't derive the area using only the length of the diagonal and the angle between diagonal and edge. How might I get to this result ?

Answer & Explanation

dominicsheq8

dominicsheq8

Expert

2022-07-22Added 15 answers

Step 1
If you use the formulas for sine and cosine in right-angled triangles, the formula can be proved rather easily: If the width and the height of the rectangle are resp. w and h, then the formulas say cos ( θ ) = w / d and sin ( θ ) = h / d.
Step 2
If you isolate w and h in these formulas and substitute in the formula "area = w h", then the formula you mention appears.
stratsticks57jl

stratsticks57jl

Expert

2022-07-23Added 3 answers

Step 1
Let ABDC be a rectangle, with long sides AB and CD of length l and short sides AC and BD of length w. And let AD be a diagonal with length d which makes an angle θ between CD and AD.
Step 2
Note that we have sin θ = w d and c o s θ = l d . Multiplying by d on both sides for both equations gives w = d sin θ
l = d cos θ

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