Area of rectangle knowing diagonal and angle between diagonal and edgeI 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 ?

Question

Area of rectangle knowing diagonal and angle between diagonal and edgeI 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&#039;t able to deduce it myself. I tried applying law of sines or generalised Pythagorean theorem but I couldn&#039;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 ?

dominicsheq8 · Accepted Answer

Step 1If 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 2If you isolate w and h in these formulas and substitute in the formula &quot;area   =  w  h&quot;, then the formula you mention appears.

stratsticks57jl · Accepted Answer

Step 1Let 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 2Note 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