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) $\theta $ is ${d}^{2}\mathrm{cos}(\theta )\mathrm{sin}(\theta )$. 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 ?

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) $\theta $ is ${d}^{2}\mathrm{cos}(\theta )\mathrm{sin}(\theta )$. 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

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 $\mathrm{cos}(\theta )=w/d$ and $\mathrm{sin}(\theta )=h/d$.

Step 2

If you isolate w and h in these formulas and substitute in the formula "area $=wh$", then the formula you mention appears.

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 $\mathrm{cos}(\theta )=w/d$ and $\mathrm{sin}(\theta )=h/d$.

Step 2

If you isolate w and h in these formulas and substitute in the formula "area $=wh$", then the formula you mention appears.

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 $\theta $ between CD and AD.

Step 2

Note that we have $\mathrm{sin}\theta =\frac{w}{d}$ and $cos\theta =\frac{l}{d}$. Multiplying by d on both sides for both equations gives $w=d\mathrm{sin}\theta $

$l=d\mathrm{cos}\theta $

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 $\theta $ between CD and AD.

Step 2

Note that we have $\mathrm{sin}\theta =\frac{w}{d}$ and $cos\theta =\frac{l}{d}$. Multiplying by d on both sides for both equations gives $w=d\mathrm{sin}\theta $

$l=d\mathrm{cos}\theta $

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