What is the smallest dimension for an inscribed rectangle of known proportions?

Question

For a circle with a diameter of d, what is the smallest dimension for an inscribed rectangle (maximum height for a horizontally-oriented rectangle or maximum width for a vertically-oriented rectangle) whose sides are proportional by 1920:1080?

In terms of application, what is the smallest dimension (maximum height if horizontally-oriented) of the largest possible camera sensor with a resolution of 1920*1080 pixels that will fit inside a telescope's image circle of diameter 1.25 inches (or any other arbitrary diameter, d)?

Answer 1

I figured it out. I got the following in WXMaxima to compute either the width or height of the inscribed rectangle:

(%i8) solve(d^2=((1920/1080)*h)^2+h^2,h);

(%o8) [h=-(9*d)/sqrt(337),h=(9*d)/sqrt(337)]

(%i9) solve(d^2=w^2+((1080/1920)*w)^2,w);

(%o9) [w=-(16*d)/sqrt(337),w=(16*d)/sqrt(337)]

or for a rectangle of proportion 1920:1080 inscribed in a circle of diameter d,

h = 9*d/SQRT(337) = ~0.490*d

w = 16*d/SQRT(337) = ~0.872*d

Answer 2

There's only one rectangle with given proportions $p:q$ inscribed in a given circle (excluding rotation symmetries). The diameter $d$ of the circle is the size of an inner diagonal of such rectangle.

Let $a$ and $b$ be the sizes of the sides of such a rectangle; and let $\theta$ be the angle between a diagonal and a side of length $b$. Now a diagonal, a side of length $b$, and an opposing side of length $a$ form a right triangle.

Suppose $a \leq b$ and $p \leq q$. Given this configuration, we have

• $\theta = \tan^{-1}(p/q)$
• $a = d*\sin(\theta)$
• $b = d*\cos(\theta)$

In your specific case,

• $\theta = \tan^{-1}(1080/1920) \approx 0.512$ radians
• $a = d*\sin(\theta) \approx 0.490 d$
• $b = d*\cos(\theta) \approx 0.872 d$