Cover a disk with thin rectangles

geometry

I remember that problem from university! (But I didn't solve it, somebody gave me a big clue)

The answer is that it is impossible, you can't cover the circle of diameter 20 using only 19 strips $1\times 20$.

To see why, suppose you have such cover and imagine a sphere of diameter 20 cut in half by our circle. Find the orthogonal projection of each strip on the sphere, it is a ring, and it is easy to compute it's area $2\pi R x$, where $R$ is the radius of the sphere and $x$ the width of the strip.

But between all the rings, they cover the whole sphere so the total area must be at least $4\pi R^2$, so $$ N \times 2 \pi R x \ge 4 \pi R^2 $$ where $N$ is the number of strips. Letting $N=19, R=10$ and $x=1$ we obtain a contradiction.

Related

Showing $1+p$ is an element of order $p^{n-1}$ in $(\mathbb{Z}/p^n\mathbb{Z})^\times$
Find three polynomials whose squares sum up to $x^4 + y^4 + x^2 + y^2$
Arc sums for a circle of $k$ positive integers whose total sum is $n$
Is the asymptotic density of powers of primes zero?
Does $ \left\lceil \frac{1}{n}\right\rceil\to 0$ or $1$ as $n\to\infty ?$
Another Series $\sum\limits_{k=2}^\infty \frac{\log(k)}{k}\sin(2k \mu \pi)$
A locally constant sheaf on a locally connected space is a covering space; Proof?
Period of derivative is the period of the original function
Cremona transformations are birational maps
Primes of ramification index 1 with inseparable residue field extension
Definition of Cartier divisors
Infinite Series $\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{n}\left\lfloor\frac{\log(n)}{\log(2)}\right\rfloor$