Where does the gap come from?

If I did this

enter image description here

Would you ask where the hole came from?

Addeudum

enter image description here


Figure out the area of the little gap in the middle:

enter image description here

(you can probably guess it, but it's easy to calculate).

Try it for yourself before peeking at the explanation at the bottom.

From there it's easy to see where the area for the square comes from.

Total area of the 13 × 5 region = 65 squares.

Total area of the two unmarked rectangular parts = (5 × 3) × 2 = 30

Total area of blue triangles = (5 × 2 × ½) × 2 = 10

Total area of red triangles = (8 × 3 × ½) × 2 = 24

Remaining area = 65 - 30 - 10 - 24 = 1 square.

If it's still not clear, rotate the upper two triangles each about the center of its own diagonal (which leaves the upper edge of the gap unchanged), you get the triangles from your two diagrams overlaid on each other in the correct positions:

enter image description here

It takes the extra square to make up for the area of that skinny parallelogram-shaped gap between the regions covered by the two arrangements of shapes.


The line over the red part is not quite as steep as that over the blue part. Just compute the two slopes and you'll see that.

Tags:

Geometry

Area

Fake Proofs

Related

Does the Lie derivative commute with $\partial$? Total number of integers relatively prime to $p^2$ Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$ Rational Question for $a + b$ and Irrationality of $a^2 + b^2$ Calculate $\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$ using residues Finding the Limit of a Sequence (No L'Hopital's Rule) Proof outer measure satisfies monotonicity: $A \subseteq B \implies m^*(A) \leq m^*(B)$ Prove that $f(x)$ is irreducible iff its reciprocal polynomial $f^*(x)$ is irreducible. Does there exist a complex function which is differentiable at one point and nowhere else continuous? Area of Mobius strip Evaluate $\int \frac{1-x}{(1+x)\sqrt{x+x^2+x^3}}dx$ Is $\frac{1}{11}+\frac{1}{111}+\frac{1}{1111}+\cdots$ an irrational number?

Recent Posts