Proof 2: Triangle area

Two brothers share a slice of beer cake. Its face has a rectangle-shape. They cut it from corner to corner to get two equal halves. Those two halves namely have the same height, width and sloped side. They are completely equal with the same area.

Also, they are triangles. And half the size of the original rectangle with the same height $h$ and length $l$:

$$A_{triangle}=\frac12 \underbrace{A_{rectangle}}_{hl}=\frac12hl\quad_\blacksquare$$

The area of a rectangle is inputted here.

Proof 10: Parallelogram and trapezium area

A trapezium consists of a rectangle and two weird triangles at the ends. \begin{align}A_\text{trapezium}&= A_\text{rectangle}+ A_{\text{triangle } a}+ A_{\text{triangle } b}\\~&=\underbrace{lw_1}_\text{rectangle}+\underbrace{\frac12 w_al}_{\text{rectangle } a}+\underbrace{\frac12 w_b…

Proof 1: Rectangle, square, box and cube area and volume

‘A room is $4\,\mathrm m$ long and $2\,\mathrm m$ wide’; a rectangle-shape. For each of the 4 metres there are 2 square metres across. That…

Proof 9: Rhombus area

A rhombus actually consists of four right-angled triangles, because the diagonals $d_1$ and $d_2$ are perpendicular. Those triangles are pair-wise equal. The rhombus area is…

Proof 4: Circle and circle sector area

A pizza slice’s edge is the pizza’s radius $r$. The slice is almost a triangle if it wasn’t for the rounding. That rounding has a…

Proof 21: Sine and cosine values

We here derive the sine and cosine values of a few chosen angles. Half ($\pi$) and quarter turns ($\frac \pi 2$, $-\frac \pi 2$): Cosine…