# 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 thus

\begin{align} A_\text{rhombus}&= A_{\text{triangle }a}+ A_{\text{triangle }b} + A_{\text{triangle }c} + A_{\text{triangle }d}\\ ~&= 2A_{\text{triangle }a} +2 A_{\text{triangle }b}\qquad\leftarrow \text{ Pairwise equal triangles}\\ ~&=\cancel2\frac1{ \cancel 2}l_aw_a+ \cancel 2\frac1{ \cancel 2}l_bw_b \qquad\leftarrow \text{Triangle area, }A=\frac12lw\\ ~&=l_aw_a+l_bw_b \\ ~&=\underbrace{\frac12 d_1}_{l_a} \underbrace{ \frac12 d_2 }_{w_a} + \underbrace{ \frac12 d_1 }_{l_b} \underbrace{ \frac12 d_2 }_{w_b} \\ ~&=\cancel 2\,\frac1{\cancel2} d_1\frac 12 d_2\\ ~&=\frac12 d_1d_2\qquad_\blacksquare \end{align}

## 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 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 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.…

## Proof 14: Motion equations

In motion along a path, $a$ is acceleration, $v_0$ and $v$ are initial and final (current) speed, and $t_0$ and $t$ are initial and final…

## 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…