Quadrilaterals: area and perimeter

The elementary table lists a square, a rectangle, a parallelogram, and a rhombus, each with its own $A$ and $C$. Four names look like four topics. They are one parallelogram with extra freezes. $C$ is the perimeter (often listed as circumference). This math handbook writes the included angle $\gamma$ in radians; a right angle is $\pi /2$. See measuring angles.

The boards below are not a general quadrilateral. The fourth vertex is constructed so opposite sides stay parallel. Drop that and these formulas do not apply.

Opposite sides stay parallel

Take $a$ as a base, $b$ as the adjacent side, $h$ as the corresponding height, and $\gamma$ as the included angle at $A$:

$A=ah=ab\sin \gamma ,C=2a+2b.$

The two expressions for $A$ are the same fact: $h=b\sin \gamma$. Drag $B$ or $D$. The fourth vertex follows; opposite sides stay equal and parallel. The dashed segment is $h$. That extra parallel constraint is what this board is.

Freeze a right angle

Keep the parallelogram and freeze $\gamma =\pi /2$. Then $\sin \gamma =1$ and the height is the adjacent side, so

$A=ab,C=2a+2b.$

That is a rectangle. Drag $D$ on the perpendicular through $A$ so the angles stay right. Drag $D$ off that line and the construction refuses: the extra freeze is the right angle, not a new area rule.

Then equal sides

Keep the right angles and freeze $a=b$. Then

$A=a^2,C=4a.$

That is a square. Drag $B$. The other two vertices follow so every side stays $a$ and every angle stays right. Two freezes stacked — equal sides on a rectangle — not a third independent figure.

Equal sides without the right angle

Go back to the parallelogram and freeze the sides instead of the angle: every side has length $a$. That is a rhombus (an equilateral parallelogram):

$A=a^2\sin \gamma ,C=4a.$

Drag $D$ on the circle through $B$ so every side stays $a$; $\gamma$ is still free. When $\gamma =\pi /2$, $\sin \gamma =1$ and $A=a^2$ — the square again, reached by freezing the angle on a rhombus instead of the sides on a rectangle. Drop equal sides and you are back at $A=ab\sin \gamma$.

Triangles from the same table live under triangles. Regular $n$-gons live under regular n-gons. Circles, circular sectors, and ellipses live under circles, sectors, and ellipses.