Geometry Essentials

A triangle's angles sum to $180^\circ$; a quadrilateral's to $360^\circ$. An exterior angle equals the sum of the two remote interior angles.

In plain terms. Angles in a shape add to a fixed total, so a missing angle is just the total minus the known ones.

Example. A triangle with angles $50^\circ$ and $60^\circ$ has third angle $180^\circ - 110^\circ = 70^\circ$.

$45\text{-}45\text{-}90$ has sides $1:1:\sqrt2$; $30\text{-}60\text{-}90$ has $1:\sqrt3:2$. Common Pythagorean triples: $3\text{-}4\text{-}5$, $5\text{-}12\text{-}13$, $8\text{-}15\text{-}17$, $7\text{-}24\text{-}25$.

In plain terms. A few triangle shapes show up constantly — memorize their side ratios so you recognize them instantly instead of using the Pythagorean theorem every time.

Example. A right triangle with legs $6$ and $8$ is a doubled $3\text{-}4\text{-}5$, so its hypotenuse is $10$.

Triangle area $= \tfrac12 bh = \tfrac12 ab\sin C$. Circle: circumference $2\pi r$, area $\pi r^2$; an arc of angle $\theta^\circ$ has length $\tfrac{\theta}{360}\cdot 2\pi r$.

In plain terms. Area of a triangle is half base times height; a circle's size formulas both start from the radius.

Example. A triangle with sides $5,5,6$: the altitude to the base $6$ makes two $3\text{-}4\text{-}5$ triangles, so height $= 4$ and area $= \tfrac12\cdot 6\cdot 4 = 12$.