Geometry Regents Reference Table & Formulas
The official Geometry Reference Sheet, plus key Geometry formulas, concepts, and vocabulary to know for the exam.
The official Geometry Reference Sheet
Provided with every Geometry Regents booklet. Current edition: Next Generation (current).
What's inside the Geometry Reference Sheet:
- Area, surface area, and volume, every common solid
- Pythagorean theorem and distance formula
- Right-triangle trigonometry (SOH-CAH-TOA)
- Special right triangles (30-60-90, 45-45-90)
- Customary ↔ metric conversions
Geometry is volume-formula heavy. If a constructed-response problem stalls, the next move is almost always re-reading the relevant formula off this sheet.
Key Geometry formulas
| Name | Formula | What it's for |
|---|---|---|
| Pythagorean theorem | $a^2 + b^2 = c^2$ | Right triangles only. c is the hypotenuse, across from the right angle. |
| SOH-CAH-TOA | $\sin\theta=\dfrac{opp}{hyp},\ \cos\theta=\dfrac{adj}{hyp},\ \tan\theta=\dfrac{opp}{adj}$ | Right-triangle trig. Label the sides relative to the angle you are using. |
| Area of a triangle | $A = \tfrac{1}{2}bh$ | Base times height over two; the height must be perpendicular to that base. |
| Distance formula | $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ | Pythagoras applied to two points in the plane. |
| Midpoint formula | $M=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)$ | Average the x's and the y's. |
| Slope | $m=\dfrac{y_2-y_1}{x_2-x_1}$ | Parallel lines share slope; perpendicular slopes are negative reciprocals. |
| Circumference & area of a circle | $C=2\pi r,\quad A=\pi r^2$ | Everything circular starts from the radius r. |
| Arc length & sector area | $s=\dfrac{\theta}{360}\,2\pi r,\quad A=\dfrac{\theta}{360}\,\pi r^2$ | A central angle of theta degrees cuts off this fraction of the whole circle. |
| Volume of a cylinder / cone | $V_{cyl}=\pi r^2 h,\quad V_{cone}=\tfrac{1}{3}\pi r^2 h$ | A cone is exactly one third of the cylinder with the same base and height. (On the official sheet.) |
| Volume of a sphere / pyramid | $V_{sph}=\tfrac{4}{3}\pi r^3,\quad V_{pyr}=\tfrac{1}{3}Bh$ | B is the area of the pyramid's base. (On the official sheet.) |
| Polygon interior angle sum | $(n-2)\cdot 180^\circ$ | Total of all interior angles of an n-sided polygon. Each angle of a regular n-gon is this over n. |
| Special right triangles | $45\text{-}45\text{-}90:\ 1:1:\sqrt2 \quad 30\text{-}60\text{-}90:\ 1:\sqrt3:2$ | Memorize these side ratios; they turn up constantly without needing a calculator. |
Key concepts
- Triangle angle sum — The three interior angles add to 180 degrees. An exterior angle equals the sum of the two remote interior angles.
- Congruence — Same size and shape. Prove it with SSS, SAS, ASA, AAS, or HL, then matching parts are congruent (CPCTC).
- Similarity — Same shape, scaled size. AA is enough; corresponding sides are in a constant ratio (the scale factor).
- Rigid motions vs dilations — Translations, reflections, rotations preserve distance and angle (congruence). A dilation by factor k scales distance and produces a similar figure.
- Inscribed angle theorem — An inscribed angle is half its intercepted arc, so it is half the central angle on the same arc. An angle in a semicircle is 90 degrees.
- Parallel lines & a transversal — Alternate interior angles are equal; same-side interior angles are supplementary; corresponding angles are equal.
- Centers of a triangle — Centroid (medians, the balance point), circumcenter (perpendicular bisectors), incenter (angle bisectors), orthocenter (altitudes).
Must-know vocabulary
- Perpendicular bisector — A line through a segment's midpoint at a right angle; every point on it is equidistant from the endpoints.
- Median (of a triangle) — A segment from a vertex to the midpoint of the opposite side.
- Altitude — A perpendicular segment from a vertex to the opposite side (the triangle's height).
- Chord / tangent / secant — Chord joins two points on a circle; a tangent touches at one point; a secant cuts through two.
- Apothem — The distance from a regular polygon's center to the midpoint of a side.
- Scale factor — The ratio of corresponding lengths in a dilation or between similar figures.
Frequently asked questions
Do you get a reference table on the Geometry Regents?
Yes. The official Geometry Reference Sheet is provided with every Geometry Regents booklet, so practice with it open.
What formulas do you need for the Geometry Regents?
The most-used Geometry formulas are on this page with what each one is for. There are 12 key formulas to know.