1.2 The Regents Reference Sheet & Key Formulas
Key Takeaways
- The Next Generation (NGLS) Geometry reference sheet prints only five volume formulas: general prism and cylinder (V = Bh), cone and pyramid (V = 1/3 Bh), and sphere (V = 4/3 pi r cubed).
- The sheet gives cylinder and cone as Bh forms, so you must compute the base area B = pi r squared yourself before substituting.
- The Pythagorean theorem, distance, midpoint, slope, circle equation, arc length, sector area, and all trigonometry are NOT on the sheet and must be memorized.
- Coordinate proofs rely on distance, midpoint, slope, and (x-h)squared + (y-k)squared = r squared; keep pi exact unless told to round.
- Arc length is (theta/360) times 2 pi r and sector area is (theta/360) times pi r squared.
What Is Actually on the Reference Sheet
A surprise for many test-takers: the Next Generation (NGLS) Geometry reference sheet is tiny. Under the heading "Reference Sheet for Geometry (NGLS)" there is exactly one section — Volume — listing five formulas. Nothing else is printed. This is a real change from the older Common Core sheet, which included conversions, the Pythagorean theorem, and more.
| Solid | Formula on the sheet | Note |
|---|---|---|
| General prism | V = Bh | B is the area of the base |
| Cylinder | V = Bh | base is a circle, so B = πr² |
| Cone | V = ⅓Bh | B is the area of the base |
| Pyramid | V = ⅓Bh | B is the area of the base |
| Sphere | V = (4/3)πr³ | r is the radius |
The "Bh" Trap
The sheet writes the cylinder and cone as V = Bh and V = ⅓Bh, not as πr²h. Here B always means the area of the base, so you must compute B yourself before substituting. For a cylinder with r = 3 and h = 10: first find B = πr² = 9π, then V = Bh = 9π(10) = 90π. The same cone gives V = ⅓(9π)(10) = 30π. Forgetting to compute B is one of the most common measurement errors.
Everything Else — You Must Memorize It
Because the sheet ends at volume, the Next Generation Geometry Regents assumes you have every other relationship memorized. None of the formulas below are printed on test day.
| Formula | Statement | Reach for it when... |
|---|---|---|
| Pythagorean theorem | a² + b² = c² | finding a missing side of a right triangle |
| Area of a triangle | A = ½bh (also ½ab·sin C) | triangle area; the sine form handles two-sides-and-angle cases |
| Area of a circle | A = πr² | circle regions; also the base B of a cylinder or cone |
| Circumference | C = 2πr = πd | distance around a circle |
| Distance | d = √((x₂−x₁)² + (y₂−y₁)²) | length of a segment on the coordinate plane |
| Midpoint | ((x₁+x₂)/2, (y₁+y₂)/2) | center of a segment; showing diagonals bisect |
| Slope | m = (y₂−y₁)/(x₂−x₁) | parallel lines (equal m) or perpendicular (m₁·m₂ = −1) |
| Equation of a circle | (x−h)² + (y−k)² = r² | center (h, k) and radius r; complete the square to find them |
| Arc length | (θ/360) · 2πr | portion of the circumference cut by a central angle θ |
| Sector area | (θ/360) · πr² | pie-slice area for central angle θ |
| Trig ratios | sin = opp/hyp, cos = adj/hyp, tan = opp/adj | right-triangle sides and angles (SOH-CAH-TOA) |
| Special right triangles | 45-45-90: x, x, x√2 · 30-60-90: x, x√3, 2x | exact side lengths without a calculator |
| Law of Sines / Cosines | a/sin A = b/sin B · c² = a²+b²−2ab·cos C | solving non-right triangles |
When to Use Each
- Coordinate geometry (12-18% of the exam): distance, midpoint, slope, and the circle equation power almost every coordinate proof. Prove a parallelogram with equal midpoints of the diagonals, or a right angle with slopes whose product is −1.
- Similarity, right triangles, trigonometry (29-37%): SOH-CAH-TOA plus the two special triangles handle right-triangle work; the Laws of Sines and Cosines handle everything non-right.
- Circles (2-8%): arc length and sector area are just the fraction θ/360 of the whole circumference or area. In radians, arc length simplifies to s = rθ, but the θ/360 form matches how Regents questions state the central angle in degrees.
- Measurement and modeling (2-8% and 8-15%): use the printed volume formulas, but remember you supply B, and add or subtract solids for composite figures. Density problems combine volume with a mass-per-unit-volume rate, so compute the volume first, then multiply.
Three Quick Worked Examples
Sphere: a ball with radius 6 has V = (4/3)π(6)³ = (4/3)π(216) = 288π cubic units. The sphere formula is already in terms of r, so substitute directly.
Square pyramid: a pyramid with a 4-by-4 square base and height 9 needs B first: B = 4·4 = 16, so V = ⅓Bh = ⅓(16)(9) = 48 cubic units. Because B is the base area, a triangular base would use ½bh and a circular base would use πr² instead.
Circle equation: given x² + y² − 6x + 4y − 12 = 0, complete the square to get (x − 3)² + (y + 2)² = 25, so the center is (3, −2) and the radius is 5. This move recurs on 4-credit items, and the circle equation is not on the sheet.
Exact vs. Approximate
Keep π symbolic (use the calculator's π key) unless a question says "to the nearest..." Round only at the final step; rounding partway through a trig or circle calculation is a classic way to lose the final credit even when the method is right.
Which formulas are actually printed on the Next Generation (NGLS) Geometry Regents reference sheet?
The reference sheet gives a cylinder's volume as V = Bh. To use it for a cylinder with radius 3 and height 10, what must you do first?
A coordinate-proof question requires the distance between two points. Where do you get the distance formula on test day?