Quick answer: A geometry formula sheet is a categorized reference of the equations used to find area, perimeter, volume, surface area, and coordinate relationships for common shapes. The formulas below are grouped by 2D shapes, 3D shapes, coordinate geometry, and angle rules, each with a worked example so you can check your own calculation method, not just the answer.
What Is a Geometry Formula Sheet?
A geometry formula sheet is a single-page reference that organizes the equations for area, perimeter, volume, surface area, and coordinate relationships by shape and topic, so you can look one up in seconds instead of re-deriving it. It’s not a substitute for understanding why a formula works — but it is the fastest way to check your work, prep for a test, or refresh a formula you learned years ago.
The NCTM’s Principles and Standards for School Mathematics groups geometry learning into four skills: analyzing shape properties, describing spatial relationships with coordinates, applying transformations, and using visualization to solve problems. This guide is organized the same way — by shape family first, then by the coordinate and angle rules that connect them.
2D Shape Formulas
Two-dimensional shapes are flat, so the two measurements you’ll need most are area (space enclosed) and perimeter (distance around the edge).
Triangles
| Formula | Expression |
|---|---|
| Area | ½ × base × height |
| Area (no height given — Heron’s formula) | √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 |
| Perimeter | a + b + c |
| Equilateral triangle area | (√3/4) × side² |
| Right triangle | a² + b² = c² |
Example: A triangle with a base of 8 cm and a height of 5 cm has an area of ½ × 8 × 5 = 20 cm².
Use Heron’s formula specifically when you know all three side lengths but not the height — it’s the only formula here that doesn’t need a perpendicular height measurement.
Quadrilaterals
| Shape | Area | Perimeter |
|---|---|---|
| Square | side² | 4 × side |
| Rectangle | length × width | 2(length + width) |
| Parallelogram | base × height | 2(a + b) |
| Rhombus | ½ × d₁ × d₂ | 4 × side |
| Trapezoid | ½ × (b₁ + b₂) × height | a + b₁ + b₂ + c |
Example: A rectangle 12 cm long and 7 cm wide has an area of 12 × 7 = 84 cm² and a perimeter of 2(12 + 7) = 38 cm.
Circles
Circles use the constant π (pi) ≈ 3.14159, the ratio of a circle’s circumference to its diameter.
| Formula | Expression |
|---|---|
| Area | π × r² |
| Circumference | 2 × π × r |
| Diameter | 2r |
| Arc length | (θ/360) × 2πr |
| Sector area | (θ/360) × πr² |
r = radius, θ = central angle in degrees.
Example: A circle with a 6 cm radius has an area of π × 6² ≈ 113.1 cm² and a circumference of 2π × 6 ≈ 37.7 cm.
3D Shape Formulas
Three-dimensional shapes add depth, so you calculate volume (space inside) and surface area (total area of every outer face).
Cube and Cuboid
| Shape | Volume | Surface Area |
|---|---|---|
| Cube | side³ | 6 × side² |
| Cuboid (rectangular prism) | l × w × h | 2(lw + wh + lh) |
Example: A cube with a 4 cm side has a volume of 4³ = 64 cm³ and a surface area of 6 × 4² = 96 cm².
Cylinder, Cone, Sphere, and Hemisphere
| Shape | Volume | Surface Area |
|---|---|---|
| Cylinder | π × r² × h | 2πr(r + h) |
| Cone | ⅓ × π × r² × h | πr(r + l), l = slant height = √(r² + h²) |
| Sphere | (4/3) × π × r³ | 4 × π × r² |
| Hemisphere | (2/3) × π × r³ | 3 × π × r² |
Example: A cylinder with a 3 cm radius and 10 cm height has a volume of π × 3² × 10 ≈ 282.7 cm³.
Coordinate Geometry Formulas
Coordinate (analytic) geometry plots shapes and lines on an x–y plane, so every formula here works from two points, (x₁, y₁) and (x₂, y₂).
| Formula | Expression |
|---|---|
| Distance between two points | √[(x₂−x₁)² + (y₂−y₁)²] |
| Midpoint of a segment | ((x₁+x₂)/2, (y₁+y₂)/2) |
| Slope of a line | (y₂−y₁) / (x₂−x₁) |
| Line equation (slope-intercept) | y = mx + c |
| Line equation (point-slope) | y − y₁ = m(x − x₁) |
| Distance from a point (x₀, y₀) to a line Ax + By + C = 0 | |Ax₀ + By₀ + C| / √(A² + B²) |
Example: The distance between A(2, 3) and B(6, 7) is √[(6−2)² + (7−3)²] = √32 ≈ 5.66 units.
Angle and Polygon Formulas
| Concept | Rule |
|---|---|
| Angles on a straight line | Sum = 180° |
| Angles around a point | Sum = 360° |
| Vertically opposite angles | Equal |
| Interior angles of a triangle | Sum = 180° |
| Interior angles of a quadrilateral | Sum = 360° |
| Interior angles of an n-sided polygon | (n − 2) × 180° |
| Each interior angle of a regular n-gon | [(n − 2) × 180°] / n |
| Exterior angle of a regular n-gon | 360° / n |
| Complementary angles | Sum = 90° |
| Supplementary angles | Sum = 180° |
Example: A regular hexagon (n = 6) has interior angles of [(6−2) × 180°] / 6 = 120° each.
The Pythagorean Theorem
a² + b² = c²
For any right triangle, the square of the hypotenuse (c, the side opposite the right angle) equals the sum of the squares of the other two sides.
Example: A right triangle with legs of 3 cm and 4 cm has a hypotenuse of √(3² + 4²) = √25 = 5 cm.
Common Pythagorean triples (whole-number solutions worth memorizing so you can skip the calculation): 3-4-5, 5-12-13, 8-15-17, 7-24-25.
What’s on the Official SAT Reference Sheet
If you’re prepping for the Digital SAT, you don’t need to memorize every formula above — the College Board provides a built-in reference sheet inside every math module. As of the current digital format, it includes:
- Area of a circle, rectangle, and triangle; circumference of a circle
- Volume of a rectangular prism, cylinder, sphere, cone, and pyramid
- The Pythagorean theorem
- Side ratios for 45-45-90 and 30-60-90 triangles
- Degrees in a circle (360°) and the sum of a triangle’s angles (180°)
Everything not on that list — including coordinate geometry, polygon angle formulas, and Heron’s formula — still needs to be memorized. You can review the current digital reference sheet through Khan Academy’s official Digital SAT prep course, built in partnership with the College Board.
Worked Examples
Volume of a cone. Radius 5 cm, height 12 cm. V = ⅓ × π × r² × h = ⅓ × π × 25 × 12 = 100π ≈ 314.16 cm³
Area of a trapezoid. Parallel sides of 8 cm and 14 cm, height 6 cm. A = ½ × (b₁ + b₂) × h = ½ × 22 × 6 = 66 cm²
Midpoint of a segment. Points P(−2, 4) and Q(6, −2). M = ((−2+6)/2, (4+(−2))/2) = (2, 1)
How to Use a Formula Sheet Without Relying on It
- Group formulas by topic — 2D, 3D, and coordinate geometry each in their own section — so you can scan instead of search.
- Apply each formula to a real problem at least once before an exam; recognition under time pressure is different from recognition while reading.
- Track your units. Area is always squared (cm², m²); volume is always cubed (cm³, m³) — a mismatched unit is one of the most common exam errors.
- Re-read what’s given before you calculate. Mixing up radius and diameter, or base and height, causes more wrong answers than not knowing the formula itself.
- Rebuild your own version by hand. Writing formulas out — not just reading them — is what moves them from a formula sheet into memory.
FAQs
What is the most important formula in geometry?
The Pythagorean theorem (a² + b² = c²) is the one most geometry educators point to first — it links algebra and geometry directly and underlies distance, slope, and trigonometric formulas used well beyond right triangles.
How many geometry formulas do I actually need to know?
There’s no official fixed count, but a typical school-level formula sheet covers roughly 40–60 formulas across 2D shapes, 3D shapes, coordinate geometry, and angle relationships — everything on this page falls within that range.
Why does π (pi) show up in so many geometry formulas?
Pi (≈ 3.14159) is the fixed ratio between a circle’s circumference and its diameter, so it appears in every formula built on a circular cross-section: circles, cylinders, cones, and spheres.
Is a formula sheet allowed during exams?
It depends on the exam. The Digital SAT provides a built-in reference sheet automatically (see the section above); most classroom exams don’t allow an outside one unless your teacher says so. Either way, knowing when to use a formula matters more than having it in front of you.
What’s the difference between area and surface area?
Area measures the space inside a flat, 2D shape. Surface area measures the combined area of every face of a 3D shape — effectively, how much material it would take to wrap it completely.
How do I find the area of an irregular shape?
Break it into regular shapes you already have formulas for — rectangles, triangles, circle segments — find each piece’s area, then add them together. This is called the decomposition method, and it’s how most real-world area problems (floor plans, land plots) actually get solved.
Which shapes should a beginner memorize first?
Triangle, square, rectangle, circle, cube, and cylinder cover the large majority of introductory geometry problems. Trapezoids, rhombuses, cones, and spheres are the natural next step once those six are solid.
Conclusion
Every formula on this page traces back to the same handful of ideas — a base and a height, a radius, or two coordinate points — reapplied to different shapes. Once you’ve worked through the examples above on paper, the fastest way to lock them in is repetition on new numbers, not re-reading the list.
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