All Geometry Formulas
in One Place
Every formula you need for 2D shapes and 3D solids perimeter, area, surface area, and volume explained with a real diagram and a fully worked example under each one. No memorizing symbols without context. Jump to any shape below, or scroll through the whole guide.
Downloadable Geometry Formulas PDFs
- Keystone Geometry Formula Sheet.
- M-G2 Geometric Formulas.
- Math Resources Geometry Formulas.
- Comprehensive Geometric Formulas.
- Simplest 1 Page Geometry Formulas PDF.
Features
- Clear diagrams that illustrate each shape.
- Quick reference for perimeter, area, surface area and volume.
- Downloadable PDFs for offline study.
How this sheet works: Formulas below are written using standard math notation that renders cleanly in all modern browsers. Use the Print / Export to PDF button to save a printable A4 copy for offline study. To keep things easy to follow, variables are used consistently throughout: a, b, c for side lengths, l, w, h for length/width/height, r, d for radius/diameter, P for perimeter, A for area, SA for surface area, V for volume, and s for semi-perimeter or slant height (depending on context).
2D Shapes
Triangle
base bheight hsides a,c
- Perimeter: P = a + b + c
- Area: A = ½bh
- Heron’s formula: s = (a+b+c)/2 → A = √[s(s−a)(s−b)(s−c)]
- Pythagorean theorem: c² = a² + b²
- Law of cosines: c² = a² + b² − 2ab·cos(γ)
Square
side adiagonal d
- Perimeter: P = 4a
- Area: A = a²
- Diagonal: d = a√2
Rectangle
length lwidth w
- Perimeter: P = 2(l + w)
- Area: A = lw
- Diagonal: d = √(l² + w²)
Parallelogram
sides a,bheight h
- Perimeter: P = 2(a + b)
- Area: A = bh
Rhombus
diagonals d1,d2
- Perimeter: P = 4a (all sides equal)
- Area: A = (d1 × d2) / 2
Trapezoid / Trapezium
bases b1,b2height h
- Perimeter: sum of all side lengths
- Area: A = [(b1 + b2)/2] × h
- Median length: m = (b1 + b2)/2
Circle
radius rdiameter d
- Circumference: C = 2πr = πd
- Area: A = πr²
- Sector (θ in radians): arc L = θr, area A = (θ/2)r²
Ellipse
semi-major asemi-minor b
- Area: A = πab
- Circumference (Ramanujan approx.): C ≈ π[3(a+b) − √((3a+b)(a+3b))]
Regular n-gon
side sapothem asides n
- Perimeter: P = ns
- Area: A = (P × a)/2 = ns² / [4tan(π/n)]
3D Shapes
Cube
edge a
- Surface area: SA = 6a²
- Volume: V = a³
- Space diagonal: d = a√3
Rectangular Prism (Cuboid)
length lwidth wheight h
- Surface area: SA = 2(lw + lh + wh)
- Volume: V = lwh
- Space diagonal: d = √(l² + w² + h²)
Right Prism (General Base)
base area A_basebase perim. P_baseheight h
- Lateral area: LA = P_base × h
- Surface area: SA = LA + 2A_base
- Volume: V = A_base × h
Right Circular Cylinder
radius rheight h
- Lateral area: LA = 2πrh
- Surface area: SA = 2πrh + 2πr²
- Volume: V = πr²h
Right Circular Cone
radius rheight hslant s
- Slant height: s = √(r² + h²)
- Surface area: SA = πrs + πr²
- Volume: V = ⅓πr²h
Right Square Pyramid
base bheight hslant s
- Slant height (face center): s = √[(b/2)² + h²]
- Surface area: SA = b² + 2bs
- Volume: V = ⅓b²h
Sphere
radius r
- Surface area: SA = 4πr²
- Volume: V = &frac43;πr³
Hemisphere
radius r
- Curved surface area (no base): CSA = 2πr²
- Total surface area (with base): SA = 3πr²
- Volume: V = ⅔πr³
Frustum of a Cone
top r r1bottom r r2height h
- Slant height: s = √[h² + (r2−r1)²]
- Lateral area: LA = π(r1+r2)s
- Volume: V = ⅓πh(r1² + r1r2 + r2²)
Frustum of a Square Pyramid
top area A1bottom area A2height h
- Lateral area (regular): LA = ½(P1+P2)s
- Surface area: SA = LA + A1 + A2
- Volume: V = ⅓h(A1 + A2 + √(A1×A2))
Torus & Ellipsoid
torus R, rellipsoid a,b,c
- Torus surface area: SA = 4π²Rr
- Torus volume: V = 2π²Rr²
- Ellipsoid volume: V = &frac43;πabc
- Ellipsoid surface area (Thomsen approx., p≈1.6075): SA ≈ 4π[(ap+bp+cp)/3]1/p
How to Actually Remember These Formulas
Most people don’t forget geometry formulas because they’re hard they forget them because they memorized symbols without a picture attached. A few things that actually help:
- Learn the shape family, not each formula in isolation. A rectangle’s area is length × width. A parallelogram is the same idea tilted base × height. A triangle is just half of that parallelogram. Once you see the pattern, you stop memorizing three formulas and start remembering one idea.
- Say the units out loud. Area is always squared (cm², m²), volume is always cubed (cm³, m³). If your answer doesn’t match the units you expect, you used the wrong formula this catches more mistakes than anything else.
- Draw the shape before you plug in numbers. Even a rough sketch stops you from mixing up radius and diameter, or height and slant height, which is the single most common source of wrong answers on cones and pyramids.
- Use π = 3.14 for estimates, the π button for final answers. Rounding too early is a common reason answers don’t match the answer key.
Common Mistakes to Avoid
- Radius vs. diameter. Circle and sphere formulas use radius (r), not diameter (d). If a problem gives you the diameter, divide by 2 first this is the most frequent error in circle and cylinder problems.
- Height vs. slant height. Cones and pyramids have two different “heights”: the vertical height (h) used for volume, and the slant height (s) used for surface area. Mixing these up gives a wrong surface area even when the volume is right.
- Forgetting the base in surface area. A cylinder or prism’s surface area includes the top and bottom, not just the “wrap-around” lateral area. A hemisphere is worse decide first whether the question wants curved surface area only or total surface area including the flat circular base.
- Using perimeter formulas for irregular shapes. A trapezoid doesn’t have a single “add up the sides” shortcut like a square does, since its sides aren’t equal you genuinely have to add all four side lengths.