By Deepak Kumar · Founder, Vriksai
42 FormulasJoinery & Furniture Formulas
Joint geometry, proportion and deflection equations.
Sag & Deflection
Simply Supported UDL
δ = (5 × w × L⁴) / (384 × E × I)Simply Supported Point
δ = (P × L³) / (48 × E × I)Cantilever Point
δ = (P × L³) / (3 × E × I)Moment of Inertia
I = (b × h³) / 12Compound Miter
Compound Miter
Miter = arctan( sin(S) x tan(C/2) ) [S = spring, C = corner angle]Compound Miter
Bevel = arcsin( cos(S) x sin(C/2) )Compound Miter
Dihedral = arccos( cos2(S) x cos(C) + sin2(S) )Compound Miter
Polygon corner = 180 - (360 / N_sides)Golden Ratio
Golden Ratio
φ = (1 + √5) / 2 = 1.6180339887…Golden Ratio
Given width a → height b = a × φGolden Ratio
Given height b → width a = b / φGolden Ratio
Diagonal d = a × √(1 + φ²) = a × 1.9021…Ring Segment
Miter angle
theta = 90 - (180 / N) where N = number of segmentsOuter chord
L = 2 x R_outer x sin(PI / N)Kerf correction
dL = kerf / sin(theta) [material lost per cut end]Cut length
L_cut = outer_chord - kerf_correctionVolume/seg
V = PI x (Rout2 - Rin2) x H / NDrawer Reveal
Face height (N equal)
FH = (OH - bot_reveal - reveal x (N+1)) / NFace width (frameless)
FW = OW - 2 x revealBox width (undermount)
BW = OW - 3mmBox width (side-mount)
BW = OW - 2 x slide_clearanceY position (drawer n)
Yn = bot_reveal + (n-1) x (FH + reveal) + revealBox Joint
Box Joint
N (odd) = nearest odd to (W / finger_width)Box Joint
Actual fw = W / NBox Joint
Glue area = 2 × N × fw × thicknessBox Joint
Strength ratio = glue_area / (W × thickness) [typically 3–6×]Mortise & Tenon
Mortise & Tenon
Tenon thickness = Rail thickness / 3Mortise & Tenon
Tenon width = Rail width − 2 × shoulder_allowanceMortise & Tenon
Tenon length = 4.5 × tenon_thickness (max = 80% of stile thickness)Mortise & Tenon
Mortise depth = Tenon length + 2mm clearanceMortise & Tenon
Cheek = (Rail thickness − Tenon thickness) / 2Dovetail
Angle
theta = arctan(1 / ratio)Tail wide end
TW = T_narrow + 2 × thickness × tan(theta)N tails
round((W − min_pin) / (TW + min_pin))Spacing
S = (W − pin_width) / N_tailsExpansion Gap
Expansion Gap
Movement = W × S × ΔMC (W=width, S=coeff/MC%, ΔMC=change %)Expansion Gap
Radial coeff ≈ Tangential × 0.55Expansion Gap
Gap per board = max(shrink, swell) × 1.15 (15% safety factor)Screw Strength
IS 883
2016 — W = k × d × L (k = 12/10/7/5 by density group)Screw Strength
NDS 2018 — W = 124.9 × G^1.5 × D^0.75 (lbs/in penetration)Screw Strength
Eurocode 5 — f_ax,k = 0.52 × d^-0.5 × L^-0.1 × ρ^0.8Screw Strength
Group factor — Cg = max(0.4, 1 − 0.05×(n−1))Common questions
Where do these formulas come from?
Our formulas on this page are based on published sources — including standards, handbooks and established trade rules for joinery and furniture making where applicable. Sources are identified where available, and derived methods are identified separately.
Can I use these formulas in my own spreadsheets or software?
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Yes. Each calculator runs the formula shown here, and automated tests check the results against worked examples from the original sources before anything goes live.
