Geometry

§1Coordinates and the straight line

Analytical geometry replaces drawing with arithmetic: a point is a pair of numbers, a line an equation, and every intersection a solvable system.

The line y = mx + b m = y₂ − y₁x₂ − x₁ = tan (slope angle) y − y₁ = m(x − x₁)

Distances between points: (x₂−x₁)² + (y₂−y₁)² point to line mx − y + b = 0: |mx₀ − y₀ + b|m² + 1

Two lines are parallel when their slopes match and perpendicular when the slopes multiply to −1. The angle between slopes m₁ and m₂ is tan β = (m₂ − m₁)/(1 + m₁m₂).

Example 1 — a slotted link as an equation

A slot runs from (20, 15) to (80, 55). Slope m = 40/60 = 0.6667, so the slot climbs at arctan 0.6667 = 33.69°, and its equation is y = 0.6667x + 1.667. A pin centre at (60, 10) sits |0.6667 × 60 − 10 + 1.667|/√(1.4444) = 26.35 mm off the slot axis — clearance settled without a drawing.

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§2Polar coordinates

Some parts are born polar — anything with holes on a pitch circle, cams, anything a rotary table touches. Converting between systems is two lines of trigonometry.

x = r cos θ y = r sin θ r = x² + y² θ = arctan(y/x) — quadrant by the signs of x, y

Example 2 — both directions

The point (86.603, 50.000) has r = √(7500 + 2500) = 100.000 and θ = arctan(50/86.603) = 30° — and rotating back, (100, 30°) returns (86.603, 50.000). The bolt-circle ordinate table on the Solution of Triangles page is this conversion run seven times.

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§3The circle

The most manufactured shape in existence. Its equation, its parts and its tangent behaviour cover most layout questions.

Equation (x − h)² + (y − k)² = r² centre (h, k), radius r

Example 3 — centre and radius from the general form

x² + y² − 12x + 8y − 48 = 0: complete the squares — (x − 6)² + (y + 4)² = 48 + 36 + 16 = 100. Centre (6, −4), radius 10.

Tangent facts that settle layout arguments: a tangent is perpendicular to the radius at its point of contact; the two tangent lengths from an external point are equal; and an angle inscribed in a semicircle is a right angle — the geometry behind checking a bore with a square.

Parts (angle θ at centre, radians) circumference = 2πr arc s = rθ chord c = 2r sin ½θ sector A = ½r²θ segment A = ½r²(θ − sin θ)

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§4Ellipse, parabola and hyperbola

Slice a cone at different angles and the three conics appear. In the workshop the ellipse turns up wherever a cylinder is cut on a slant; the parabola wherever something is focused or thrown.

Fig. 1. Ellipse: the sum of distances to the two foci is constant (= 2a); the foci sit at ±√(a² − b²) from centre.

Ellipse x²a² + y²b² = 1 A = πab

Perimeter (Ramanujan) C ≈ π(a+b)[1 + 3h/(10 + 4 − 3h)], h = (a−b)²(a+b)²

No exact closed form exists for the perimeter — the approximation above is astonishingly good. Parabola: y² = 4ax, focus at (a, 0); every ray parallel to the axis reflects through the focus. Hyperbola: x²/a² − y²/b² = 1, asymptotes y = ±(b/a)x.

Example 4 — perimeter of a 100 × 60 elliptical cover

a = 50, b = 30: h = (20/80)² = 0.0625, so C ≈ π × 80 × [1 + 0.1875/11.953] = 255.27 mm. Numerical integration of the exact perimeter integral gives 255.270 — Ramanujan’s formula is right to the last figure shown.

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§5Areas of plane figures

The mensuration table — every flat figure a drawing office meets, one line each.

Areas of plane figures
Figure	Area	Notes
Triangle	½ × base × height	or Heron’s rule from three sides
Rectangle	l × w	diagonal = √(l² + w²)
Parallelogram	base × height	height ⊥ base, not the slant side
Trapezium	½(a + b) × h	a, b the parallel sides
Regular polygon, n sides s	¼ n s² cot(180°/n)	coefficients tabulated in §6
Circle	πr² = πd²/4	C = πd
Annulus (ring)	π(R² − r²)	= π(R−r)(R+r)
Sector	½ r² θ	θ in radians
Segment	½ r² (θ − sin θ)	tabulated in §7
Ellipse	π a b	perimeter: §4
Parabolic segment	⅔ × chord × height	chord ⊥ axis
Cycloid (one arch)	3πr²	arch length = 8r
Fillet (square corner, radius r)	r²(1 − π/4) ≈ 0.2146 r²	corner area outside the arc

Irregular flat areas yield to the trapezoidal rule — divide into strips of equal width w and sum A ≈ w(½y₀ + y₁ + ··· + yₙ₋₁ + ½yₙ) — or, with an odd trick more accuracy, Simpson’s rule. Both are the Geometry page’s handshake with the Integrals section of Algebra and Equations.

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§6Regular polygons

For a regular polygon everything scales from the side: circumscribed radius, inscribed radius and area follow the coefficients below, computed for unit side.

Regular polygons, side s = 1 (computed)
n — name	Interior ∠	Circumradius R	Inradius r	Area
3 — triangle	60.0°	0.57735	0.28868	0.43301
4 — square	90.0°	0.70711	0.50000	1.00000
5 — pentagon	108.0°	0.85065	0.68819	1.72048
6 — hexagon	120.0°	1.00000	0.86603	2.59808
7 — heptagon	128.6°	1.15238	1.03826	3.63391
8 — octagon	135.0°	1.30656	1.20711	4.82843
9 — nonagon	140.0°	1.46190	1.37374	6.18182
10 — decagon	144.0°	1.61803	1.53884	7.69421
11 — hendecagon	147.3°	1.77473	1.70284	9.36564
12 — dodecagon	150.0°	1.93185	1.86603	11.19615
For side s: multiply R and r by s, area by s². R is centre-to-corner (across corners = 2R), r is centre-to-flat (across flats = 2r).

Example 5 — hexagon across corners, square across diagonal

Hex bar of 24 mm across flats: across corners = AF × 2/√3 = 27.71 mm — the minimum round it can be milled from. A 50 mm square’s diagonal is 50√2 = 70.71 mm.

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§7Circular segments and tank contents

The segment — the sliver between chord and arc — prices every partial fill, every flat on a bar, every sight-glass reading. Its table below is computed for unit radius; scale areas by r².

Circular segments for radius r = 1 (computed)
θ	Arc s	Chord c	Height h	Area
10°	0.17453	0.17431	0.00381	0.00044
20°	0.34907	0.34730	0.01519	0.00352
30°	0.52360	0.51764	0.03407	0.01180
40°	0.69813	0.68404	0.06031	0.02767
50°	0.87266	0.84524	0.09369	0.05331
60°	1.04720	1.00000	0.13397	0.09059
70°	1.22173	1.14715	0.18085	0.14102
80°	1.39626	1.28558	0.23396	0.20573
90°	1.57080	1.41421	0.29289	0.28540
100°	1.74533	1.53209	0.35721	0.38026
110°	1.91986	1.63830	0.42642	0.49008
120°	2.09440	1.73205	0.50000	0.61418
130°	2.26893	1.81262	0.57738	0.75144
140°	2.44346	1.87939	0.65798	0.90034
150°	2.61799	1.93185	0.74118	1.05900
160°	2.79253	1.96962	0.82635	1.22525
170°	2.96706	1.99239	0.91284	1.39671
180°	3.14159	2.00000	1.00000	1.57080
Scale s, c, h by r and area by r². Given chord and height only: r = (c²/4 + h²)/2h.

Fig. 2. Part-filled horizontal tank: depth h fixes the half-angle through cos ½θ = (r − h)/r; the wetted cross-section is the segment ½r²(θ − sin θ).

Example 6 — litres in a part-filled tank

A horizontal tank Ø1.2 m × 3.0 m long holds fluid 400 mm deep. r = 0.6, h = 0.4: cos ½θ = 0.2/0.6, so θ = 141.06° = 2.4619 rad. Segment area = ½ × 0.36 × (2.4619 − sin 141.06°) = 0.330 0 m²; volume = 0.3300 × 3.0 = 0.990 m³ = 990 litres.

The same segment arithmetic gives the depth of a milled flat (worked on the Solution of Triangles page, §9) and the area lost to a keyway in a shaft section.

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§8Volumes of solids

One table for the standard solids, then two rules — prismoidal and Pappus — that dispatch the shapes the table forgot.

Volumes and surfaces of solids
Solid	Volume	Surface (curved / total)
Prism / cylinder	A_base × h	cyl: 2πrh (curved)
Pyramid / cone	⅓ A_base × h	cone: πr × slant
Frustum of cone	⅓πh(R² + Rr + r²)	π(R + r) × slant
Sphere	4⁄3 πr³	4πr²
Spherical cap, height h	⅓πh²(3r − h)	zone: 2πrh
Torus (mean R, section r)	2π²Rr²	4π²Rr

Example 7 — a 60 mm ball and its cap

Sphere r = 60: V = 4⁄3π × 216 000 = 904 779 mm³, A = 4π × 3600 = 45 239 mm². Grinding a cap 15 mm high off it removes ⅓π × 225 × 165 = 38 877 mm³ and exposes a zone of 2π × 60 × 15 = 5 655 mm².

Prismoidal formula V = h6(A₁ + 4Aₘ + A₂) A₁, A₂ the end sections, Aₘ the true mid-height section

Example 8 — a square hopper transition

A hopper tapers from a 400 mm square to a 200 mm square over 300 mm. Aₘ (side 300) = 90 000: V = 50 × (160 000 + 360 000 + 40 000) = 28.0 litres — and the exact frustum formula returns the identical figure, because the prismoidal rule is exact for any solid whose section varies quadratically.

Pappus (Guldinus) rules revolve a plane area: V = 2πȳ · A revolve a curve: S = 2πȳ · L (ȳ = centroid radius from the axis)

Example 9 — an O-ring groove’s ring of metal

A full torus of section radius 6 mm on mean radius 40 mm: V = 2π × 40 × π × 36 = 28 425 mm³, surface 4π² × 40 × 6 = 9 475 mm². Pappus turns every ring, rim and flange into one multiplication.

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§9Circles in circles; rollers on a shaft

A ring of equal circles — rollers round a shaft, wires in a cable, pins in a bore — obeys one relation between count, circle size and ring size.

Ring of n circles, diameter d, centres on circle E E = dsin(180°/n) around a shaft: D_shaft = E − d · inside a bore: D_bore = E + d

A ring of n touching circles of diameter d (computed)
n	Centre circle E/d	Enclosing bore D/d	Central shaft D/d
3	1.15470	2.15470	0.15470
4	1.41421	2.41421	0.41421
5	1.70130	2.70130	0.70130
6	2.00000	3.00000	1.00000
7	2.30476	3.30476	1.30476
8	2.61313	3.61313	1.61313
9	2.92380	3.92380	1.92380
10	3.23607	4.23607	2.23607
11	3.54947	4.54947	2.54947
12	3.86370	4.86370	2.86370
E = d / sin(180°/n). Bore = E + d encloses the ring; shaft = E − d fills its centre.

Example 10 — eight rollers round a shaft

Eight 12 mm rollers touching each other and a central shaft: E = 12/sin 22.5° = 31.358 mm, so the shaft is Ø19.358 mm and the measurement over the rollers is E + d = 43.358 mm — a needle-bearing layout settled in two lines.

Packing circles into a circle without the ring constraint is a genuinely irregular problem — optimal counts jump unpredictably as the ratio grows, and layout work should rely on published packing data or a drawing trial rather than a formula. The single-ring case above, and rings nested with a centre circle (add one: the centre takes D_bore ≥ 3d at n = 6), cover most machine-element situations.

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§10Quick reference

The working core of the page on one card rack.

Line

y = mx + b

⊥ slopes: m₁m₂ = −1

Circle parts

s = rθ A_seg = ½r²(θ − sinθ)

c = 2r sin ½θ

Ellipse

A = πab

C ≈ π(a+b)(1 + 3h/(10+√(4−3h)))

Polygon

A = ¼ns² cot(180°/n)

hex AC = AF × 2/√3

Solids

sphere V = 4⁄3πr³

cap V = ⅓πh²(3r−h)

Two big rules

V = (h/6)(A₁+4Aₘ+A₂)

Pappus: V = 2πȳA

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§1Coordinates and the straight line

§2Polar coordinates

§3The circle

§4Ellipse, parabola and hyperbola

§5Areas of plane figures

§6Regular polygons

§7Circular segments and tank contents

§8Volumes of solids

§9Circles in circles; rollers on a shaft

§10Quick reference

Line

Circle parts

Ellipse

Polygon

Solids

Two big rules

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