Geometry

Part of Mathematics

Practical geometry for building, surveying, and design using only simple tools.

Why This Matters

Geometry is the mathematics of shape, space, and measurement. Every building, every field boundary, every road, every water channel, every piece of furniture requires geometric understanding. Without geometry, you cannot lay out a square foundation, calculate the area of a field for planting estimates, design a roof that sheds water properly, or survey a straight line across uneven terrain.

The remarkable thing about geometry is how much you can accomplish with almost nothing. A length of cord, a few stakes, and a straightedge let you construct perfect right angles, bisect any angle, draw parallel lines, and divide segments into equal parts. The ancient Greeks developed virtually all of practical geometry using only an unmarked straightedge and a compass (a cord on a stake). These same constructions work identically today.

In a rebuilding community, the person who understands geometry becomes indispensable. They lay out buildings, survey boundaries, design irrigation channels, calculate material needs, and resolve disputes about land area. This knowledge requires no technology beyond the ability to stretch a cord taut and scratch a line in the ground.

Essential Constructions with Cord and Stakes

All of these require only a length of cord (or rope) and stakes (or a scratching tool). The cord serves as both straightedge and compass.

Constructing a Right Angle

Method 1: The 3-4-5 Triangle

1. Drive a stake at the corner point
2. Measure 3 units along the baseline and stake
3. From the corner, measure 4 units in roughly the perpendicular direction
4. Adjust until the distance between the 3-mark and 4-mark is exactly 5 units
5. The angle at the corner is exactly 90 degrees

Method 2: The Semicircle Method

1. Draw a circle (swing cord from a center stake)
2. Mark any point A on the circle
3. Draw a diameter through the center to point B on the opposite side
4. Mark any point C on the circle (not at A or B)
5. The angle ACB is always exactly 90 degrees (Thales’ theorem)

Bisecting a Line Segment

To find the exact midpoint of any line:

1. Set cord length to more than half the line
2. From each endpoint, swing arcs on both sides of the line
3. The arcs intersect at two points
4. A line through those two intersection points crosses the original line at its exact midpoint — and is perfectly perpendicular to it

Constructing Parallel Lines

1. Draw your first line
2. Choose two points on it, well apart
3. At each point, construct a perpendicular (using the semicircle or 3-4-5 method)
4. Measure the same distance along each perpendicular
5. Connect those two points — this line is parallel to the first

Dividing a Segment into Equal Parts

To divide a line into (say) 7 equal parts without measuring:

1. From one end of the line, draw a second line at any convenient angle
2. Along this second line, mark 7 equal spaces (any convenient unit)
3. Connect the last mark to the other end of the original line
4. Draw lines parallel to this connection through each of the other 6 marks
5. Where these parallel lines cross the original line, they divide it into 7 equal parts

This is the geometric version of the carpenter's diagonal line trick — both use the same principle of similar triangles.

Area Calculations

Standard Shapes

Shape	Formula	Example
Rectangle	length × width	30 ft × 20 ft = 600 sq ft
Triangle	base × height ÷ 2	24 ft × 10 ft ÷ 2 = 120 sq ft
Circle	π × radius²	π × 7² = 153.9 sq ft
Trapezoid	(top + bottom) ÷ 2 × height	(12 + 20) ÷ 2 × 8 = 128 sq ft
Parallelogram	base × perpendicular height	15 ft × 9 ft = 135 sq ft

Remember Pi

π ≈ 3.14159. For field estimates, use 3.14 or even just 3 plus 1/7 (22/7 = 3.1429, accurate to 0.04%).

Composite Shapes

Most real areas are not perfect rectangles. Break them into standard shapes:

1. Sketch the area on paper
2. Divide it into rectangles, triangles, and trapezoids using straight lines
3. Calculate each piece separately
4. Add the areas together

For areas with curved boundaries, approximate the curve with straight-line segments. The more segments you use, the more accurate the result.

The Cross-Multiplication Method for Irregular Polygons

For any polygon where you know the coordinates (or can measure them from a baseline):

1. List all corners in order, going around the perimeter
2. For each pair of adjacent corners, multiply x₁ × y₂ and x₂ × y₁
3. Sum all the (x₁ × y₂) products and all the (x₂ × y₁) products
4. Subtract the second sum from the first
5. Divide by 2 — this is the area

This is called the Shoelace Formula and works for any polygon, no matter how irregular.

Volume Calculations

Standard Volumes

Shape	Formula	Common Use
Rectangular box	l × w × h	Buildings, tanks, raised beds
Cylinder	π × r² × h	Water tanks, silos, wells
Cone	π × r² × h ÷ 3	Grain piles, sand heaps
Sphere	4/3 × π × r³	Rarely needed in practice
Trapezoidal prism	((top + bottom) ÷ 2 × depth) × length	Ditches, trenches, embankments

Trench Volume

A drainage trench is 50 feet long, 3 feet deep, 4 feet wide at the top, and 2 feet wide at the bottom.

• Cross-section area: (4 + 2) ÷ 2 × 3 = 9 sq ft
• Volume: 9 × 50 = 450 cubic feet
• In cubic yards (÷27): 450 ÷ 27 = 16.7 cubic yards of soil to remove

Liquid Volume Conversions

Unit	Cubic Inches	Cubic Feet	Liters
1 US gallon	231	0.134	3.785
1 cubic foot	1,728	1.000	28.32
1 liter	61.02	0.0353	1.000
1 barrel (55 gal)	12,705	7.35	208.2

Quick Estimate

A cubic foot of water is about 7.5 gallons and weighs about 62.4 pounds. A 4-foot diameter, 3-foot deep circular tank holds: π × 2² × 3 = 37.7 cubic feet = 282 gallons = 1,068 liters.

Angles and Circles

Measuring Angles Without a Protractor

The Clock Method: A clock face is divided into 30-degree increments (360 ÷ 12). Use a round object and mark it like a clock to estimate angles.

The Folding Method:

• Fold a square paper corner in half = 45 degrees
• Fold that in half again = 22.5 degrees
• A full square corner = 90 degrees
• Two square corners = 180 degrees (straight line)

The Chord Method: For a circle of radius r, the chord length for various angles:

Angle	Chord Length (× radius)
30°	0.518
45°	0.765
60°	1.000 (chord = radius)
90°	1.414
120°	1.732

The fact that a 60-degree angle produces a chord equal to the radius is extremely useful — you can step a radius around a circle exactly 6 times to create a regular hexagon.

Circle Geometry

Key relationships:

• Circumference = 2 × π × radius = π × diameter
• Area = π × radius²
• Arc length = (angle/360) × circumference
• Sector area = (angle/360) × full area

Water Wheel Calculation

A water wheel has a diameter of 8 feet. How far does the rim travel in one revolution?

• Circumference = π × 8 = 25.13 feet
• If the wheel turns at 5 RPM, the rim speed is 25.13 × 5 = 125.7 feet per minute

Surveying Applications

Running a Straight Line Over a Hill

You cannot see from point A to point B because a hill is in the way:

1. Place a stake C on the hilltop where you can see both A and B
2. Stand at A and sight toward C — have an assistant place stake D between A and C, on the line
3. Stand at B and sight toward C — place stake E between B and C, on the line
4. Now stand at D and sight toward E. If A, D, E, and B are all on the same line, you are done
5. If not, adjust C and repeat. Two or three iterations will converge on a straight line

Determining True North

The geometry of shadows gives you true north (not magnetic north):

1. Plant a vertical pole on level ground
2. In the morning, mark the tip of the shadow and draw a circle with that radius
3. In the afternoon, mark where the shadow tip touches the same circle
4. A line connecting the two marks runs exactly east-west
5. A perpendicular to this line is true north-south

Measuring Across Obstacles

To measure the width of a river or ravine you cannot cross:

1. Stand at point A on your side, directly across from a visible landmark B on the far side
2. Walk perpendicular to AB along the bank for a measured distance to point C
3. From C, measure the angle ACB (using the chord method or a simple angle gauge)
4. Calculate: AB = AC × tan(angle ACB)

For a simpler approach without trigonometry:

1. Walk from A to C (any distance, perpendicular to the river)
2. From C, walk perpendicular to AC until you sight B at exactly 45 degrees from your path
3. You are now at point D. The distance CD equals the river width AB

The 45-Degree Trick

At 45 degrees, the opposite and adjacent sides of a right triangle are equal. This eliminates all calculation — just measure CD and you have the river width. The only skill needed is recognizing a 45-degree angle, which is half of a square corner.

Practical Geometry Toolkit

The minimum tools for field geometry:

1. Cord/rope — at least 50 feet, non-stretch (sisal or hemp, not nylon). Mark at regular intervals with knots or paint
2. Stakes — sharpened hardwood, at least 12
3. Plumb bob — any weight on a string, for establishing true vertical
4. Straightedge — a planed board or taut string
5. Square — a rigid right-angle made from three sticks in a 3-4-5 ratio, joined firmly at the corners
6. Level — a trough of water establishes a perfectly horizontal reference. A tube filled with water (water level) can transfer level marks around corners and across distances

With these six items, you can perform every geometric construction described in this article — and every construction used to build the cathedrals, aqueducts, and cities of the pre-industrial world.