Geometrical Constructions: Lines, Circles & Regular Polygons (General Method)

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1. Line and Angle Constructions

A. Dividing a Straight Line into $n$ Equal Parts

To divide a given line segment $AB$ of length $L$ into $n$ equal parts (e.g., $n = 7$) without arithmetic division:

1. From end $A$, draw an auxiliary ray $AC$ at any convenient acute angle (typically $20^\circ\text{ to }30^\circ$) to $AB$.
2. Using a divider or scale, mark $n$ equal divisions of any convenient length on ray $AC$: labeled $1, 2, 3, \dots, n$.
3. Join the last point $n$ to endpoint $B$ with a straight line.
4. Using a mini-drafter or parallel set-squares, draw lines through points $(n-1), (n-2), \dots, 1$ parallel to line $nB$.
5. These parallel lines intersect $AB$ at points $1', 2', \dots, (n-1)'$, dividing $AB$ into $n$ exactly equal segments.

  C
  + n
 /|
/ | 
| | (parallel lines)
| v
A-----------------------------B
  1' 2' 3' 4' 5' 6'

B. Bisecting an Angle

1. With vertex $O$ as center and any radius, draw an arc intersecting arms $OA$ and $OB$ at points $P$ and $Q$.
2. With centers $P$ and $Q$ and equal radii greater than half the distance $PQ$, draw intersecting arcs at point $R$.
3. The ray $OR$ is the exact angular bisector.

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2. Tangency Constructions

• Tangent from an External Point $P$ to a Circle of Radius $R$ and Center $O$:
  1. Join line $OP$ and locate its midpoint $M$ by perpendicular bisection.
  2. With $M$ as center and radius $MO = MP$, draw a semicircle intersecting the given circle at point $T$.
  3. Join $PT$. Line $PT$ is the required tangent ($\angle PTO = 90^\circ$ by Thales' theorem).
• External and Internal Common Tangents:
  • External tangent: Draw auxiliary circle with radius $(R_1 - R_2)$ concentric with the larger circle.
  • Internal (transverse) tangent: Draw auxiliary circle with radius $(R_1 + R_2)$ concentric with the larger circle.

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3. Regular Polygons: Geometry Fundamentals

A regular polygon has all sides equal and all interior angles equal.

Polygon	Number of Sides ($n$)	Exterior Angle ($\phi = \frac{360^\circ}{n}$)	Interior Angle ($\theta = \frac{(n-2) \times 180^\circ}{n}$)
Equilateral Triangle	3	$120^\circ$	$60^\circ$
Square	4	$90^\circ$	$90^\circ$
Regular Pentagon	5	$72^\circ$	$108^\circ$
Regular Hexagon	6	$60^\circ$	$120^\circ$
Regular Heptagon	7	$51.43^\circ$ ($51^\circ 26'$)	$128.57^\circ$
Regular Octagon	8	$45^\circ$	$135^\circ$

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4. The General Method for Constructing Any Regular Polygon

The General Method (Universal Centers / 4–6 Method) is a standard engineering procedure to draw any regular polygon (pentagon, hexagon, heptagon, octagon, etc.) on a given side length $AB$.

                   8 (Octagon center)
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                   7 (Heptagon center)
                   |
                   6 (Hexagon center: R = AB)
                   |
                   5 (Pentagon center: mid of 4-6)
                   |
                   4 (Square center: semi-circle top)
                   |
                   | Perpendicular Bisector
                   v
         A---------M---------B

Step-by-Step Construction Procedure:

1. Base Line: Draw horizontal line $AB$ equal to the specified side length of the polygon.
2. Perpendicular Bisector: Erect the perpendicular bisector of line $AB$, meeting $AB$ at its midpoint $M$ and extending upward.
3. Locating Center 4 (Square):
   • With midpoint $M$ as center and radius $MA = MB$, draw a semicircle above $AB$.
   • The semicircle intersects the perpendicular bisector at Point 4.
   • (A circle drawn with center 4 and radius $4A$ circumscribes a square of side $AB$).
4. Locating Center 6 (Hexagon):
   • With endpoint $A$ (or $B$) as center and radius $AB$, draw an arc cutting the perpendicular bisector at Point 6.
   • (A circle drawn with center 6 and radius $6A = AB$ circumscribes a regular hexagon).
5. Locating Center 5 (Pentagon):
   • Bisect the distance between points 4 and 6 on the bisector to find Point 5.
   • (A circle drawn with center 5 and radius $5A$ circumscribes a regular pentagon).
6. Locating Higher Order Centers (7, 8, 9, ...):
   • The distance $4\text{–}5 = 5\text{–}6$.
   • Set divider to step distance $5\text{–}6$ and mark off points 7, 8, 9, 10 above point 6 along the perpendicular bisector.
7. Drawing the Polygon:
   • To draw an $n$-sided polygon, place compass needle at Point $n$ on the bisector.
   • Set compass radius equal to distance $n\text{–}A$ ($= n\text{–}B$).
   • Draw a circumscribing circle.
   • Take side length $AB$ on the compass and step it off along the circumference of the circle.
   • Connect the stepped points sequentially with dark, continuous lines ($2\text{H}/\text{H}$) to complete the polygon.

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5. Alternative General Method: Semicircle Method

1. Draw given side $AB$. Produce line $AB$ to point $P$ such that $AP = AB$.
2. With center $A$ and radius $AB$, draw a semicircle above line $PB$.
3. Divide the semicircle ($180^\circ$) into $n$ equal sectors using a protractor (each angle $= 180^\circ / n$). Label divisions $1, 2, 3, \dots, (n-1)$.
4. Join $A$ to point $2$. Line $A2$ is always the second side of the polygon.
5. Draw the perpendicular bisectors of $AB$ and $A2$ to intersect at center $O$.
6. With center $O$ and radius $OA$, draw the circumscribing circle and step off side length $AB$.

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6. Summary of Key Exam Points

1. Exterior Angle Formula: $\phi = \frac{360^\circ}{n}$.
2. Interior Angle Formula: $\theta = 180^\circ - \phi = \frac{(n-2) \times 180^\circ}{n}$.
3. Hexagon Geometry: In a regular hexagon, the distance from center to any vertex equals the side length ($R = a$).
4. General 4–6 Method: Point 4 is found by radius $MA$ from midpoint; Point 6 is found by radius $AB$ from vertex; Point 5 is exactly midway between 4 and 6.