Sunday, May 3, 2020

Construction Master 5: Area of Regular Polygons

Construction Master 5: Area of Regular Polygons



Introduction:  Area of a Regular Polygon



The area of a regular polygon is:

Area = 1/2 * P * A

where:
P = perimeter = n * S 
n = number of sides
S = side length
A = apothem length

Divided the regular polygon from drawing lines from the center to each of the polygon's vertices.  Triangles about the center are formed with a central angle, 360° ÷ n. 

The apothem bisects each of the triangles, with the angle nearest to the center of 180° ÷ n.   A right angle is formed between the side and apothem.   The small triangle formed has the base of the s ÷ 2 (since the apothem bisects the side) and A.  The triangle has angles half-central angle (HCA), 90°, and 180° - 90° - HCA.  I will refer to the third angle as the Pitch angle (see the diagram above). 

The trigonometric relationships are:

tan( Pitch ) = A / ( s / 2 )

A = s / 2 * tan(Pitch)

s = 2 * A / tan(Pitch)

It is possible to determine the area of regular polygon knowing the number of sides and either the side length or the apothem.

The Calculated Industries Construction Master 5 does not have trigonometric keys.  However, the Construction Master 5 has four keys that deal with right triangles:

[ Pitch ]:  angle in degrees.  Enter the angle as a unit-less measure.  For example, to enter 60°, press 60 [ Pitch ].  The angle considers the rise of the opposite side, and run the adjacent side.

[ Rise ]:  rise - opposite side.  The Rise is considered the apothem of length A.  You can enter either unit-less or measured amounts (feet/inches/yards/m). 

[ Run ]:  run - adjacent side.  You can enter either unit-less or measured amounts (feet/inches/yards/m).  This represents the side or half-side.

[ Diag ]:  hypotenuse.  The algorithms will not use this key in this blog. 

Known:  Number of Sides and Side Length

The Math:

Area

= 1/2 * P * A
= 1/2 * n * s * (s/2 * tan(Pitch))
= 1/4 * n *  (s * tan(Pitch))

Keystrokes:

180 [ ÷ ] n [ Conv ] (+/-) [ + ] 90 [ Pitch ]
s [ Run ]
[ Rise ] [ × ] [ Rcl ] [ Run ] [ × ] n [ ÷ ] 4 [ = ]

Example:
s = 40 ft, n = 6

180 [ ÷ ] 6 [ Conv ] (+/-) [ + ] 90 [ Pitch ]    (Display:  PTCH 60.00°)
40 [ Feet ] [ Run ]
[ Rise ] [ × ] [ Rcl ] [ Run ] [ × ] 6 [ ÷ ] 4 [ = ]   (Display:  4156.922 SQ FEET)

Area ≈ 4156.922 ft^2

Known:  Number of Sides and Apothem

The Math

Area

= 1/2 * P * A
= 1/2 * (n * s) * A
= 1/2 * n * (2 * A) / tan(Pitch) * A
= n * A  * (A / tan(Pitch))

Keystrokes:

180 [ ÷ ] n [ Conv ] (+/-) [ + ] 90 [ Pitch ]
s [ Rise ]
[ Run ] [ × ] [ Rcl ] [ Rise ] [ × ] n [ = ]

Example:
A = 40 ft, n = 6

180 [ ÷ ] 6 [ Conv ] (+/-) [ + ] 90 [ Pitch ]   (Display:  PTCH 60.00°)
40 [ Feet ] [ Rise ]
[ Run ] [ × ] [ Rcl ] [ Rise ] [ × ] 6 [ = ]   (Display:  5542.563 SQ FEET)

Area ≈ 5542.563 ft^2

And that is how to find the area of a regular polygon using the Construction Master 5.

Eddie

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