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
All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
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
All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.