A new geometric shape proposal: basic formula of smooth polyarcs

Emin Yaşar

Article Details

A new geometric shape proposal: basic formula of smooth polyarcs

Authors:

Emin Yaşar

Publisher:

Gumushane University

Main Topics:

Temel Matematik (Diğer)

Journal:

Gümüşhane Üniversitesi Fen Bilimleri Dergisi Vol.15 Issue 3, pp.711-727

Document Type:

Article

Publication Date:

9/2025

Document Language:

Turkish

Document ID (Org No.):

2001358135

pISSN:

2146-538X

Keywords:

Abstract:

 In this study, a new geometric structure is proposed using regular polygons and circles. In the proposed structure, called a regular polyarc, circles with diameters equal to the side length of a regular polygon are used. The centers of these circles are placed at the vertices of the corresponding polygon, and the regions formed by subtracting the circles from the polygon are analyzed. These regions are classified based on the number of arcs they contain and are named as regular tri-arc, regular quad-arc, regular penta-arc, and so on. Using the perimeter and area formulas of the circle along with the area formulas of regular polygons, perimeter and area calculations for these new shapes are carried out. Based on the identified patterns, general formulas are developed to calculate the perimeter and area of any regular polyarc. Additionally, spheres with diameters equal to the edge lengths of Platonic solids regular tetrahedron, cube, and dodecahedron are centered at the vertices of these solids, and new three-dimensional structures are formed by subtracting the spheres from the solids. These resulting bodies are named regular polyarc solids and are classified according to their face structure as 44, 68, and 1220 regular polyarc solids. General formulas for calculating the surface area and volume of these solids are derived using the surface area and volume formulas of spheres, the volume formulas of Platonic solids, and the surface area formulas of regular polyarcs. As a result, new geometric shapes and solids are proposed based on existing geometric structures.

 In this study, a new geometric structure is proposed using regular polygons and circles. In the proposed structure, called a regular polyarc, circles with diameters equal to the side length of a regular polygon are used. The centers of these circles are placed at the vertices of the corresponding polygon, and the regions formed by subtracting the circles from the polygon are analyzed. These regions are classified based on the number of arcs they contain and are named as regular tri-arc, regular quad-arc, regular penta-arc, and so on. Using the perimeter and area formulas of the circle along with the area formulas of regular polygons, perimeter and area calculations for these new shapes are carried out. Based on the identified patterns, general formulas are developed to calculate the perimeter and area of any regular polyarc. Additionally, spheres with diameters equal to the edge lengths of Platonic solids regular tetrahedron, cube, and dodecahedron are centered at the vertices of these solids, and new three-dimensional structures are formed by subtracting the spheres from the solids. These resulting bodies are named regular polyarc solids and are classified according to their face structure as 44, 68, and 1220 regular polyarc solids. General formulas for calculating the surface area and volume of these solids are derived using the surface area and volume formulas of spheres, the volume formulas of Platonic solids, and the surface area formulas of regular polyarcs. As a result, new geometric shapes and solids are proposed based on existing geometric structures.

 In this study, a new geometric structure is proposed using regular polygons and circles. In the proposed structure, called a regular polyarc, circles with diameters equal to the side length of a regular polygon are used. The centers of these circles are placed at the vertices of the corresponding polygon, and the regions formed by subtracting the circles from the polygon are analyzed. These regions are classified based on the number of arcs they contain and are named as regular tri-arc, regular quad-arc, regular penta-arc, and so on. Using the perimeter and area formulas of the circle along with the area formulas of regular polygons, perimeter and area calculations for these new shapes are carried out. Based on the identified patterns, general formulas are developed to calculate the perimeter and area of any regular polyarc. Additionally, spheres with diameters equal to the edge lengths of Platonic solids regular tetrahedron, cube, and dodecahedron are centered at the vertices of these solids, and new three-dimensional structures are formed by subtracting the spheres from the solids. These resulting bodies are named regular polyarc solids and are classified according to their face structure as 44, 68, and 1220 regular polyarc solids. General formulas for calculating the surface area and volume of these solids are derived using the surface area and volume formulas of spheres, the volume formulas of Platonic solids, and the surface area formulas of regular polyarcs. As a result, new geometric shapes and solids are proposed based on existing geometric structures.

 Bu çalışmada, geometrik şekillerden düzgün çokgenler ve daireler kullanılarak yeni bir geometrik yapı önerilmiştir. Düzgün çokyay olarak önerilen bu yapıda çap uzunluğu düzgün çokgenin bir kenar uzunluğuna eşit daireler kullanılmıştır. Bu dairelerin merkezleri, ilgili çokgenin köşelerine yerleştirilmiş ve çokgenin içerisinden dairelerin çıkarılmasıyla oluşan bölgeler incelenmiştir. Elde edilen bu bölgeler yay sayısına bağlı olarak düzgün üçyay, düzgün dörtyay, düzgün beşyay vb. biçiminde sınıflandırılmıştır. Dairenin çevre ve alan formülleri ile düzgün çokgenlerin alan formülleri kullanılarak bu yeni yapıların çevre ve alan hesaplamaları yapılmış; elde edilen örüntüler doğrultusunda genel formüller türetilmiştir. Ayrıca, çapları platonik cisimlerin düzgün dört yüzlü, altı yüzlü ve on iki yüzlü ayrıt uzunluğuna eşit kürelerin merkezleri, bu cisimlerin köşelerine yerleştirilmiş ve bu kürelerin cisimlerin içerisinden çıkarımasıyla oluşan üç boyutlu yapılar incelenmiştir. Ulaşılan bu yapılar düzgün çokyay cisim olarak isimlendirilmiş ve cismin yüzeylerine göre 44, 68 ve 1220 düzgün çokyay cismi şeklinde sınıflandırılmıştır. Çalışma kapsamında, bu cisimlerin yüzey alanı ve hacim hesaplamalarına yönelik genel formüller geliştirilmiş; hesaplamalarda kürelerin yüzey alanı ve hacim formülleri, platonik cisimlerin hacim formülleri ile beraber düzgün çokyayların yüzey alan formülleri de kullanılmıştır. Sonuç olarak var olan geometrik yapılardan yola çıkarak yeni bir geometrik şekil ve cisim önerilmiştir.

Article Details

A new geometric shape proposal: basic formula of smooth polyarcs

Authors:

Emin Yaşar

Publisher:

Gumushane University

Main Topics:

Temel Matematik (Diğer)

Journal:

Gümüşhane Üniversitesi Fen Bilimleri Dergisi Vol.15 Issue 3, pp.711-727

Document Type:

Article

Publication Date:

9/2025

Document Language:

Turkish

Document ID (Org No.):

2001358135

pISSN:

2146-538X

Area

Analytic Geometry

Perimeter

PolyarcAlan

Analitik Geometri

Çevre

Çokyay

Issues

Article Details

Authors: Emin Yaşar

Publisher: Gumushane University

Main Topics: Temel Matematik (Diğer)

Journal: Gümüşhane Üniversitesi Fen Bilimleri Dergisi Vol.15 Issue 3, pp.711-727

Document Type: Article

Publication Date: 9/2025

Document Language: Turkish

Document ID (Org No.): 2001358135

pISSN: 2146-538X

Issues

Authors:

Emin Yaşar

Publisher:

Gumushane University

Main Topics:

Temel Matematik (Diğer)

Journal:

Gümüşhane Üniversitesi Fen Bilimleri Dergisi Vol.15 Issue 3, pp.711-727

Document Type:

Article

Publication Date:

9/2025

Document Language:

Turkish

Document ID (Org No.):

2001358135

pISSN:

2146-538X