Overview
Regular Polygons and Circles
Solved Examples
Resources

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka

# Polygons and Circles Welcome to our site.
I greet you this day,
The polygons discussed in this site are regular polygons only.
First: read the notes. Second: view the videos. Third: solve the questions/solved examples. Fourth: check your solutions with my thoroughly-explained solutions. Fifth: check your answers with the calculators as applicable.
I wrote the codes for the calculators using JavaScript, a client-side scripting language. Please use the latest Internet browsers. The calculators should work. Only integers and decimals are allowed. Fractions are not allowed.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me. If you are my student, please do not contact me here. Contact me via the school's system.
Thank you for visiting!!!

Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

• ## Symbols and Meanings

• $h$ = perpendicular height of the equilateral triangle
• $r$ = radius of the circle
• $b$ = base of the equilateral triangle
• $a$ = apothem of the equilateral triangle
• $A_T$ = area of the equilateral triangle
• $A_C$ = area of the circle
• $P_T$ = perimeter of the square
• $P_C$ = circumference or perimeter of the circle
• $A_r$ = the sum of the remaining areas
• $A_{ep}$ = area of each part of the remaining areas

Equilateral Triangle inscribed in a Circle OR a Circle circumscribed about an Equilateral Triangle

### Formulas

 $h = \dfrac{3r}{2} \\[5ex] b = r\sqrt{3} \\[3ex] a = \dfrac{r}{2} \\[5ex] r = 2a \\[3ex] A_T = \dfrac{bh}{2} \\[5ex] A_C = \pi r^2 \\[3ex] \dfrac{A_T}{A_C} = \dfrac{3\sqrt{3}}{4\pi} \\[5ex] \dfrac{A_C}{A_T} = \dfrac{4\pi \sqrt{3}}{9} \\[5ex] A_r = A_C - A_T \\[3ex] A_{ep} = \dfrac{A_r}{3}$ $h = \dfrac{b\sqrt{3}}{2} \\[5ex] b = \dfrac{2h\sqrt{3}}{3} \\[5ex] a = \dfrac{b\sqrt{3}}{6} \\[5ex] r = \dfrac{2h}{3} \\[5ex] A_T = \dfrac{3r^2\sqrt{3}}{4} \\[5ex]$ $h = 3a \\[3ex] b = 2a\sqrt{3} \\[3ex] a = \dfrac{h}{3} \\[5ex] r = \dfrac{b\sqrt{3}}{3} \\[5ex]$

### Calculators

• Equilateral Triangle inscribed in Circle

To find: other details

• Equilateral Triangle inscribed in Circle

Given: perpendicular height of the equilateral triangle

To find: other details

• Equilateral Triangle inscribed in Circle

Given: base of the triangle (or any side)

To find: other details

• Equilateral Triangle inscribed in Circle

Given: apothem of the triangle

To find: other details

Circle inscribed in an Equilateral Triangle OR an Equilateral Triangle circumscribed about a Circle

### Formulas

 $h = 3r \\[3ex] b = 2r\sqrt{3} \\[3ex] a = r \\[3ex] r = a \\[3ex] A_T = \dfrac{bh}{2} \\[5ex] A_C = \pi r^2 \\[3ex] \dfrac{A_T}{A_C} = \dfrac{3\sqrt{3}}{\pi} \\[5ex] \dfrac{A_C}{A_T} = \dfrac{\pi \sqrt{3}}{9} \\[5ex] A_r = A_T - A_C \\[3ex] A_{ep} = \dfrac{A_r}{3}$ $h = \dfrac{b\sqrt{3}}{2} \\[5ex] b = \dfrac{2h\sqrt{3}}{3} \\[5ex] a = \dfrac{b\sqrt{3}}{6} \\[5ex] r = \dfrac{h}{3} \\[5ex] A_T = 3r^2\sqrt{3} \\[3ex]$ $h = 3a \\[3ex] b = 2a\sqrt{3} \\[3ex] a = \dfrac{h}{3} \\[5ex] r = \dfrac{b\sqrt{3}}{6} \\[5ex]$

### Calculators

• Circle inscribed in Equilateral Triangle

To find: other details

• Circle inscribed in Equilateral Triangle

Given: perpendicular height of the equilateral triangle

To find: other details

• Circle inscribed in Equilateral Triangle

Given: base of the triangle (or any side)

To find: other details

• Circle inscribed in Equilateral Triangle

Given: apothem of the triangle

To find: other details

• ## Symbols and Meanings

• $d$ = diameter of the circle
• $d_S$ = diagonal of the square
• $l$ = length of a side of the square (side length of the square)
• $r$ = radius of the circle
• $a$ = apothem of the square
• $A_S$ = area of the square
• $A_C$ = area of the circle
• $P_S$ = perimeter of the square
• $P_C$ = circumference or perimeter of the circle
• $A_r$ = the sum of the remaining areas
• $A_{ep}$ = area of each part of the remaining areas

Square inscribed in a Circle OR a Circle circumscribed about a Square

The diameter of the circle is equal to the diagonal of the square

### Formulas

 $d = d_S \\[3ex] l = r\sqrt{2} \\[3ex] a = \dfrac{l}{2} \\[5ex] r = \dfrac{l\sqrt{2}}{2} \\[5ex] A_S = l^2 \\[3ex] A_C = \pi r^2 \\[3ex] \dfrac{A_S}{A_C} = \dfrac{2}{\pi} \\[5ex] \dfrac{A_C}{A_S} = \dfrac{\pi}{2} \\[5ex] A_r = A_C - A_S \\[3ex] A_{ep} = \dfrac{A_r}{4}$ $l = 2a \\[3ex] a = \dfrac{r\sqrt{2}}{2} \\[5ex] r = a\sqrt{2} \\[3ex] A_S = 2r^2 \\[3ex]$

### Calculators

• Square inscribed in Circle

To find: other details

• Square inscribed in Circle

Given: side length of the square

To find: other details

• Square inscribed in Circle

Given: apothem of the square

To find: other details

Circle inscribed in a Square OR a Square circumscribed about a Circle

### Calculators

• Circle inscribed in Square

To find: other details

• Circle inscribed in Square

Given: side length of the square

To find: other details

• Circle inscribed in Square

Given: apothem of the square

To find: other details