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.
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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