Polygons and Circles

Equilateral Triangle and Circle

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

Equilateral Triangle inscribed in Circle

Equilateral Triangle inscribed in Circle

Equilateral Triangle inscribed in Circle

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

Circle inscribed in Equilateral Triangle

Circle inscribed in Equilateral Triangle

Circle inscribed in Equilateral Triangle

Square and Circle

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

Square inscribed in Circle

Square inscribed in Circle

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

Formulas

Calculators

Circle inscribed in Square

Circle inscribed in Square

Circle inscribed in Square

Polygons and Circles

Equilateral Triangle and Circle

Symbols and Meanings

Formulas

Calculators

Formulas

Calculators

Square and Circle

Symbols and Meanings

Formulas

Calculators

Formulas

Calculators