Solved Examples: Circles

$ A = \pi \\[3ex] r = ? \\[3ex] r = \dfrac{\sqrt{A\pi}}{\pi} \\[5ex] = \dfrac{\sqrt{\pi * \pi}}{\pi} \\[5ex] = \dfrac{\sqrt{\pi^2}}{\pi} \\[5ex] = \dfrac{\pi}{\pi} \\[5ex] = 1\:\:inch $

$ C = \pi \\[3ex] d = ? \\[3ex] d = \dfrac{C}{\pi} \\[5ex] d = \dfrac{\pi}{\pi} \\[5ex] d = 1\;inch $

$ d = 2r \\[3ex] C = 2\pi r \\[3ex] \dfrac{d}{C} = \dfrac{2r}{2\pi r} \\[5ex] \dfrac{d}{C} = \dfrac{1}{\pi} $

The equation of a circle that is in the standard $(x, y)$ coordinate plane with center $(h, k)$ is:

$ (x - h)^2 + (y - k)^2 = r^2 \\[3ex] Compare:\;\; (x - 3)^2 + (y - 5)^2 = 4 \\[3ex] -h = -3 \\[3ex] h = 3 \\[3ex] -k = -5 \\[3ex] k = 5 \\[3ex] Center:\;\; (h, k) = (3, 5) $

Let:
radius = r
diameter = d
circumference = C
area = A

Radius (cm)	Diameter (cm)	Circumference (cm)	Area (cm²)
17	$ d = 2r \\[3ex] d = 2(17) \\[3ex] d = 34 $	$ C = 2\pi r \\[3ex] C = 2 * \pi * 17 \\[3ex] C = 34\pi $	$ A = \pi r^2 \\[3ex] A = \pi * (17)^2 \\[3ex] A = 289\pi $
$ r = \dfrac{d}{2} \\[5ex] r = \dfrac{28}{2} \\[5ex] r = 14 $	28	$ C = \pi d \\[3ex] C = \pi * 28 \\[3ex] C = 28\pi $	$ A = \dfrac{\pi d^2}{4} \\[5ex] A = \dfrac{\pi (28)^2}{4} \\[5ex] A = 196\pi $
$ r = \sqrt{\dfrac{A}{\pi}} \\[5ex] r = \sqrt{\dfrac{169\pi}{\pi}} \\[5ex] r = \sqrt{169} \\[3ex] r = 13 $	$ d = \sqrt{\dfrac{4A}{\pi}} \\[5ex] d = \sqrt{\dfrac{4 * 169\pi}{\pi}} \\[5ex] d = \sqrt{4 * 169} \\[3ex] d = \sqrt{4} * \sqrt{169} \\[3ex] d = 2 * 13 \\[3ex] d = 26 $	$ C = 2\sqrt{A\pi} \\[3ex] C = 2\sqrt{169\pi * \pi} \\[3ex] C = 2\sqrt{169 * \pi^2} \\[3ex] C = 2 * \sqrt{169} * \sqrt{\pi^2} \\[3ex] C = 2 * 13 * \pi \\[3ex] C = 26\pi $	169π
$ r = \dfrac{C}{2\pi} \\[5ex] r = \dfrac{32\pi}{2\pi} \\[5ex] r = 16 $	$ d = \dfrac{C}{\pi} \\[5ex] d = \dfrac{32\pi}{\pi} \\[5ex] d = 32 $	32π	$ A = \dfrac{C^2}{4\pi} \\[5ex] A = \dfrac{(32\pi)^2}{4\pi} \\[5ex] A = \dfrac{32 \cdot 32 \cdot \pi \cdot \pi}{4 \cdot \pi} \\[5ex] A = 8 \codt 32 \cdot \pi \\[3ex] A = 256\pi $

Let:
circumference = C
area = A

$ C = 4\;m \\[3ex] A = \dfrac{C^2}{4\pi} \\[5ex] A = \dfrac{4^2}{4\pi} \\[5ex] A = \dfrac{4}{\pi}\;m^2 $

The equation of a circle that is in the standard $(x, y)$ coordinate plane with center $(h, k)$ is:

$ (x - h)^2 + (y - k)^2 = r^2 \\[3ex] Center:\;\; (h, k) = (8, 5) \\[3ex] h = 8 \\[3ex] k = 5 \\[3ex] r = 9 \\[3ex] \implies \\[3ex] (x - 8)^2 + (y - 5)^2 = 9^2 \\[3ex] (x - 8)^2 + (y - 5)^2 = 81 $

$ r = 18\;cm \\[3ex] \theta = 120^\circ \\[3ex] P_{sec} = \dfrac{r(\pi\theta + 360)}{180} \\[5ex] P_{sec} = \dfrac{18(120\pi + 360)}{180} \\[5ex] P_{sec} = \dfrac{120\pi + 360}{10} \\[5ex] P_{sec} = \dfrac{120(\pi + 3)}{10} \\[5ex] P_{sec} = 12(\pi + 3)\;cm $

$ d = 10 \\[3ex] r = \dfrac{d}{2} = \dfrac{10}{2} = 5 \\[5ex] center = (h, k) = (1, 4) \\[3ex] h = 1 \\[3ex] k = 4 \\[3ex] Standard\;\;Form:\;\; (x - h)^2 + (y - k)^2 = r^2 \\[3ex] \implies \\[3ex] (x - 1)^2 + (y - 4)^2 = 5^2 \\[3ex] [(x - 1)(x - 1)] + [(y - 4)(y - 4)] = 25 \\[3ex] (x^2 - x - x + 1) + (y^2 - 4y - 4y + 16) = 25 \\[3ex] x^2 - 2x + 1 + y^2 - 8y + 16 = 25 \\[3ex] x^2 - 2x + y^2 - 8y = 25 - 1 - 16 \\[3ex] x^2 - 2x + y^2 - 8y = 8 $

$ \underline{Standard\;\;Form\;\;of\;\;the\;\;Equation\;\;of\;\;a\;\;Circle} \\[3ex] (x - h)^2 + (y - k)^2 = r^2 \\[3ex] $ (a.) To adapt the formula for a circle to describe the interior of a circle: rewrite the formula: $ (x - h)^2 + (y - k)^2 \lt r^2 \\[3ex] $ (b.) To adapt the formula for a circle to describe the exterior of a circle: rewrite the formula: $ (x - h)^2 + (y - k)^2 \gt r^2 \\[3ex] $

$ \underline{General\;\;Form\;\;of\;\;the\;\;Equation\;\;of\;\;a\;\;Circle} \\[3ex] x^2 + y^2 + 2gx + 2fy + c = 0 \\[3ex] Point\;1:\;\;(3, 2) \implies x = 3;\;\; y = 2 \\[3ex] 3^2 + 2^2 + 2g(3) + 2f(2) + c = 0 \\[3ex] 9 + 4 + 6g + 4f + c = 0 \\[3ex] 13 + 6g + 4f + c = 0 \\[3ex] 6g + 4f + c = -13 ...eqn.(1) \\[3ex] Point\;2:\;\;(-1, -2) \implies x = -1;\;\; y = -2 \\[3ex] (-1)^2 + (-2)^2 + 2g(-1) + 2f(-2) + c = 0 \\[3ex] 1 + 4 - 2g - 4f + c = 0 \\[3ex] 5 - 2g - 4f + c = 0 \\[3ex] -2g - 4f + c = -5 ...eqn.(2) \\[3ex] Point\;3:\;\;(5, -4) \implies x = 5;\;\; y = -4 \\[3ex] 5^2 + (-4)^2 + 2g(5) + 2f(-4) + c = 0 \\[3ex] 25 + 16 + 10g - 8f + c = 0 \\[3ex] 41 + 10g - 8f + c = 0 \\[3ex] 10g - 8f + c = -41 ...eqn.(3) \\[3ex] eqn.(1) - eqn.(2) \implies \\[3ex] (6g + 4f + c) - (-2g - 4f + c) = -13 - (-5) \\[3ex] 6g + 4f + c + 2g + 4f - c = -13 + 5 \\[3ex] 8g + 8f = -8 \\[3ex] Divide\;\;each\;\;term\;\;by\;\;8 \\[3ex] g + f = -1 ...eqn.(4) \\[3ex] eqn.(1) - eqn.(3) \implies \\[3ex] (6g + 4f + c) - (10g - 8f + c) = -13 - (-41) \\[3ex] 6g + 4f + c - 10g + 8f - c = -13 + 41 \\[3ex] 6g + 4f + c - 10g + 8f - c = 28 \\[3ex] -4g + 12f = 28 \\[3ex] Divide\;\;each\;\;term\;\;by\;\;4 \\[3ex] -g + 3f = 7 ... eqn.(5) \\[3ex] eqn.(4) + eqn.(5) \implies \\[3ex] (g + f) + (-g + 3f) = -1 + 7 \\[3ex] g + f - g + 3f = 6 \\[3ex] 4f = 6 \\[3ex] f = \dfrac{6}{4} \\[5ex] f = \dfrac{3}{2} \\[5ex] eqn.(5) - 3 * eqn.(4) \implies \\[3ex] (-g + 3f) - 3(g + f) = 7 - 3(-1) \\[3ex] -g + 3f - 3g - 3f = 7 + 3 \\[3ex] -4g = 10 \\[3ex] g = \dfrac{10}{-4} \\[5ex] g = -\dfrac{5}{2} \\[5ex] h = -g \\[3ex] h = -1 * -\dfrac{5}{2} \\[5ex] h = \dfrac{5}{2} \\[5ex] k = -f \\[3ex] k = -\dfrac{3}{2} \\[5ex] Center = (h, k) = \left(\dfrac{5}{2}, -\dfrac{3}{2}\right) \\[5ex] (b) \\[3ex] From\;\;eqn.(2) \\[3ex] -2g - 4f + c = -5 ...eqn.(2) \\[3ex] c = -5 + 2g + 4f \\[3ex] c = -5 + 2\left(-\dfrac{5}{2}\right) + 4\left(\dfrac{3}{2}\right) \\[5ex] c = -5 + -5 + 2(3) \\[3ex] c = -5 - 5 + 6 \\[3ex] c = -4 \\[3ex] r^2 = g^2 + f^2 - c \\[3ex] r^2 = \left(-\dfrac{5}{2}\right)^2 + \left(\dfrac{3}{2}\right)^2 + 4 \\[5ex] r^2 = \dfrac{25}{4} + \dfrac{9}{4} + 4 \\[5ex] r^2 = \dfrac{25}{4} + \dfrac{9}{4} + \dfrac{16}{4} \\[5ex] r^2 = \dfrac{25 + 9 + 16}{4} \\[5ex] r^2 = \dfrac{50}{4} \\[5ex] r^2 = \dfrac{25}{2} \\[5ex] r = \sqrt{r^2} \\[3ex] r = \sqrt{\dfrac{25}{2}} \\[5ex] r = 3.535533906\;units \\[5ex] (c) \\[3ex] \underline{General\;\;Form\;\;of\;\;the\;\;Equation\;\;of\;\;a\;\;Circle} \\[3ex] x^2 + y^2 + 2gx + 2fy + c = 0 \\[3ex] x^2 + y^2 + 2 * -\dfrac{5}{2} * x + 2 * \dfrac{3}{2} * y + -4 = 0 \\[5ex] x^2 + y^2 - 5x + 3y - 4 = 0 \\[3ex] \underline{Standard\;\;Form\;\;of\;\;the\;\;Equation\;\;of\;\;a\;\;Circle} \\[3ex] (x - h)^2 + (y - k)^2 = r^2 \\[3ex] \left(x - \dfrac{5}{2}\right)^2 + \left(y - -\dfrac{3}{2}\right)^2 = \dfrac{25}{2} \\[5ex] \left(x - \dfrac{5}{2}\right)^2 + \left(y + \dfrac{3}{2}\right)^2 = \dfrac{25}{2} $

The equation of a circle that is in the standard $(x, y)$ coordinate plane with center $(h, k)$ is:

$ (x - h)^2 + (y - k)^2 = r^2 \\[3ex] Center:\;\; (h, k) = (1, -4) \\[3ex] h = 1 \\[3ex] k = -4 \\[3ex] r = 5 \\[3ex] \implies \\[3ex] (x - 1)^2 + (y - -4)^2 = 5^2 \\[3ex] (x - 1)^2 + (y + 4)^2 = 25 $

$ \underline{Endpoints} \\[3ex] A(2, 2) \rightarrow x_1 = 2,\;\;\;y_1 = 2 \\[3ex] B(5, 4) \rightarrow x_2 = 5,\;\;\;y_2 = 4 \\[3ex] (a) \\[3ex] Midpoint\;\;of\;\;\overrightarrow{AB} \\[3ex] = \left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right) \\[5ex] = \left(\dfrac{2 + 5}{2}, \dfrac{2 + 4}{2}\right) \\[5ex] = \left(\dfrac{7}{2}, \dfrac{6}{2}\right) \\[5ex] = \left(\dfrac{7}{2}, 3\right) \\[5ex] (b) \\[3ex] diameter \\[3ex] = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\[3ex] = \sqrt{(5 - 2)^2 + (4 - 2)^2} \\[3ex] = \sqrt{3^2 + 2^2} \\[3ex] = \sqrt{9 + 4} \\[3ex] = \sqrt{13} $

The length of a semicircle is the circumference of the semicircle

$ C = \pi r \\[3ex] (a.) \\[3ex] r = 4\;units \\[3ex] C = \pi * 4 \\[3ex] C = 4\pi\;units \\[5ex] (b.) \\[3ex] r = \dfrac{1}{4}\;unit \\[5ex] C = \pi * \dfrac{1}{4} \\[5ex] C = \dfrac{1}{4}\pi\;unit \\[5ex] $ Now, we move to Circle

$ r = \dfrac{C}{2\pi} \\[5ex] (c.) \\[3ex] C = 4\pi\;cm \\[3ex] r = \dfrac{4\pi}{2\pi} \\[5ex] r = 2\;cm \\[5ex] (d.) \\[3ex] C = 20\;m \\[3ex] r = \dfrac{20}{2\pi} \\[5ex] r = \dfrac{10}{\pi}\;m \\[5ex] $ Still on Circle

$ (e.) \\[3ex] d = 21\;ft \\[3ex] C = \pi d \\[3ex] C = \pi (21) \\[3ex] C = 21\pi \;ft \\[5ex] (f.) \\[3ex] r = \dfrac{18}{\pi}\;ft \\[5ex] C = 2\pi r \\[3ex] C = 2\pi * \dfrac{18}{\pi} \\[5ex] C = 36\;ft $

(a.) What is the relationship between the circumference of a circle and it's radius?

$ C = 2\pi r \\[3ex] r = 4r...radius\;\;is\;\;quadrupled​ \\[3ex] \implies \\[3ex] C = 2 * \pi * 4r \\[3ex] C = 2 * \pi * r * 4 \\[3ex] C = 4 * 2\pi r \\[3ex] C = 4 * C \\[3ex] C = 4C ...circumference\;\;is\;\;quadrupled​ \\[3ex] $ (b.) What is the relationship between the area of a circle and it's diameter?

$ A = \dfrac{\pi d^2}{4} \\[5ex] d = 2d...diameter\;\;is\;\;doubled​ \\[3ex] \implies \\[3ex] A = \dfrac{\pi * (2d)^2}{4} \\[5ex] A = \dfrac{4\pi d^2}{4} \\[5ex] A = \pi d^2 \\[3ex] A = \dfrac{1}{4} * A \\[5ex] A = \dfrac{1}{4}A ...area\;\;is\;\;four\;\;times\;\;smaller​ \\[3ex] $ (c.) What is the relationship between the area of a circle and it's radius?

$ A = \pi r^2 \\[3ex] r = r + 20\%r...radius\;\;is\;\;increased\;\;by\;\;20\%​ \\[3ex] r = r + 0.2r \\[3ex] r = 1.2r \\[3ex] \implies \\[3ex] A = \pi * (1.2r)^2 \\[3ex] A = \pi * 1.44r^2 \\[3ex] A = 1.44 * \pi r^2 \\[3ex] A = 1.44 * A \\[3ex] A = 1.44A \\[3ex] A = A + 0.44A \\[3ex] A = A + 44\%A...area\;\;is\;\;increased\;\;by\;\;44\% \\[3ex] $ (d.) What is the relationship between the area of a circle and it's circumference?

$ A = \dfrac{C^2}{4\pi} \\[5ex] C = 4C ...circumference\;\;is\;\;quadrupled​ \\[3ex] \implies \\[3ex] A = \dfrac{(4C)^2}{4\pi} \\[5ex] A = \dfrac{16C^2}{4\pi} \\[5ex] A = \dfrac{4C^2}{\pi} \\[5ex] $ What will you do to $\dfrac{C^2}{4\pi}$ to get $\dfrac{4C^2}{\pi}$?
Let us multiply by a variable, p and then solve for p

$ \dfrac{C^2}{4\pi} * p = \dfrac{4C^2}{\pi} \\[5ex] p = \dfrac{4C^2}{\pi} \div \dfrac{C^2}{4\pi} \\[5ex] p = \dfrac{4C^2}{\pi} \times \dfrac{4\pi}{C^2} \\[5ex] p = 16 \\[3ex] \implies \\[3ex] A = 16 * \dfrac{C^2}{4\pi} \\[5ex] A = 16 * A \\[3ex] A = 16A ...area\;\;is\;\;6\;\;times\;\;larger \\[3ex] $

Length of the rope is the circumference

$ \underline{1\;\;circular\;\;loop} \\[3ex] d = 10\;in \\[3ex] C = \pi d \\[3ex] C = \pi (10) \\[3ex] C = 31.41592654 \\[5ex] \underline{Rope = 6\;\;circular\;\;loops} \\[3ex] C = 6(31.41592654) \\[3ex] C = 188.4955592\;inches \\[3ex] $ The closest to the length, in inches, of the rope is 190 inches

$ C = \pi d \\[3ex] \underline{Small\;\;circle} \\[3ex] d = 30\;feet \\[3ex] C = 30\pi \\[3ex] \underline{Large\;\;circle} \\[3ex] d = 50\;feet \\[3ex] C = 50\pi \\[3ex] \underline{Figure\;\;8\;\;pattern} \\[3ex] C = 30\pi + 50\pi = 80\pi \\[3ex] $

Proportional Reasooning Method to calculate speed

Distance (feet)	Time (minute)
$75$	$1$
$80\pi$	$p$

$ \dfrac{x}{1} = \dfrac{80\pi}{75} \\[5ex] x = 80 * \dfrac{22}{7} * \dfrac{1}{75} \\[5ex] x = \dfrac{1760}{525} \\[5ex] x = 3.352380952\; feet \\[3ex] x \approx 3\;minutes $

On the x-axis, all y-values are zero.

$ x^2 + y^2 = 144 \\[3ex] y = 0 \\[3ex] \implies \\[3ex] x^2 + 0^2 = 144 \\[3ex] x^2 = 144 \\[3ex] x = \pm \sqrt{144} \\[3ex] x = \pm 12 \\[3ex] Points = (-12, 0) \;\;and\;\; (12, 0) $

$ x^2 + y^2 - 6x = 56 - 8y \\[3ex] x^2 - 6x + y^2 + 8y = 56 \\[3ex] \underline{Completing\;\;the\;\;Square\;\;method} \\[3ex] Coefficient\;\;of\;\;x = -6 \\[3ex] Half\;\;of\;\;it = \dfrac{1}{2} * -6 = -3 \\[5ex] Square\;\;it = (-3)^2 \\[3ex] Coefficient\;\;of\;\;y = 8 \\[3ex] Half\;\;of\;\;it = \dfrac{1}{2} * 8 = 4 \\[5ex] Square\;\;it = 4^2 \\[3ex] \implies \\[3ex] x^2 - 6x + (-3)^2 + y^2 + 8y + 4^2 = 56 + (-3)^2 + 4^2 \\[3ex] (x - 3)^2 + (y + 4)^2 = 56 + 9 + 16 \\[3ex] (x - 3)^2 + (y + 4)^2 = 81 \\[3ex] (x - 3)^2 + (y + 4)^2 = 9^2 \\[3ex] Standard\;\;Form:\;\; (x - h)^2 + (y - k)^2 = r^2 \\[3ex] Compare:\;\;\implies \\[3ex] -h = -3 \\[3ex] h = 3 \\[3ex] -k = 4 \\[3ex] k = -4 \\[3ex] Center = (h, k) = (3, -4) \\[3ex] r^2 = 9^2 \\[3ex] r = 9 \\[3ex] Radius:\;\;r = 9\;units $

$ x^2 + 6x + y^2 + 10y - 66 = 0 \\[3ex] x^2 + 6x + y^2 + 10y = 66 \\[3ex] \underline{Completing\;\;the\;\;Square\;\;method} \\[3ex] Coefficient\;\;of\;\;x = 6 \\[3ex] Half\;\;of\;\;it = \dfrac{1}{2} * 6 = 3 \\[5ex] Square\;\;it = 3^2 \\[3ex] Coefficient\;\;of\;\;y = 10 \\[3ex] Half\;\;of\;\;it = \dfrac{1}{2} * 10 = 5 \\[5ex] Square\;\;it = 5^2 \\[3ex] \implies \\[3ex] x^2 + 6x + 3^2 + y^2 + 10y + 5^2 = 66 + 3^2 + 5^2 \\[3ex] (x + 3)^2 + (y + 5)^2 = 66 + 9 + 25 \\[3ex] (x + 3)^2 + (y + 5)^2 = 100 \\[3ex] (x + 3)^2 + (y + 5)^2 = 10^2 \\[3ex] Standard\;\;Form:\;\; (x - h)^2 + (y - k)^2 = r^2 \\[3ex] Compare:\;\;\implies \\[3ex] -h = 3 \\[3ex] h = -3 \\[3ex] -k = 5 \\[3ex] k = -5 \\[3ex] Center = (h, k) = (-3, -5) \\[3ex] r^2 = 10^2 \\[3ex] r = 10 \\[3ex] Radius:\;\;r = 10\;units $

The equation of a circle that is in the standard $(x, y)$ coordinate plane with center $(h, k)$ is:

$ (x - h)^2 + (y - k)^2 = r^2 \\[3ex] Center:\;\; (h, k) = (4, -3) \\[3ex] h = 4 \\[3ex] k = -3 \\[3ex] r = 5 \\[3ex] \implies \\[3ex] (x - 4)^2 + (y - -3)^2 = 5^2 \\[3ex] (x - 4)^2 + (y + 3)^2 = 25 $