Question

h
$\huge \bf \pink{hlo \: guys} \\ \\ \small \bf \green{my \: question \: is \: } \\ \\ \small \bf \green{how \: to \: prove} \\ \\ \small \bf \green{(1).the \: area \: of \: circle = \pi \: {r}^{2} } \\ \\ \small \bf \green{(2).perimeter \: of \: circle = 2\pi \: r}$

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$\small \bf \pink{thanks \: in \: advance}$

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Answer 1

$\huge{\mathfrak{Question:-}$

Prove the area of circle = πr²

$\huge{\mathfrak{Solution:-}$

Consider a circle divided into large number of sectors as shown in (1) figure in the attachment.

Cut the sectors and arrange them as shone in figure (2) in attachment. So that it form a rectangle.

$\bold{Length\;of\;rectangle = half\;the\;circumference\;of\;the\;circle}\\ \\ \\ \bold{= 2\pi \frac{r}{2}}\\ \\ \\ \bold{Breadth\;of\;rectangle = r(radius\;of\;the\;circle)} \\ \\ \\ \bold{Area\;of\;rectangle = length\times breadth}\\ \\ \\ \bold{= \pi r\times r}\\ \\ \\ \bold{=\pi r^{2}}\\ \\ \\ \bold{Hence\;area\;of\;a\;circle\;is\;\pi r^{2} sq.\;units}$

Answer 2

✍️ $\huge\mathfrak\purple{area\: of \:the\: circle}$

Consider a circle divided into large number of sectors as shown.

(ATTACHMENT 1)

Cut the sectors and arrange them as shown so that it forms a rectangle.

(ATTACHMENT 2)

Length of rectangle = half the circumference of the circle = 2πr/2= πr

Breadth of the rectangle = r (radius of the circle)

Area of rectangle = length x breadth

= πr x r

$\boxed{= πr2}$

Hence area of a circle is πr2 sq units.

✍️ $\huge\mathfrak\purple{perimeter\: of \:the\: circle}$

Recall that the definition of pi (π) is the circumference c of any circle divided by its diameter d. Put as an equation, pi is defined as

π = c/d

Rearranging this to solve for c we get

c = πd

The diameter of a circle is twice its radius, so substituting 2r for d,

$\boxed{c = 2 π r}$

$\huge\mathfrak{brainiest\:plz}$ ☺️❤️