Question

Prove that circumference of a circle is 2πr.
Please try to give a mathematical proof :D

Remember:
⇒ Should be explained properly.
⇒ Mention formulas if you used any.
⇒ Give reasons if required.

Answer 1

Answer:

Circumference of circle: The distance around the circular region is called its circumference. The ratio of circumference of any circle to its diameter is constant. This constant is denoted by π and is read as pie.

Circumference/Diameter = Pie

i.e., c/d = π or c = πd

We know that diameter is twice the radius, i.e., d = 2r

C = π × 2r

⇒ C = 2πr

Therefore approximate value of π = 22/7 or 3.14.

Step-by-step explanation:

I hope it will help you plz follow

Answer 2

Consider an elemental strip of circumferential length $\sf{dS}$ and span $\sf{d\theta}$, located at angle $\sf{\theta}$ as shown in the figure below. (Figure In Attachment)

• The radius of the circle is denoted by $\sf{r}$

Consider the formula,

$\textbf{ Angle }=\dfrac{\bf{Arc}}{\textbf{Radius}}$

$\to\textsf{ Angle }=\dfrac{\sf{dS}}{\textsf{r}}$

$\to\sf{ d\theta }=\dfrac{\sf{dS}}{\textsf{r}}$

$\to\sf{\sf{ d\theta\times r }=\dfrac{\sf{dS}}{\textsf{r}}\times r}$

$\to\sf{\sf{ rd\theta}=dS}$

The Circumference of the full circle is calculated by integrating the elemental arc length between the limits $\sf{\theta=0}$ to $\sf{\theta=2\pi}$

$\sf{C=\int \mathsf{d} S}$

$\to\sf{\displaystyle C=\int_{\theta=0}^{\theta=2 \pi} r \mathsf{~d} \theta}$

$\to\sf{\displaystyle C=r\cdot \int_{\theta=0}^{\theta=2 \pi} \mathsf{d} \theta\quad \bf{\{\because}\textbf{ r is constant for a given circle\}}}$

$\to\sf{\displaystyle C=r\cdot [\theta]_{\theta=\sf{0}}^{\theta=\sf{2} \pi}}$

$\to\sf{\displaystyle C=r\cdot \{2 \pi-0\}}$

$\to\sf{\displaystyle C=r\cdot \{2 \pi\}}$

$\boxed{\boxed{\to\sf{\displaystyle C=2\pi r}}}$

Hence proved that the circumference of the full circle is given by 2πr.