Angle inscirbed in a circle

Question

Can someone please derive and explain how to find the angel size inscribed in a circle when knowing the radius and arc. This would be a non central angle

Bunuel · Answer

If you know the length  L  of the minor arc and radius, the inscribed angle is:  Angle=\frac{90L}{\pi{r}} .

The way to derive the above formula:

Length of minor arc is  L= \frac{Central-Angle}{360}* Circumference  -->  L= \frac{Central-Angle}{360}* 2\pi{r} .

The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle.

So,  L= \frac{2*Inscribed-Angle}{360}* 2\pi{r}  -->  L= \frac{Inscribed-Angle}{90}*\pi{r}  -->  Inscribed-Angle=\frac{90L}{\pi{r}} .

For more on this issue please check Circles chapter of Math Book (link in my signature).

Hope it helps.

Jinglander · Answer

Thanks sooo much. I am a bit confused what is 2X inscribed in the third step and how does the 360 become 60 in the denominator

Jinglander · Answer

Also I am a bit confused about l=C-A/360 * circum can you explain this I dont see it visually

Bunuel · Answer

The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle.

So  Central Angle =2*Inscribed Angle  --> than I substituted  Central Angle  with  2*Inscribed Angle .

We would have  L= \frac{2*Inscribed-Angle*2\pi{r}}{360}  --> reduce by 2*2=4 -->  L= \frac{Inscribed-Angle*\pi{r}}{90}  -->  Inscribed-Angle=\frac{90L}{\pi{r}} .

About  L= \frac{Central-Angle}{360}* Circumference : it describes the fact that the ratio of length of arc to the circumference equals to the ratio of central angle to 360 degrees (the whole circle) -->  \frac{L}{Circumference}= \frac{Central-Angle}{360} .

I do advice to check Circles chapter of Math Book (link in my signature) and to revise basics of geometry.

Hope it helps.

Jinglander · Answer

Yes i read the math book on my the app and i didnt see that much detail there. So your varible "inscribed" and "angle" are really the same. I am still confued by Central-Angle/360 shouldn't this just be Central/360 because you said "length of arc to the circumference equals to the ratio of central angle to 360 degrees" why then do you need the subtraction

Jinglander · Answer

Of wait thats a dash not a minus

Bunuel · Answer

OK:
Inscribed angle when formatted looks like  Inscribed-Angle  it's not Inscribed minus angle.
Central angle when formatted looks like  Central-Angle  it's not Central minus angle.

Jinglander · Answer

Thanks for all your help. BTW is there a section on here where I can find a good tutor. I am really struggling

Angle inscirbed in a circle

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