.A,B,C,D, and E are points on a circle of radius 2 in counterclockwise order. We know AB=BC=CD=DE=2. Find the area of pentagon ABCDE.

Guest Feb 4, 2022

#1**0 **

Pentagon inscribed in a circle => 5 isosceles triangles with central angle 72^{o}.

Circle centre O with r = 2, angle subtended between A and B, B and C, C and D, D and E, E and A with centre O = 360^{o} / 5 = 72^{o}

Area of triangle ABO = (1/2)(2*2*sin 72^{o}) = 1.902 sq units

[ABCDE] = There are 5 triangles of equal area = (1.902 sq units)5 = 9.51 sq units.

Guest Feb 4, 2022

#2**+2 **

.A,B,C,D, and E are points on a circle of radius 2 in counterclockwise order. We know AB=BC=CD=DE=2. Find the area of pentagon ABCDE.

**Hello Guest!**

\(360^o-4\cdot 60^o=120^o\)

\(ABCDE=4\cdot MAB+MEA\\ =4\cdot \frac{a\sqrt{3}}{2}+MEA\\ =4\cdot \frac{2\sqrt{3}}{2}+4\cdot sin(60^o)cos(60^o)\\ =4\cdot \frac{2\sqrt{3}}{2}+4\cdot \frac{1}{2} \sqrt{3}\ \cdot \ \frac{1}{2}\\ =5\cdot\sqrt{3}\)

\(ABCDE=8.660254\)

!

asinus Feb 4, 2022

#3**0 **

\(ABCDE\) forms an irregular pentagon, which can be split into \(5\) triangles, each with an area of \(\sqrt3\), so the answer is \(\color{brown}\boxed{5\sqrt3}\).

BuilderBoi Feb 4, 2022

edited by
BuilderBoi
Feb 4, 2022

edited by BuilderBoi Feb 4, 2022

edited by BuilderBoi Feb 4, 2022

edited by BuilderBoi Feb 4, 2022

edited by BuilderBoi Feb 4, 2022

edited by BuilderBoi Feb 4, 2022

edited by BuilderBoi Feb 4, 2022

#4**+2 **

Equal chords are subtended from equal angles. at the centre

We have 4 equilateral triangles.

together that make an angles at the centre of 4*60 = 240degrees

The angle at the centre remaining for the last tirangle is 120 degrees.

The Apothem for the 4 congruent trianlges is sqrt3

So their total area is 4sqrt3

The last triangle is isosceles with angles 120, 30,30

The length of the base is 2sqrt3 and the height is 1 so ther area is sqrt3

So the total area is 5sqrt3 units (which agrees with asin's answer.)

Here is the pic

Melody Feb 4, 2022

#7**0 **

Hi BilderBoi,

Thanks!

I use a free program called GeoGebra. It takes a while to get the hang of, just do simple things to start and your knowledge will build.

Here is a download page. I didn't know that there were all these ones to choose from. I just use Classic 6.

I don't think there used to be so many when i dowloaded mine. ://

https://www.geogebra.org/download?lang=en-AU

Anyway, I do almost all my diagrams from GeoGebra6 and I really enjoy using it.

I often prefer it even over Desmos.

If you give it a go I wll be more than happy to help when you get stuck :)

Melody
Feb 4, 2022

#8**0 **

Thanks! I've used it in class before, but I didn't know you do such stuff with it lol! (You misspelled my name btw.)

BuilderBoi
Feb 4, 2022

#9**0 **

oh if you have a free geogebra account you can save you pics online.

For posting all of my pics I used Gyazo. It is like an onle version of 'Snip'

Gyazo has a free and a paid version. They are the same except the paid version lets you store as many pics as you want but the free one stores only the most recent few. I am not sure how many becasue I have the paid version.

I do not think Gyazo is the best online storage of pics, I expect there are many to chose from. This forum saves in Tinypics I think if you use the provided one. But i used to find the link to tinypics was often broken, so Gyazo has been a better for me.

Melody
Feb 4, 2022