Question

Hey friends. This is not a question but a tutorial on Heron's formula .. please see the attachment

Answer 1

any suggestions and feedbacks will be welcomed. ..

Answer 2

I can't get it to download right, sorry :c

Answer 3

is their any problem with file or downloading speed is low @rebeccaskell94 ?

Answer 4

The file. Some files when I convert them/download them only upload as symbols and stuff.

Answer 5

wait i am going to type that all soon

Answer 6

Okay :) Just tag me again and I'll come back. I must go study now c:

Answer 7

you made this maths?

Answer 8

interesting

Answer 9

yes @lgbasallote ..

Answer 10

one of the most useful formulas in geometry
thank u @mathslover
very useful tutorial

Answer 11

gr8 to know @mukushla thanks a lot

Answer 12

why does herons formula work?

Answer 13

great job mathslover ! i think it's going to help me a lot

Answer 14

gr8 to know @kritima
@UnkleRhaukus do u mean for proof

Answer 15

yeah,

Answer 16

it is very long .. can u just wait for some time i will upload soon

Answer 17

i have got it upto very nearer ... for the proof

Answer 18

HERONS' FORMULA :

Basically Herons' formula is :

Area of a triangle = \(\sqrt{s(s-a)(s-b)(s-c)}\)

where s =\(\frac{a+b+c}{2}\) and a , b and c are the sides of a triangle

s can also be said as : semi perimeter as a + b + c = perimeter of a triangle and when we half it ..then it becomes semi perimeter .

I should also introduce you all with : a basic formula for the triangle area :
(base*corresponding height) / 2

In some cases we are not able to find the height .. but we are given with all sides of the triangle..
Hence in that case we generally use : heron's formula to find the area of a triangle

For example :

Find the area of a triangle having sides : 5 cm , 6 cm and 10 cm .

In this case we are unable to find the height ..

Hence we will be going to use : herons' formula

\[\large{s=\frac{5 cm + 6 cm + 10 cm}{2}=\frac{21 cm }{2}}\]

now applying the formula :

area of the triangle : \(\sqrt{\frac{21}{2}(\frac{21}{2}-5)(\frac{21}{2}-6)(\frac{21}{2}-10)}\)
\[\large{\sqrt{\frac {21}{2}*\frac{11}{2}*\frac{9}{2}*\frac{1}{2}}}\]

\[\large{\sqrt{\frac{21*11*9*1}{2^4}}}\]

\[\large{\sqrt{\frac{21*11*3^2*1^2}{(2^2)^2}}}\]

\[\large{\frac{3}{4}\sqrt{231}}\]

hence the area of the triangle with the given information will be \(\frac{3}{4}\sqrt{231}\)

Now coming to the main point : area of an equilateral triangle :

(base*corresponding height)/2

Since this is an equilateral triangle : having all sides equal ( let it be : a )

\[\large{\frac{a*h}{2}}\]

Now we will calculate h ( height )

as we know that whenever we draw a perpendicual bisector on a base of an equilateral triangle , it will divide the base into 2 equal parts .

hence the equal divided lengths of the base = \(\frac{a}{2}\)

as per pythagoras theorem :
\[\large{h^2+\frac{a^2}{4}=a^2}\]
\[\large{h^2=a^2-\frac{a^2}{4}}\]
\[\large{h^2=\frac{3a^2}{4}}\]
\[\large{h=\sqrt{\frac{3a^2}{4}}}\]
\[\large{h=\frac{\sqrt{3}}{2}a}\]
\[\large{h=\frac{\sqrt{3}a}{2}}\]
\[\large{\textbf{Area of the equilateral triangle}=\frac{a*h}{2}}\]
\[\large{\textbf{Area of the equilateral triangle}=\frac{a*\frac{\sqrt{3}a}{2}}{2}}\]
\[\large{\textbf{Area of the equilateral triangle}=\frac{\sqrt{3}a^2}{4}}\]

Now prooving this formula by heron's formula

\[\sqrt{s(s-a)(s-b)(s-c)}=\textbf{Area of the equilateral triangle}\]
\[\sqrt{\frac{3a}{2}(\frac{3a}{2}-a)(\frac{3a}{2}-a)(\frac{3a}{2}-a)}\]
\[\sqrt{\frac{3a}{2}*\frac{a}{2}*\frac{a}{2}*\frac{a}{2}}\]
\[\sqrt{\frac{3a^4}{2^4}}\]
\[\frac{a^2}{4}\sqrt{3}\]

Answer 19

@estudier @Diyadiya @annas @rebeccaskell94 @Romero @satellite73 @ujjwal

Answer 20

@ParthKohli

Answer 21

You really wrote this?

Answer 22

yes ..r u talking about that pdf file or this. . latex file ?

Answer 23

This Latex one. Your English is really good for this, so I was kinda surprised! Good job :)

Answer 24

thanks

Answer 25

awesome @mathslover your work is clearly appreciable ... keep up the good work and god bless you bro!!

Answer 26

thanks a lot annas ... just needed all of ur's wishes . . . that is what i got ! thanks a lot
I promise that i will continue to maintain this ...

Answer 27

@rebeccaskell94 you can download it by pressing right mouse button a box will appear with some options there is an option save as click it ... file will be downloaded as .pdf

Answer 28

Well it has that, but sometimes it downloads weird. It's not really a big deal, it's just frustrating.

Answer 29

sometimes your system cant identify some symbols because there ASCII codes are unknown to CPU ... btw .pdf files never create problems

Answer 30

@lalaly @jiteshmeghwal9 @TheViper @maheshmeghwal9 @ash2326 @goformit100 @robtobey @waterineyes @CarlosGP

Answer 31

@amistre64 sir please have a look

Answer 32

nice work:)
latex one is a very very nice one :D

Answer 33

thanks @jiteshmeghwal9 that's why i put up latex here also .... in the place of that pdf ..so that all can view easily ...

Answer 34

yeah it is really better.

Answer 35

& i think the best.

Answer 36

thanks @jiteshmeghwal9 ...more comments and suggestions will be appreciated and welcomed

Answer 37

draw more picture , less words

Answer 38

Thanks @mathslover

Answer 39

So here i go with the explanation for :
\[\textbf{How to find the area of a quadrilateral using heron's formula}\]

In the above diagram we have : a quadrilateral ..
So how to find the area of a quadrilateral ..having sides a , b , c and d

as i drew the diagonals of the quadrilateral .. we can find the area of the quadrilateral very easily .. let me show u all how .