Question

You have given an equal sided triangle with side length a. A straight line connects the center

of the bottom side to the border of the triangle with an angle of α. Derive an expression for

the enclosed area A(α) with respect to the angle (see drawing)​

Answer 1

Answer:

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Answer 2

Given : You have given an equal-sided triangle with side length a. A straight line connects the center of the bottom side to the border of the triangle with an angle of α.  

To Find : Derive an expression for the enclosed area A(α) with respect to the angle

Solution:

ABC is a triangle , AB = BC = AC = a

D is center of BC   => BD  = CD  = a/2

case 1 - α is between 0 and 90 deg

DE meets AC at E making angle α

A(α) = Area of Triangle DCE

∠DCE = 60°

∠CDE = α

∠DEC = 120° - α

Using Sine rule of triangle

CD/Sin(120° - α )  = CE/Sinα  = DE/Sin60°

=> a/2Sin(120° - α )  = CE/Sinα  = DE/Sin60°

=> DE = aSin60°/2Sin(120° - α )

A(α) = Area of Triangle DCE  = (1/2) CD. DE.Sin α

= (1/2) (a/2). DE.Sin α

= (1/2) (a/2). (aSin60°/2Sin(120° - α )).Sin α

Sin60° = √3/2

= (√3a²/16)Sin α /Sin(120° - α )

Sin(120° - α ) = Sin(60° + α )

= (√3 a²/16)Sin α /Sin(60° + α )

A(α) =  (√3 a²/16)Sin α /Sin(60° + α )  for  α is between 0 and 90°

case 2 - α is between 90 and 180°

DF meets AB at F making angle α

A(α) = Area of Triangle ABC - Area of Triangle BDF

∠DBF = 60°

∠BDF = 180°- α

∠DFB =  α - 60°

Using Sine rule of triangle

BD/Sin(α - 60°)  = BF/Sin(180°- α)  = DF/Sin60°

=> a/2Sin(α - 60°)  =  DF/Sin60°

=> DE = aSin60°/2Sin(α - 60°)

A(α) = Area of Triangle ABC - Area of Triangle BDF

= (√3 /4 )a² - (1/2)BD. DF Sin(180°- α)

= (√3 /4 )a² -  (1/2) (a/2). DF.Sin α

= (√3 /4 )a² -  (1/2) (a/2). (aSin60°/2Sin(α - 60°)).Sin α

Sin60° = √3/2

= (√3 /4 )a² -   (√3a²/16)Sin α /Sin(α - 60°)  

for α  between 90 and 180 deg

A(α) =  (√3 a²/16)Sin α /Sin(60° + α ) for  α is between 0 and 90°

(√3 /4 )a² -   (√3a²/16)Sin α /Sin(α - 60°)  for α  between 90° and 180°

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Derive an expression for the enclosed area A(α) with respect to the angle

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