Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
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Obtuse scalene triangle.

Sides: a = 15.15215158987   b = 17.48881513194   c = 26.19

Area: T = 127.0330192715
Perimeter: p = 58.83296672182
Semiperimeter: s = 29.41548336091

Angle ∠ A = α = 33.69° = 33°41'24″ = 0.5888001425 rad
Angle ∠ B = β = 39.81° = 39°48'36″ = 0.69548155752 rad
Angle ∠ C = γ = 106.5° = 106°30' = 1.85987756534 rad

Height: ha = 16.76879846114
Height: hb = 14.52875724569
Height: hc = 9.70106638194

Median: ma = 20.94400014274
Median: mb = 19.5266479659
Median: mc = 9.80993277191

Inradius: r = 4.31985759404
Circumradius: R = 13.65774160124

Vertex coordinates: A[26.19; 0] B[0; 0] C[11.63989671145; 9.70106638194]
Centroid: CG[12.61096557048; 3.23435546065]
Coordinates of the circumscribed circle: U[13.095; -3.87989157165]
Coordinates of the inscribed circle: I[11.92766822896; 4.31985759404]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.31° = 146°18'36″ = 0.5888001425 rad
∠ B' = β' = 140.19° = 140°11'24″ = 0.69548155752 rad
∠ C' = γ' = 73.5° = 73°30' = 1.85987756534 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 33° 41'24" ; ; beta = 39° 48'36" ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 33° 41'24" - 39° 48'36" = 106° 30' ; ;

2. By using the law of sines, we calculate unknown side a

c = 26.19 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 26.19 * fraction{ sin(33° 41'24") }{ sin (106° 30') } = 15.15 ; ;

3. By using the law of sines, we calculate last unknown side b

 fraction{ b }{ c } = fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = c * fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = 26.19 * fraction{ sin(39° 48'36") }{ sin (106° 30') } = 17.49 ; ;


Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 15.15 ; ; b = 17.49 ; ; c = 26.19 ; ;

4. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 15.15+17.49+26.19 = 58.83 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58.83 }{ 2 } = 29.41 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.41 * (29.41-15.15)(29.41-17.49)(29.41-26.19) } ; ; T = sqrt{ 16136.67 } = 127.03 ; ;

7. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 127.03 }{ 15.15 } = 16.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 127.03 }{ 17.49 } = 14.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 127.03 }{ 26.19 } = 9.7 ; ;

8. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 15.15**2-17.49**2-26.19**2 }{ 2 * 17.49 * 26.19 } ) = 33° 41'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17.49**2-15.15**2-26.19**2 }{ 2 * 15.15 * 26.19 } ) = 39° 48'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 26.19**2-15.15**2-17.49**2 }{ 2 * 17.49 * 15.15 } ) = 106° 30' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 127.03 }{ 29.41 } = 4.32 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15.15 }{ 2 * sin 33° 41'24" } = 13.66 ; ;




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