Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
°
°


Acute scalene triangle.

Sides: a = 4.99554075996   b = 9.7954622944   c = 10.6

Area: T = 24.37109737842
Perimeter: p = 25.39900305436
Semiperimeter: s = 12.69550152718

Angle ∠ A = α = 28° = 0.48986921906 rad
Angle ∠ B = β = 67° = 1.16993705988 rad
Angle ∠ C = γ = 85° = 1.48435298642 rad

Height: ha = 9.75773514466
Height: hb = 4.97663985655
Height: hc = 4.59882969404

Median: ma = 9.8954887318
Median: mb = 6.6843815444
Median: mc = 5.68880900002

Inradius: r = 1.92197278036
Circumradius: R = 5.3220245139

Vertex coordinates: A[10.6; 0] B[0; 0] C[1.95218612487; 4.59882969404]
Centroid: CG[4.18439537496; 1.53327656468]
Coordinates of the circumscribed circle: U[5.3; 0.46436899167]
Coordinates of the inscribed circle: I[2.99003923278; 1.92197278036]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152° = 0.48986921906 rad
∠ B' = β' = 113° = 1.16993705988 rad
∠ C' = γ' = 95° = 1.48435298642 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 28° ; ; beta = 67° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 28° - 67° = 85° ; ;

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

c = 10.6 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 10.6 * fraction{ sin(28° ) }{ sin (85° ) } = 5 ; ;

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 = 10.6 * fraction{ sin(67° ) }{ sin (85° ) } = 9.79 ; ;


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 = 5 ; ; b = 9.79 ; ; c = 10.6 ; ;

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

p = a+b+c = 5+9.79+10.6 = 25.39 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25.39 }{ 2 } = 12.7 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.7 * (12.7-5)(12.7-9.79)(12.7-10.6) } ; ; T = sqrt{ 593.94 } = 24.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 24.37 }{ 5 } = 9.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 24.37 }{ 9.79 } = 4.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 24.37 }{ 10.6 } = 4.6 ; ;

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{ 5**2-9.79**2-10.6**2 }{ 2 * 9.79 * 10.6 } ) = 28° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.79**2-5**2-10.6**2 }{ 2 * 5 * 10.6 } ) = 67° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.6**2-5**2-9.79**2 }{ 2 * 9.79 * 5 } ) = 85° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 24.37 }{ 12.7 } = 1.92 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 28° } = 5.32 ; ;




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