Triangle calculator ASA

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

Sides: a = 138.3299689501   b = 132.153272577   c = 125

Area: T = 7485.699027773
Perimeter: p = 395.4522415272
Semiperimeter: s = 197.7266207636

Angle ∠ A = α = 65° = 1.13444640138 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 55° = 0.96599310886 rad

Height: ha = 108.2533175473
Height: hb = 113.288847338
Height: hc = 119.7711044444

Median: ma = 108.4577228598
Median: mb = 114.0610581824
Median: mc = 119.9565506428

Inradius: r = 37.85988674068
Circumradius: R = 76.29884117976

Vertex coordinates: A[125; 0] B[0; 0] C[69.15498447507; 119.7711044444]
Centroid: CG[64.71766149169; 39.92436814812]
Coordinates of the circumscribed circle: U[62.5; 43.76329711381]
Coordinates of the inscribed circle: I[65.57334818656; 37.85988674068]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115° = 1.13444640138 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 125° = 0.96599310886 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 65° ; ; beta = 60° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 65° - 60° = 55° ; ;

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

c = 125 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 125 * fraction{ sin(65° ) }{ sin (55° ) } = 138.3 ; ;

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 = 125 * fraction{ sin(60° ) }{ sin (55° ) } = 132.15 ; ;


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 = 138.3 ; ; b = 132.15 ; ; c = 125 ; ;

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

p = a+b+c = 138.3+132.15+125 = 395.45 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 395.45 }{ 2 } = 197.73 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 197.73 * (197.73-138.3)(197.73-132.15)(197.73-125) } ; ; T = sqrt{ 56035558.93 } = 7485.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7485.69 }{ 138.3 } = 108.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7485.69 }{ 132.15 } = 113.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7485.69 }{ 125 } = 119.77 ; ;

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{ 138.3**2-132.15**2-125**2 }{ 2 * 132.15 * 125 } ) = 65° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 132.15**2-138.3**2-125**2 }{ 2 * 138.3 * 125 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 125**2-138.3**2-132.15**2 }{ 2 * 132.15 * 138.3 } ) = 55° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7485.69 }{ 197.73 } = 37.86 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 138.3 }{ 2 * sin 65° } = 76.3 ; ;




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