Triangle calculator ASA

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


Obtuse isosceles triangle.

Sides: a = 69.28220323028   b = 69.28220323028   c = 120

Area: T = 2078.461096908
Perimeter: p = 258.5644064605
Semiperimeter: s = 129.2822032303

Angle ∠ A = α = 30° = 0.52435987756 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: ha = 60
Height: hb = 60
Height: hc = 34.64110161514

Median: ma = 91.65215138991
Median: mb = 91.65215138991
Median: mc = 34.64110161514

Inradius: r = 16.07769515459
Circumradius: R = 69.28220323028

Vertex coordinates: A[120; 0] B[0; 0] C[60; 34.64110161514]
Centroid: CG[60; 11.54770053838]
Coordinates of the circumscribed circle: U[60; -34.64110161514]
Coordinates of the inscribed circle: I[60; 16.07769515459]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150° = 0.52435987756 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 60° = 2.09443951024 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 30° ; ; beta = 30° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 30° - 30° = 120° ; ;

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

c = 120 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 120 * fraction{ sin(30° ) }{ sin (120° ) } = 69.28 ; ;

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 = 120 * fraction{ sin(30° ) }{ sin (120° ) } = 69.28 ; ;


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 = 69.28 ; ; b = 69.28 ; ; c = 120 ; ;

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

p = a+b+c = 69.28+69.28+120 = 258.56 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 258.56 }{ 2 } = 129.28 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 129.28 * (129.28-69.28)(129.28-69.28)(129.28-120) } ; ; T = sqrt{ 4320000 } = 2078.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2078.46 }{ 69.28 } = 60 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2078.46 }{ 69.28 } = 60 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2078.46 }{ 120 } = 34.64 ; ;

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{ 69.28**2-69.28**2-120**2 }{ 2 * 69.28 * 120 } ) = 30° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 69.28**2-69.28**2-120**2 }{ 2 * 69.28 * 120 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 120**2-69.28**2-69.28**2 }{ 2 * 69.28 * 69.28 } ) = 120° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2078.46 }{ 129.28 } = 16.08 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 69.28 }{ 2 * sin 30° } = 69.28 ; ;




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