Triangle calculator ASA

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


Obtuse scalene triangle.

Sides: a = 58.05498098697   b = 16.82441275087   c = 61

Area: T = 488.0211230994
Perimeter: p = 135.8743937378
Semiperimeter: s = 67.93769686892

Angle ∠ A = α = 72° = 1.25766370614 rad
Angle ∠ B = β = 16° = 0.27992526803 rad
Angle ∠ C = γ = 92° = 1.60657029118 rad

Height: ha = 16.81438787048
Height: hb = 58.0144447494
Height: hc = 16.00106960982

Median: ma = 34.05326140955
Median: mb = 58.94659701451
Median: mc = 29.93660292319

Inradius: r = 7.18334413635
Circumradius: R = 30.51985911011

Vertex coordinates: A[61; 0] B[0; 0] C[55.80110586843; 16.00106960982]
Centroid: CG[38.93436862281; 5.33435653661]
Coordinates of the circumscribed circle: U[30.5; -1.06550834695]
Coordinates of the inscribed circle: I[51.11328411805; 7.18334413635]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108° = 1.25766370614 rad
∠ B' = β' = 164° = 0.27992526803 rad
∠ C' = γ' = 88° = 1.60657029118 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 72° ; ; beta = 16° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 72° - 16° = 92° ; ;

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

c = 61 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 61 * fraction{ sin(72° ) }{ sin (92° ) } = 58.05 ; ;

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 = 61 * fraction{ sin(16° ) }{ sin (92° ) } = 16.82 ; ;


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 = 58.05 ; ; b = 16.82 ; ; c = 61 ; ;

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

p = a+b+c = 58.05+16.82+61 = 135.87 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 135.87 }{ 2 } = 67.94 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 67.94 * (67.94-58.05)(67.94-16.82)(67.94-61) } ; ; T = sqrt{ 238164.72 } = 488.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 488.02 }{ 58.05 } = 16.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 488.02 }{ 16.82 } = 58.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 488.02 }{ 61 } = 16 ; ;

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{ 58.05**2-16.82**2-61**2 }{ 2 * 16.82 * 61 } ) = 72° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16.82**2-58.05**2-61**2 }{ 2 * 58.05 * 61 } ) = 16° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 61**2-58.05**2-16.82**2 }{ 2 * 16.82 * 58.05 } ) = 92° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 488.02 }{ 67.94 } = 7.18 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 58.05 }{ 2 * sin 72° } = 30.52 ; ;




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