Triangle calculator ASA

Acute scalene triangle.

Sides: a = 129.0599195015 b = 99.3743802605 c = 93

Area: T = 4597.226618162
Perimeter: p = 321.433299762
Semiperimeter: s = 160.716649881

Angle ∠ A = α = 84.2° = 84°12' = 1.47695672302 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 45.8° = 45°48' = 0.79993607974 rad

Height: h_a = 71.24221332101
Height: h_b = 92.52439059211
Height: h_c = 98.86550791746

Median: m_a = 71.44003317052
Median: m_b = 100.9155062047
Median: m_c = 105.3732976759

Inradius: r = 28.60545690123
Circumradius: R = 64.86216431446

Vertex coordinates: A[93; 0] B[0; 0] C[82.95876514718; 98.86550791746]
Centroid: CG[58.65325504906; 32.95550263915]
Coordinates of the circumscribed circle: U[46.5; 45.21992741142]
Coordinates of the inscribed circle: I[61.3432696205; 28.60545690123]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 95.8° = 95°48' = 1.47695672302 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 134.2° = 134°12' = 0.79993607974 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

$alpha = 84° 12' ; ; beta = 50° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 84° 12' - 50° = 45° 48' ; ;$

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

$c = 93 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 93 * fraction{ sin 84° 12' }{ sin 45° 48' } = 129.06 ; ;$

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

$fraction{ b }{ c } = fraction{ sin beta }{ sin gamma } ; ; ; ; b = c * fraction{ sin beta }{ sin gamma } ; ; ; ; b = 93 * fraction{ sin 50° }{ sin 45° 48' } = 99.37 ; ;$

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.

$a = 129.06 ; ; b = 99.37 ; ; c = 93 ; ;$

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

$p = a+b+c = 129.06+99.37+93 = 321.43 ; ;$

5. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 321.43 }{ 2 } = 160.72 ; ;$

6. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 160.72 * (160.72-129.06)(160.72-99.37)(160.72-93) } ; ; T = sqrt{ 21134488.56 } = 4597.23 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4597.23 }{ 129.06 } = 71.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4597.23 }{ 99.37 } = 92.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4597.23 }{ 93 } = 98.87 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 99.37**2+93**2-129.06**2 }{ 2 * 99.37 * 93 } ) = 84° 12' ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 129.06**2+93**2-99.37**2 }{ 2 * 129.06 * 93 } ) = 50° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 129.06**2+99.37**2-93**2 }{ 2 * 129.06 * 99.37 } ) = 45° 48' ; ;$

9. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 4597.23 }{ 160.72 } = 28.6 ; ;$

10. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 129.06 }{ 2 * sin 84° 12' } = 64.86 ; ;$

11. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 99.37**2+2 * 93**2 - 129.06**2 } }{ 2 } = 71.4 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 93**2+2 * 129.06**2 - 99.37**2 } }{ 2 } = 100.915 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 99.37**2+2 * 129.06**2 - 93**2 } }{ 2 } = 105.373 ; ;$
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