Triangle calculator ASA

Acute scalene triangle.

Sides: a = 247.7099088639 b = 183.7087504454 c = 184

Area: T = 16828.83331473
Perimeter: p = 615.4176593093
Semiperimeter: s = 307.7088296546

Angle ∠ A = α = 84.7° = 84°42' = 1.47882938764 rad
Angle ∠ B = β = 47.6° = 47°36' = 0.83107767239 rad
Angle ∠ C = γ = 47.7° = 47°42' = 0.83325220532 rad

Height: h_a = 135.8765782675
Height: h_b = 183.2133344466
Height: h_c = 182.9222099427

Median: m_a = 135.8765956106
Median: m_b = 197.9166104698
Median: m_c = 197.7122214831

Inradius: r = 54.69108657847
Circumradius: R = 124.3866333436

Vertex coordinates: A[184; 0] B[0; 0] C[167.0310829895; 182.9222099427]
Centroid: CG[117.0110276632; 60.97440331423]
Coordinates of the circumscribed circle: U[92; 83.71435589124]
Coordinates of the inscribed circle: I[124.0010792092; 54.69108657847]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 95.3° = 95°18' = 1.47882938764 rad
∠ B' = β' = 132.4° = 132°24' = 0.83107767239 rad
∠ C' = γ' = 132.3° = 132°18' = 0.83325220532 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

$alpha = 84° 42' ; ; beta = 47° 36' ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 84° 42' - 47° 36' = 47° 42' ; ;$

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

$c = 184 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 184 * fraction{ sin 84° 42' }{ sin 47° 42' } = 247.71 ; ;$

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

$fraction{ b }{ c } = fraction{ sin beta }{ sin gamma } ; ; ; ; b = c * fraction{ sin beta }{ sin gamma } ; ; ; ; b = 184 * fraction{ sin 47° 36' }{ sin 47° 42' } = 183.71 ; ;$
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.

$a = 247.71 ; ; b = 183.71 ; ; c = 184 ; ;$

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

$p = a+b+c = 247.71+183.71+184 = 615.42 ; ;$

5. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 615.42 }{ 2 } = 307.71 ; ;$

6. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 307.71 * (307.71-247.71)(307.71-183.71)(307.71-184) } ; ; T = sqrt{ 283209625.1 } = 16828.83 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16828.83 }{ 247.71 } = 135.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16828.83 }{ 183.71 } = 183.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16828.83 }{ 184 } = 182.92 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 183.71**2+184**2-247.71**2 }{ 2 * 183.71 * 184 } ) = 84° 42' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 247.71**2+184**2-183.71**2 }{ 2 * 247.71 * 184 } ) = 47° 36' ; ; gamma = 180° - alpha - beta = 180° - 84° 42' - 47° 36' = 47° 42' ; ;$

9. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 16828.83 }{ 307.71 } = 54.69 ; ;$

10. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 247.71 }{ 2 * sin 84° 42' } = 124.39 ; ;$

11. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 183.71**2+2 * 184**2 - 247.71**2 } }{ 2 } = 135.876 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 184**2+2 * 247.71**2 - 183.71**2 } }{ 2 } = 197.916 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 183.71**2+2 * 247.71**2 - 184**2 } }{ 2 } = 197.712 ; ;$
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