Triangle calculator SSA

Please enter two sides and a non-included angle
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Obtuse scalene triangle.

Sides: a = 312   b = 527   c = 678.5122297075

Area: T = 79397.6955247
Perimeter: p = 1517.512229708
Semiperimeter: s = 758.7566148538

Angle ∠ A = α = 26.36550377232° = 26°21'54″ = 0.46601567157 rad
Angle ∠ B = β = 48.6° = 48°36' = 0.84882300165 rad
Angle ∠ C = γ = 105.0354962277° = 105°2'6″ = 1.83332059214 rad

Height: ha = 508.9659584916
Height: hb = 301.3219526554
Height: hc = 234.0354653725

Median: ma = 587.1276876102
Median: mb = 457.6344372224
Median: mc = 269.1550080215

Inradius: r = 104.6421913479
Circumradius: R = 351.2811311086

Vertex coordinates: A[678.5122297075; 0] B[0; 0] C[206.3299301981; 234.0354653725]
Centroid: CG[294.9477199685; 78.01215512416]
Coordinates of the circumscribed circle: U[339.2566148537; -91.1255326874]
Coordinates of the inscribed circle: I[231.7566148537; 104.6421913479]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.6354962277° = 153°38'6″ = 0.46601567157 rad
∠ B' = β' = 131.4° = 131°24' = 0.84882300165 rad
∠ C' = γ' = 74.96550377232° = 74°57'54″ = 1.83332059214 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 312 ; ; b = 527 ; ; beta = 48° 36' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 527**2 = 312**2 + c**2 -2 * 312 * c * cos (48° 36') ; ; ; ; c**2 -412.659c -180385 =0 ; ; p=1; q=-412.659; r=-180385 ; ; D = q**2 - 4pr = 412.659**2 - 4 * 1 * (-180385) = 891827.123424 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 412.66 ± sqrt{ 891827.12 } }{ 2 } ; ; c_{1,2} = 206.32930198 ± 472.182995094 ; ;
c_{1} = 678.512297074 ; ; c_{2} = -265.853693114 ; ; ; ; text{ Factored form: } ; ; (c -678.512297074) (c +265.853693114) = 0 ; ; ; ; c>0 ; ;
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 = 312 ; ; b = 527 ; ; c = 678.51 ; ;

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

p = a+b+c = 312+527+678.51 = 1517.51 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1517.51 }{ 2 } = 758.76 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 758.76 * (758.76-312)(758.76-527)(758.76-678.51) } ; ; T = sqrt{ 6303994010.53 } = 79397.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 79397.7 }{ 312 } = 508.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 79397.7 }{ 527 } = 301.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 79397.7 }{ 678.51 } = 234.03 ; ;

6. 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{ 527**2+678.51**2-312**2 }{ 2 * 527 * 678.51 } ) = 26° 21'54" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 312**2+678.51**2-527**2 }{ 2 * 312 * 678.51 } ) = 48° 36' ; ; gamma = 180° - alpha - beta = 180° - 26° 21'54" - 48° 36' = 105° 2'6" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 79397.7 }{ 758.76 } = 104.64 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 312 }{ 2 * sin 26° 21'54" } = 351.28 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 527**2+2 * 678.51**2 - 312**2 } }{ 2 } = 587.127 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 678.51**2+2 * 312**2 - 527**2 } }{ 2 } = 457.634 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 527**2+2 * 312**2 - 678.51**2 } }{ 2 } = 269.15 ; ;
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