Triangle calculator SSA

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Triangle has two solutions with side c=311.9990016873 and with side c=76.89547681098

#1 Obtuse scalene triangle.

Sides: a = 242.2   b = 186.2   c = 311.9990016873

Area: T = 22526.56328857
Perimeter: p = 740.3990016873
Semiperimeter: s = 370.1955008436

Angle ∠ A = α = 50.85440734067° = 50°51'15″ = 0.8887571019 rad
Angle ∠ B = β = 36.6° = 36°36' = 0.63987905062 rad
Angle ∠ C = γ = 92.54659265933° = 92°32'45″ = 1.61552311284 rad

Height: ha = 186.0166208801
Height: hb = 241.9610933252
Height: hc = 144.4065664717

Median: ma = 226.5810880293
Median: mb = 263.3099125011
Median: mc = 149.4366265153

Inradius: r = 60.85105311319
Circumradius: R = 156.149913753

Vertex coordinates: A[311.9990016873; 0] B[0; 0] C[194.4422392491; 144.4065664717]
Centroid: CG[168.8110803121; 48.13552215722]
Coordinates of the circumscribed circle: U[155.9955008436; -6.93661728921]
Coordinates of the inscribed circle: I[183.9955008436; 60.85105311319]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.1465926593° = 129°8'45″ = 0.8887571019 rad
∠ B' = β' = 143.4° = 143°24' = 0.63987905062 rad
∠ C' = γ' = 87.45440734067° = 87°27'15″ = 1.61552311284 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 242.2 ; ; b = 186.2 ; ; beta = 36° 36' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 186.2**2 = 242.2**2 + c**2 -2 * 242.2 * c * cos (36° 36') ; ; ; ; c**2 -388.885c +23990.4 =0 ; ; p=1; q=-388.885; r=23990.4 ; ; D = q**2 - 4pr = 388.885**2 - 4 * 1 * 23990.4 = 55269.7759909 ; ;
D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 388.88 ± sqrt{ 55269.78 } }{ 2 } ; ; c_{1,2} = 194.44239249 ± 117.547624382 ; ; c_{1} = 311.990016873 ; ; c_{2} = 76.8947681098 ; ; ; ; text{ Factored form: } ; ; (c -311.990016873) (c -76.8947681098) = 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 = 242.2 ; ; b = 186.2 ; ; c = 311.99 ; ;

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

p = a+b+c = 242.2+186.2+311.99 = 740.39 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 740.39 }{ 2 } = 370.2 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 370.2 * (370.2-242.2)(370.2-186.2)(370.2-311.99) } ; ; T = sqrt{ 507446035.45 } = 22526.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22526.56 }{ 242.2 } = 186.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22526.56 }{ 186.2 } = 241.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22526.56 }{ 311.99 } = 144.41 ; ;

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{ 186.2**2+311.99**2-242.2**2 }{ 2 * 186.2 * 311.99 } ) = 50° 51'15" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 242.2**2+311.99**2-186.2**2 }{ 2 * 242.2 * 311.99 } ) = 36° 36' ; ;
gamma = 180° - alpha - beta = 180° - 50° 51'15" - 36° 36' = 92° 32'45" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22526.56 }{ 370.2 } = 60.85 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 242.2 }{ 2 * sin 50° 51'15" } = 156.15 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 186.2**2+2 * 311.99**2 - 242.2**2 } }{ 2 } = 226.581 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 311.99**2+2 * 242.2**2 - 186.2**2 } }{ 2 } = 263.309 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 186.2**2+2 * 242.2**2 - 311.99**2 } }{ 2 } = 149.436 ; ;



#2 Obtuse scalene triangle.

Sides: a = 242.2   b = 186.2   c = 76.89547681098

Area: T = 5552.022005106
Perimeter: p = 505.295476811
Semiperimeter: s = 252.6477384055

Angle ∠ A = α = 129.1465926593° = 129°8'45″ = 2.25440216346 rad
Angle ∠ B = β = 36.6° = 36°36' = 0.63987905062 rad
Angle ∠ C = γ = 14.25440734067° = 14°15'15″ = 0.24987805128 rad

Height: ha = 45.84765735018
Height: hb = 59.63550166602
Height: hc = 144.4065664717

Median: ma = 75.00994172843
Median: mb = 153.6855434187
Median: mc = 212.5733372414

Inradius: r = 21.97553712148
Circumradius: R = 156.149913753

Vertex coordinates: A[76.89547681098; 0] B[0; 0] C[194.4422392491; 144.4065664717]
Centroid: CG[90.44657202004; 48.13552215722]
Coordinates of the circumscribed circle: U[38.44773840549; 151.3421837609]
Coordinates of the inscribed circle: I[66.44773840549; 21.97553712148]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 50.85440734067° = 50°51'15″ = 2.25440216346 rad
∠ B' = β' = 143.4° = 143°24' = 0.63987905062 rad
∠ C' = γ' = 165.7465926593° = 165°44'45″ = 0.24987805128 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 242.2 ; ; b = 186.2 ; ; beta = 36° 36' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 186.2**2 = 242.2**2 + c**2 -2 * 242.2 * c * cos (36° 36') ; ; ; ; c**2 -388.885c +23990.4 =0 ; ; p=1; q=-388.885; r=23990.4 ; ; D = q**2 - 4pr = 388.885**2 - 4 * 1 * 23990.4 = 55269.7759909 ; ; : Nr. 1
D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 388.88 ± sqrt{ 55269.78 } }{ 2 } ; ; c_{1,2} = 194.44239249 ± 117.547624382 ; ; c_{1} = 311.990016873 ; ; c_{2} = 76.8947681098 ; ; ; ; text{ Factored form: } ; ; (c -311.990016873) (c -76.8947681098) = 0 ; ; ; ; c>0 ; ; : Nr. 1
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 = 242.2 ; ; b = 186.2 ; ; c = 76.89 ; ;

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

p = a+b+c = 242.2+186.2+76.89 = 505.29 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 505.29 }{ 2 } = 252.65 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 252.65 * (252.65-242.2)(252.65-186.2)(252.65-76.89) } ; ; T = sqrt{ 30824926.65 } = 5552.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5552.02 }{ 242.2 } = 45.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5552.02 }{ 186.2 } = 59.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5552.02 }{ 76.89 } = 144.41 ; ;

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{ 186.2**2+76.89**2-242.2**2 }{ 2 * 186.2 * 76.89 } ) = 129° 8'45" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 242.2**2+76.89**2-186.2**2 }{ 2 * 242.2 * 76.89 } ) = 36° 36' ; ;
gamma = 180° - alpha - beta = 180° - 129° 8'45" - 36° 36' = 14° 15'15" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5552.02 }{ 252.65 } = 21.98 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 242.2 }{ 2 * sin 129° 8'45" } = 156.15 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 186.2**2+2 * 76.89**2 - 242.2**2 } }{ 2 } = 75.009 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 76.89**2+2 * 242.2**2 - 186.2**2 } }{ 2 } = 153.685 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 186.2**2+2 * 242.2**2 - 76.89**2 } }{ 2 } = 212.573 ; ;
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