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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

5. 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{ 242.2**2-186.2**2-311.99**2 }{ 2 * 186.2 * 311.99 } ) = 50° 51'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 186.2**2-242.2**2-311.99**2 }{ 2 * 242.2 * 311.99 } ) = 36° 36' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 311.99**2-242.2**2-186.2**2 }{ 2 * 186.2 * 242.2 } ) = 92° 32'45" ; ;

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 186.2**2 = 242.2**2 + c**2 -2 * 186.2 * c * cos (36° 36') ; ; ; ; c**2 -388.885c +23990.4 =0 ; ; p=1; q=-388.884784983; 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} = 194.442392491 ± 117.547624382 ; ;
c_{1} = 311.990016873 ; ; c_{2} = 76.8947681098 ; ; ; ; (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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 242.2**2-186.2**2-76.89**2 }{ 2 * 186.2 * 76.89 } ) = 129° 8'45" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 186.2**2-242.2**2-76.89**2 }{ 2 * 242.2 * 76.89 } ) = 36° 36' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 76.89**2-242.2**2-186.2**2 }{ 2 * 186.2 * 242.2 } ) = 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 ; ;




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