Triangle calculator SSA

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Triangle has two solutions with side c=9.78332535706 and with side c=4.97327833025

#1 Acute scalene triangle.

Sides: a = 8.7   b = 5.2   c = 9.78332535706

Area: T = 22.55218552227
Perimeter: p = 23.68332535706
Semiperimeter: s = 11.84216267853

Angle ∠ A = α = 62.44985318839° = 62°26'55″ = 1.09899324944 rad
Angle ∠ B = β = 32° = 0.55985053606 rad
Angle ∠ C = γ = 85.55114681161° = 85°33'5″ = 1.49331547985 rad

Height: ha = 5.1844334534
Height: hb = 8.67437904703
Height: hc = 4.61102975988

Median: ma = 6.51656369768
Median: mb = 8.8854876207
Median: mc = 5.23880327789

Inradius: r = 1.90444558346
Circumradius: R = 4.90664077785

Vertex coordinates: A[9.78332535706; 0] B[0; 0] C[7.37880184366; 4.61102975988]
Centroid: CG[5.72204240024; 1.53767658663]
Coordinates of the circumscribed circle: U[4.89216267853; 0.38105583816]
Coordinates of the inscribed circle: I[6.64216267853; 1.90444558346]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 117.5511468116° = 117°33'5″ = 1.09899324944 rad
∠ B' = β' = 148° = 0.55985053606 rad
∠ C' = γ' = 94.44985318839° = 94°26'55″ = 1.49331547985 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 8.7 ; ; b = 5.2 ; ; beta = 32° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 5.2**2 = 8.7**2 + c**2 -2 * 8.7 * c * cos (32° ) ; ; ; ; c**2 -14.756c +48.65 =0 ; ; p=1; q=-14.756; r=48.65 ; ; D = q**2 - 4pr = 14.756**2 - 4 * 1 * 48.65 = 23.1406242009 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 14.76 ± sqrt{ 23.14 } }{ 2 } ; ; c_{1,2} = 7.37801844 ± 2.40523513408 ; ; c_{1} = 9.78325357408 ; ; c_{2} = 4.97278330592 ; ; ; ; text{ Factored form: } ; ; (c -9.78325357408) (c -4.97278330592) = 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 = 8.7 ; ; b = 5.2 ; ; c = 9.78 ; ;

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

p = a+b+c = 8.7+5.2+9.78 = 23.68 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.68 }{ 2 } = 11.84 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.84 * (11.84-8.7)(11.84-5.2)(11.84-9.78) } ; ; T = sqrt{ 508.59 } = 22.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 22.55 }{ 8.7 } = 5.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 22.55 }{ 5.2 } = 8.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 22.55 }{ 9.78 } = 4.61 ; ;

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{ 5.2**2+9.78**2-8.7**2 }{ 2 * 5.2 * 9.78 } ) = 62° 26'55" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.7**2+9.78**2-5.2**2 }{ 2 * 8.7 * 9.78 } ) = 32° ; ; gamma = 180° - alpha - beta = 180° - 62° 26'55" - 32° = 85° 33'5" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 22.55 }{ 11.84 } = 1.9 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.7 }{ 2 * sin 62° 26'55" } = 4.91 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.2**2+2 * 9.78**2 - 8.7**2 } }{ 2 } = 6.516 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.78**2+2 * 8.7**2 - 5.2**2 } }{ 2 } = 8.885 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.2**2+2 * 8.7**2 - 9.78**2 } }{ 2 } = 5.238 ; ;







#2 Obtuse scalene triangle.

Sides: a = 8.7   b = 5.2   c = 4.97327833025

Area: T = 11.46330054595
Perimeter: p = 18.87327833025
Semiperimeter: s = 9.43663916512

Angle ∠ A = α = 117.5511468116° = 117°33'5″ = 2.05216601592 rad
Angle ∠ B = β = 32° = 0.55985053606 rad
Angle ∠ C = γ = 30.44985318839° = 30°26'55″ = 0.53114271338 rad

Height: ha = 2.63551736688
Height: hb = 4.40988482536
Height: hc = 4.61102975988

Median: ma = 2.63985198287
Median: mb = 6.59216073068
Median: mc = 6.7221819438

Inradius: r = 1.21547657583
Circumradius: R = 4.90664077785

Vertex coordinates: A[4.97327833025; 0] B[0; 0] C[7.37880184366; 4.61102975988]
Centroid: CG[4.1176933913; 1.53767658663]
Coordinates of the circumscribed circle: U[2.48663916512; 4.23297392172]
Coordinates of the inscribed circle: I[4.23663916512; 1.21547657583]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 62.44985318839° = 62°26'55″ = 2.05216601592 rad
∠ B' = β' = 148° = 0.55985053606 rad
∠ C' = γ' = 149.5511468116° = 149°33'5″ = 0.53114271338 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 8.7 ; ; b = 5.2 ; ; beta = 32° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 5.2**2 = 8.7**2 + c**2 -2 * 8.7 * c * cos (32° ) ; ; ; ; c**2 -14.756c +48.65 =0 ; ; p=1; q=-14.756; r=48.65 ; ; D = q**2 - 4pr = 14.756**2 - 4 * 1 * 48.65 = 23.1406242009 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 14.76 ± sqrt{ 23.14 } }{ 2 } ; ; c_{1,2} = 7.37801844 ± 2.40523513408 ; ; c_{1} = 9.78325357408 ; ; c_{2} = 4.97278330592 ; ; ; ; text{ Factored form: } ; ; (c -9.78325357408) (c -4.97278330592) = 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 = 8.7 ; ; b = 5.2 ; ; c = 4.97 ; ;

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

p = a+b+c = 8.7+5.2+4.97 = 18.87 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.87 }{ 2 } = 9.44 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.44 * (9.44-8.7)(9.44-5.2)(9.44-4.97) } ; ; T = sqrt{ 131.4 } = 11.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11.46 }{ 8.7 } = 2.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11.46 }{ 5.2 } = 4.41 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11.46 }{ 4.97 } = 4.61 ; ;

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{ 5.2**2+4.97**2-8.7**2 }{ 2 * 5.2 * 4.97 } ) = 117° 33'5" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.7**2+4.97**2-5.2**2 }{ 2 * 8.7 * 4.97 } ) = 32° ; ; gamma = 180° - alpha - beta = 180° - 117° 33'5" - 32° = 30° 26'55" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11.46 }{ 9.44 } = 1.21 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.7 }{ 2 * sin 117° 33'5" } = 4.91 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.2**2+2 * 4.97**2 - 8.7**2 } }{ 2 } = 2.639 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.97**2+2 * 8.7**2 - 5.2**2 } }{ 2 } = 6.592 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.2**2+2 * 8.7**2 - 4.97**2 } }{ 2 } = 6.722 ; ;
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