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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 5.2**2 = 8.7**2 + c**2 -2 * 5.2 * c * cos (32° ) ; ; ; ; c**2 -14.756c +48.65 =0 ; ; p=1; q=-14.7560368731; 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} = 7.37801843656 ± 2.40523513408 ; ; c_{1} = 9.78325357065 ; ;
c_{2} = 4.97278330248 ; ; ; ; (c -9.78325357065) (c -4.97278330248) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 8.7**2-5.2**2-4.97**2 }{ 2 * 5.2 * 4.97 } ) = 117° 33'5" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.2**2-8.7**2-4.97**2 }{ 2 * 8.7 * 4.97 } ) = 32° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.97**2-8.7**2-5.2**2 }{ 2 * 5.2 * 8.7 } ) = 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 ; ;




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