Triangle calculator SSA

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Triangle has two solutions with side c=12.08217110387 and with side c=3.86994850299

#1 Acute scalene triangle.

Sides: a = 10.6   b = 8.1   c = 12.08217110387

Area: T = 42.17879073232
Perimeter: p = 30.78217110387
Semiperimeter: s = 15.39108555193

Angle ∠ A = α = 59.541061001° = 59°32'26″ = 1.03991796833 rad
Angle ∠ B = β = 41.2° = 41°12' = 0.71990756518 rad
Angle ∠ C = γ = 79.259938999° = 79°15'34″ = 1.38333373184 rad

Height: ha = 7.95880957214
Height: hb = 10.41442981045
Height: hc = 6.98221082773

Median: ma = 8.81546962971
Median: mb = 10.61989157079
Median: mc = 7.24552097688

Inradius: r = 2.74404524245
Circumradius: R = 6.1498572651

Vertex coordinates: A[12.08217110387; 0] B[0; 0] C[7.97655980343; 6.98221082773]
Centroid: CG[6.6865769691; 2.32773694258]
Coordinates of the circumscribed circle: U[6.04108555193; 1.14658665888]
Coordinates of the inscribed circle: I[7.29108555193; 2.74404524245]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120.459938999° = 120°27'34″ = 1.03991796833 rad
∠ B' = β' = 138.8° = 138°48' = 0.71990756518 rad
∠ C' = γ' = 100.741061001° = 100°44'26″ = 1.38333373184 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 = 10.6 ; ; b = 8.1 ; ; c = 12.08 ; ;

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

p = a+b+c = 10.6+8.1+12.08 = 30.78 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30.78 }{ 2 } = 15.39 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.39 * (15.39-10.6)(15.39-8.1)(15.39-12.08) } ; ; T = sqrt{ 1778.98 } = 42.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 42.18 }{ 10.6 } = 7.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 42.18 }{ 8.1 } = 10.41 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 42.18 }{ 12.08 } = 6.98 ; ;

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{ 10.6**2-8.1**2-12.08**2 }{ 2 * 8.1 * 12.08 } ) = 59° 32'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.1**2-10.6**2-12.08**2 }{ 2 * 10.6 * 12.08 } ) = 41° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.08**2-10.6**2-8.1**2 }{ 2 * 8.1 * 10.6 } ) = 79° 15'34" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 42.18 }{ 15.39 } = 2.74 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.6 }{ 2 * sin 59° 32'26" } = 6.15 ; ;





#2 Obtuse scalene triangle.

Sides: a = 10.6   b = 8.1   c = 3.86994850299

Area: T = 13.50985817281
Perimeter: p = 22.56994850299
Semiperimeter: s = 11.2854742515

Angle ∠ A = α = 120.459938999° = 120°27'34″ = 2.10224129703 rad
Angle ∠ B = β = 41.2° = 41°12' = 0.71990756518 rad
Angle ∠ C = γ = 18.341061001° = 18°20'26″ = 0.32201040315 rad

Height: ha = 2.54987890053
Height: hb = 3.33554522785
Height: hc = 6.98221082773

Median: ma = 3.49330584304
Median: mb = 6.87548787043
Median: mc = 9.23326470419

Inradius: r = 1.19770660128
Circumradius: R = 6.1498572651

Vertex coordinates: A[3.86994850299; 0] B[0; 0] C[7.97655980343; 6.98221082773]
Centroid: CG[3.94883610214; 2.32773694258]
Coordinates of the circumscribed circle: U[1.9354742515; 5.83662416884]
Coordinates of the inscribed circle: I[3.1854742515; 1.19770660128]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 59.541061001° = 59°32'26″ = 2.10224129703 rad
∠ B' = β' = 138.8° = 138°48' = 0.71990756518 rad
∠ C' = γ' = 161.659938999° = 161°39'34″ = 0.32201040315 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 10.6 ; ; b = 8.1 ; ; beta = 41° 12' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 8.1**2 = 10.6**2 + c**2 -2 * 8.1 * c * cos (41° 12') ; ; ; ; c**2 -15.951c +46.75 =0 ; ; p=1; q=-15.9511960686; r=46.75 ; ; D = q**2 - 4pr = 15.951**2 - 4 * 1 * 46.75 = 67.4406560186 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.95 ± sqrt{ 67.44 } }{ 2 } ; ; c_{1,2} = 7.97559803429 ± 4.10611300437 ; ;
c_{1} = 12.0817110387 ; ; c_{2} = 3.86948502993 ; ; ; ; (c -12.0817110387) (c -3.86948502993) = 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 = 10.6 ; ; b = 8.1 ; ; c = 3.87 ; ;

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

p = a+b+c = 10.6+8.1+3.87 = 22.57 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.57 }{ 2 } = 11.28 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.28 * (11.28-10.6)(11.28-8.1)(11.28-3.87) } ; ; T = sqrt{ 182.48 } = 13.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13.51 }{ 10.6 } = 2.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13.51 }{ 8.1 } = 3.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13.51 }{ 3.87 } = 6.98 ; ;

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{ 10.6**2-8.1**2-3.87**2 }{ 2 * 8.1 * 3.87 } ) = 120° 27'34" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.1**2-10.6**2-3.87**2 }{ 2 * 10.6 * 3.87 } ) = 41° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.87**2-10.6**2-8.1**2 }{ 2 * 8.1 * 10.6 } ) = 18° 20'26" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13.51 }{ 11.28 } = 1.2 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.6 }{ 2 * sin 120° 27'34" } = 6.15 ; ;




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