Triangle calculator SSA

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Triangle has two solutions with side c=27.85223197195 and with side c=1.4376777992

#1 Obtuse scalene triangle.

Sides: a = 16.51   b = 15.25   c = 27.85223197195

Area: T = 106.1665656398
Perimeter: p = 59.61223197195
Semiperimeter: s = 29.80661598598

Angle ∠ A = α = 29.99333608679° = 29°59'36″ = 0.52334829009 rad
Angle ∠ B = β = 27.5° = 27°30' = 0.48799655443 rad
Angle ∠ C = γ = 122.5076639132° = 122°30'24″ = 2.13881442084 rad

Height: ha = 12.86107700059
Height: hb = 13.92333647736
Height: hc = 7.62334696045

Median: ma = 20.88109023243
Median: mb = 21.58876187172
Median: mc = 7.65772430783

Inradius: r = 3.56218696571
Circumradius: R = 16.51333143478

Vertex coordinates: A[27.85223197195; 0] B[0; 0] C[14.64545488558; 7.62334696045]
Centroid: CG[14.16656228584; 2.54111565348]
Coordinates of the circumscribed circle: U[13.92661598598; -8.87442110809]
Coordinates of the inscribed circle: I[14.55661598598; 3.56218696571]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0076639132° = 150°24″ = 0.52334829009 rad
∠ B' = β' = 152.5° = 152°30' = 0.48799655443 rad
∠ C' = γ' = 57.49333608679° = 57°29'36″ = 2.13881442084 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 = 16.51 ; ; b = 15.25 ; ; c = 27.85 ; ;

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

p = a+b+c = 16.51+15.25+27.85 = 59.61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59.61 }{ 2 } = 29.81 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.81 * (29.81-16.51)(29.81-15.25)(29.81-27.85) } ; ; T = sqrt{ 11271.15 } = 106.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 106.17 }{ 16.51 } = 12.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 106.17 }{ 15.25 } = 13.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 106.17 }{ 27.85 } = 7.62 ; ;

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{ 16.51**2-15.25**2-27.85**2 }{ 2 * 15.25 * 27.85 } ) = 29° 59'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15.25**2-16.51**2-27.85**2 }{ 2 * 16.51 * 27.85 } ) = 27° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27.85**2-16.51**2-15.25**2 }{ 2 * 15.25 * 16.51 } ) = 122° 30'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 106.17 }{ 29.81 } = 3.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16.51 }{ 2 * sin 29° 59'36" } = 16.51 ; ;





#2 Obtuse scalene triangle.

Sides: a = 16.51   b = 15.25   c = 1.4376777992

Area: T = 5.47766166753
Perimeter: p = 33.1976777992
Semiperimeter: s = 16.5988388996

Angle ∠ A = α = 150.0076639132° = 150°24″ = 2.61881097527 rad
Angle ∠ B = β = 27.5° = 27°30' = 0.48799655443 rad
Angle ∠ C = γ = 2.49333608679° = 2°29'36″ = 0.04435173566 rad

Height: ha = 0.66334302453
Height: hb = 0.71882448099
Height: hc = 7.62334696045

Median: ma = 7.01220175769
Median: mb = 8.89884038175
Median: mc = 15.8766246951

Inradius: r = 0.33299486882
Circumradius: R = 16.51333143478

Vertex coordinates: A[1.4376777992; 0] B[0; 0] C[14.64545488558; 7.62334696045]
Centroid: CG[5.36604422826; 2.54111565348]
Coordinates of the circumscribed circle: U[0.7188388996; 16.49876806854]
Coordinates of the inscribed circle: I[1.3488388996; 0.33299486882]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.99333608679° = 29°59'36″ = 2.61881097527 rad
∠ B' = β' = 152.5° = 152°30' = 0.48799655443 rad
∠ C' = γ' = 177.5076639132° = 177°30'24″ = 0.04435173566 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 16.51 ; ; b = 15.25 ; ; beta = 27° 30' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 15.25**2 = 16.51**2 + c**2 -2 * 15.25 * c * cos (27° 30') ; ; ; ; c**2 -29.289c +40.018 =0 ; ; p=1; q=-29.2890977115; r=40.0176 ; ; D = q**2 - 4pr = 29.289**2 - 4 * 1 * 40.018 = 697.780844756 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 29.29 ± sqrt{ 697.78 } }{ 2 } ; ; c_{1,2} = 14.6445488558 ± 13.2077708637 ; ;
c_{1} = 27.8523197195 ; ; c_{2} = 1.43677799203 ; ; ; ; (c -27.8523197195) (c -1.43677799203) = 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 = 16.51 ; ; b = 15.25 ; ; c = 1.44 ; ;

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

p = a+b+c = 16.51+15.25+1.44 = 33.2 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33.2 }{ 2 } = 16.6 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.6 * (16.6-16.51)(16.6-15.25)(16.6-1.44) } ; ; T = sqrt{ 29.99 } = 5.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5.48 }{ 16.51 } = 0.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5.48 }{ 15.25 } = 0.72 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5.48 }{ 1.44 } = 7.62 ; ;

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{ 16.51**2-15.25**2-1.44**2 }{ 2 * 15.25 * 1.44 } ) = 150° 24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15.25**2-16.51**2-1.44**2 }{ 2 * 16.51 * 1.44 } ) = 27° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.44**2-16.51**2-15.25**2 }{ 2 * 15.25 * 16.51 } ) = 2° 29'36" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5.48 }{ 16.6 } = 0.33 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16.51 }{ 2 * sin 150° 24" } = 16.51 ; ;




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