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?

1. Use Law of Cosines

a = 16.51 ; ; b = 15.25 ; ; beta = 27° 30' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 15.25**2 = 16.51**2 + c**2 -2 * 16.51 * c * cos (27° 30') ; ; ; ; c**2 -29.289c +40.018 =0 ; ; p=1; q=-29.289; r=40.018 ; ; 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} = 14.64454886 ± 13.2077708637 ; ;
c_{1} = 27.8523197237 ; ; c_{2} = 1.43677799626 ; ; ; ; text{ Factored form: } ; ; (c -27.8523197237) (c -1.43677799626) = 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 = 27.85 ; ;

2. 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 ; ;

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 15.25**2+27.85**2-16.51**2 }{ 2 * 15.25 * 27.85 } ) = 29° 59'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 16.51**2+27.85**2-15.25**2 }{ 2 * 16.51 * 27.85 } ) = 27° 30' ; ; gamma = 180° - alpha - beta = 180° - 29° 59'36" - 27° 30' = 122° 30'24" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.25**2+2 * 27.85**2 - 16.51**2 } }{ 2 } = 20.881 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.85**2+2 * 16.51**2 - 15.25**2 } }{ 2 } = 21.588 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.25**2+2 * 16.51**2 - 27.85**2 } }{ 2 } = 7.657 ; ;







#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 - 2ac cos beta ; ; 15.25**2 = 16.51**2 + c**2 -2 * 16.51 * c * cos (27° 30') ; ; ; ; c**2 -29.289c +40.018 =0 ; ; p=1; q=-29.289; r=40.018 ; ; 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} = 14.64454886 ± 13.2077708637 ; ; : Nr. 1
c_{1} = 27.8523197237 ; ; c_{2} = 1.43677799626 ; ; ; ; text{ Factored form: } ; ; (c -27.8523197237) (c -1.43677799626) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 15.25**2+1.44**2-16.51**2 }{ 2 * 15.25 * 1.44 } ) = 150° 24" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 16.51**2+1.44**2-15.25**2 }{ 2 * 16.51 * 1.44 } ) = 27° 30' ; ; gamma = 180° - alpha - beta = 180° - 150° 24" - 27° 30' = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.25**2+2 * 1.44**2 - 16.51**2 } }{ 2 } = 7.012 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.44**2+2 * 16.51**2 - 15.25**2 } }{ 2 } = 8.898 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.25**2+2 * 16.51**2 - 1.44**2 } }{ 2 } = 15.876 ; ;
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