Triangle calculator SSA

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Triangle has two solutions with side c=86.26216875406 and with side c=6.39991328681

#1 Obtuse scalene triangle.

Sides: a = 49   b = 43   c = 86.26216875406

Area: T = 688.0599431118
Perimeter: p = 178.2621687541
Semiperimeter: s = 89.13108437703

Angle ∠ A = α = 21.77770728969° = 21°46'37″ = 0.38800816235 rad
Angle ∠ B = β = 19° = 0.33216125579 rad
Angle ∠ C = γ = 139.2232927103° = 139°13'23″ = 2.43298984722 rad

Height: ha = 28.0844058413
Height: hb = 32.0032764238
Height: hc = 15.95328395684

Median: ma = 63.59986585446
Median: mb = 66.77441669261
Median: mc = 16.2710535199

Inradius: r = 7.72196557557
Circumradius: R = 66.03883999653

Vertex coordinates: A[86.26216875406; 0] B[0; 0] C[46.33304102044; 15.95328395684]
Centroid: CG[44.1977365915; 5.31876131895]
Coordinates of the circumscribed circle: U[43.13108437703; -50.00880052155]
Coordinates of the inscribed circle: I[46.13108437703; 7.72196557557]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.2232927103° = 158°13'23″ = 0.38800816235 rad
∠ B' = β' = 161° = 0.33216125579 rad
∠ C' = γ' = 40.77770728969° = 40°46'37″ = 2.43298984722 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 49 ; ; b = 43 ; ; beta = 19° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 43**2 = 49**2 + c**2 -2 * 49 * c * cos (19° ) ; ; ; ; c**2 -92.661c +552 =0 ; ; p=1; q=-92.661; r=552 ; ; D = q**2 - 4pr = 92.661**2 - 4 * 1 * 552 = 6378.02763882 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 92.66 ± sqrt{ 6378.03 } }{ 2 } ; ; c_{1,2} = 46.3304102 ± 39.9312773363 ; ; c_{1} = 86.2616875363 ; ;
c_{2} = 6.39913286375 ; ; ; ; text{ Factored form: } ; ; (c -86.2616875363) (c -6.39913286375) = 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 = 49 ; ; b = 43 ; ; c = 86.26 ; ;

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

p = a+b+c = 49+43+86.26 = 178.26 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 178.26 }{ 2 } = 89.13 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 89.13 * (89.13-49)(89.13-43)(89.13-86.26) } ; ; T = sqrt{ 473425.78 } = 688.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 688.06 }{ 49 } = 28.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 688.06 }{ 43 } = 32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 688.06 }{ 86.26 } = 15.95 ; ;

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{ 43**2+86.26**2-49**2 }{ 2 * 43 * 86.26 } ) = 21° 46'37" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 49**2+86.26**2-43**2 }{ 2 * 49 * 86.26 } ) = 19° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 49**2+43**2-86.26**2 }{ 2 * 49 * 43 } ) = 139° 13'23" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 688.06 }{ 89.13 } = 7.72 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 49 }{ 2 * sin 21° 46'37" } = 66.04 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 43**2+2 * 86.26**2 - 49**2 } }{ 2 } = 63.599 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 86.26**2+2 * 49**2 - 43**2 } }{ 2 } = 66.774 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 43**2+2 * 49**2 - 86.26**2 } }{ 2 } = 16.271 ; ;







#2 Obtuse scalene triangle.

Sides: a = 49   b = 43   c = 6.39991328681

Area: T = 51.04221700109
Perimeter: p = 98.39991328681
Semiperimeter: s = 49.21995664341

Angle ∠ A = α = 158.2232927103° = 158°13'23″ = 2.76215110301 rad
Angle ∠ B = β = 19° = 0.33216125579 rad
Angle ∠ C = γ = 2.77770728969° = 2°46'37″ = 0.04884690656 rad

Height: ha = 2.0833353878
Height: hb = 2.37440544191
Height: hc = 15.95328395684

Median: ma = 18.56767566024
Median: mb = 27.54549532716
Median: mc = 45.98765499318

Inradius: r = 1.03774516222
Circumradius: R = 66.03883999653

Vertex coordinates: A[6.39991328681; 0] B[0; 0] C[46.33304102044; 15.95328395684]
Centroid: CG[17.57765143575; 5.31876131895]
Coordinates of the circumscribed circle: U[3.21995664341; 65.96108447839]
Coordinates of the inscribed circle: I[6.21995664341; 1.03774516222]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 21.77770728969° = 21°46'37″ = 2.76215110301 rad
∠ B' = β' = 161° = 0.33216125579 rad
∠ C' = γ' = 177.2232927103° = 177°13'23″ = 0.04884690656 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 49 ; ; b = 43 ; ; beta = 19° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 43**2 = 49**2 + c**2 -2 * 49 * c * cos (19° ) ; ; ; ; c**2 -92.661c +552 =0 ; ; p=1; q=-92.661; r=552 ; ; D = q**2 - 4pr = 92.661**2 - 4 * 1 * 552 = 6378.02763882 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 92.66 ± sqrt{ 6378.03 } }{ 2 } ; ; c_{1,2} = 46.3304102 ± 39.9312773363 ; ; c_{1} = 86.2616875363 ; ; : Nr. 1
c_{2} = 6.39913286375 ; ; ; ; text{ Factored form: } ; ; (c -86.2616875363) (c -6.39913286375) = 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 = 49 ; ; b = 43 ; ; c = 6.4 ; ;

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

p = a+b+c = 49+43+6.4 = 98.4 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 98.4 }{ 2 } = 49.2 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 49.2 * (49.2-49)(49.2-43)(49.2-6.4) } ; ; T = sqrt{ 2605.3 } = 51.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 51.04 }{ 49 } = 2.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 51.04 }{ 43 } = 2.37 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 51.04 }{ 6.4 } = 15.95 ; ;

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{ 43**2+6.4**2-49**2 }{ 2 * 43 * 6.4 } ) = 158° 13'23" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 49**2+6.4**2-43**2 }{ 2 * 49 * 6.4 } ) = 19° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 49**2+43**2-6.4**2 }{ 2 * 49 * 43 } ) = 2° 46'37" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 51.04 }{ 49.2 } = 1.04 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 49 }{ 2 * sin 158° 13'23" } = 66.04 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 43**2+2 * 6.4**2 - 49**2 } }{ 2 } = 18.567 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.4**2+2 * 49**2 - 43**2 } }{ 2 } = 27.545 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 43**2+2 * 49**2 - 6.4**2 } }{ 2 } = 45.987 ; ;
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