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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 43**2 = 49**2 + c**2 -2 * 43 * c * cos (19° ) ; ; ; ; c**2 -92.661c +552 =0 ; ; p=1; q=-92.6608204087; 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} = 46.3304102044 ± 39.9312773363 ; ; c_{1} = 86.2616875406 ; ;
c_{2} = 6.39913286811 ; ; ; ; (c -86.2616875406) (c -6.39913286811) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 49**2-43**2-6.4**2 }{ 2 * 43 * 6.4 } ) = 158° 13'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 43**2-49**2-6.4**2 }{ 2 * 49 * 6.4 } ) = 19° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.4**2-49**2-43**2 }{ 2 * 43 * 49 } ) = 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 ; ;




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