Triangle calculator SSA

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Triangle has two solutions with side c=43.94443174473 and with side c=11.62883521894

#1 Acute scalene triangle.

Sides: a = 40   b = 33   c = 43.94443174473

Area: T = 632.2187930772
Perimeter: p = 116.9444317447
Semiperimeter: s = 58.47221587237

Angle ∠ A = α = 60.68333509748° = 60°41' = 1.05991242757 rad
Angle ∠ B = β = 46° = 0.80328514559 rad
Angle ∠ C = γ = 73.31766490252° = 73°19' = 1.2879616922 rad

Height: ha = 31.61108965386
Height: hb = 38.31662382286
Height: hc = 28.77435920135

Median: ma = 33.3177435645
Median: mb = 38.64332596704
Median: mc = 29.35551399421

Inradius: r = 10.81222898927
Circumradius: R = 22.93876992518

Vertex coordinates: A[43.94443174473; 0] B[0; 0] C[27.78663348184; 28.77435920135]
Centroid: CG[23.91102174219; 9.59111973378]
Coordinates of the circumscribed circle: U[21.97221587237; 6.58550047826]
Coordinates of the inscribed circle: I[25.47221587237; 10.81222898927]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 119.3176649025° = 119°19' = 1.05991242757 rad
∠ B' = β' = 134° = 0.80328514559 rad
∠ C' = γ' = 106.6833350975° = 106°41' = 1.2879616922 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 40 ; ; b = 33 ; ; beta = 46° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 33**2 = 40**2 + c**2 -2 * 40 * c * cos (46° ) ; ; ; ; c**2 -55.573c +511 =0 ; ; p=1; q=-55.573; r=511 ; ; D = q**2 - 4pr = 55.573**2 - 4 * 1 * 511 = 1044.32161055 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 55.57 ± sqrt{ 1044.32 } }{ 2 } ; ; c_{1,2} = 27.78633482 ± 16.157982629 ; ; c_{1} = 43.944317449 ; ; c_{2} = 11.628352191 ; ; ; ; text{ Factored form: } ; ; (c -43.944317449) (c -11.628352191) = 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 = 40 ; ; b = 33 ; ; c = 43.94 ; ;

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

p = a+b+c = 40+33+43.94 = 116.94 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 116.94 }{ 2 } = 58.47 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 58.47 * (58.47-40)(58.47-33)(58.47-43.94) } ; ; T = sqrt{ 399699.51 } = 632.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 632.22 }{ 40 } = 31.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 632.22 }{ 33 } = 38.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 632.22 }{ 43.94 } = 28.77 ; ;

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{ 33**2+43.94**2-40**2 }{ 2 * 33 * 43.94 } ) = 60° 41' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 40**2+43.94**2-33**2 }{ 2 * 40 * 43.94 } ) = 46° ; ; gamma = 180° - alpha - beta = 180° - 60° 41' - 46° = 73° 19' ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 632.22 }{ 58.47 } = 10.81 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 40 }{ 2 * sin 60° 41' } = 22.94 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 43.94**2 - 40**2 } }{ 2 } = 33.317 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 43.94**2+2 * 40**2 - 33**2 } }{ 2 } = 38.643 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 40**2 - 43.94**2 } }{ 2 } = 29.355 ; ;







#2 Obtuse scalene triangle.

Sides: a = 40   b = 33   c = 11.62883521894

Area: T = 167.2954730844
Perimeter: p = 84.62883521894
Semiperimeter: s = 42.31441760947

Angle ∠ A = α = 119.3176649025° = 119°19' = 2.08224683779 rad
Angle ∠ B = β = 46° = 0.80328514559 rad
Angle ∠ C = γ = 14.68333509748° = 14°41' = 0.25662728197 rad

Height: ha = 8.36547365422
Height: hb = 10.13990745966
Height: hc = 28.77435920135

Median: ma = 14.5643972237
Median: mb = 24.43999853959
Median: mc = 36.20435268495

Inradius: r = 3.95436331859
Circumradius: R = 22.93876992518

Vertex coordinates: A[11.62883521894; 0] B[0; 0] C[27.78663348184; 28.77435920135]
Centroid: CG[13.13882290026; 9.59111973378]
Coordinates of the circumscribed circle: U[5.81441760947; 22.18985872309]
Coordinates of the inscribed circle: I[9.31441760947; 3.95436331859]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 60.68333509748° = 60°41' = 2.08224683779 rad
∠ B' = β' = 134° = 0.80328514559 rad
∠ C' = γ' = 165.3176649025° = 165°19' = 0.25662728197 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 40 ; ; b = 33 ; ; beta = 46° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 33**2 = 40**2 + c**2 -2 * 40 * c * cos (46° ) ; ; ; ; c**2 -55.573c +511 =0 ; ; p=1; q=-55.573; r=511 ; ; D = q**2 - 4pr = 55.573**2 - 4 * 1 * 511 = 1044.32161055 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 55.57 ± sqrt{ 1044.32 } }{ 2 } ; ; c_{1,2} = 27.78633482 ± 16.157982629 ; ; c_{1} = 43.944317449 ; ; c_{2} = 11.628352191 ; ; ; ; text{ Factored form: } ; ; (c -43.944317449) (c -11.628352191) = 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 = 40 ; ; b = 33 ; ; c = 11.63 ; ;

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

p = a+b+c = 40+33+11.63 = 84.63 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 84.63 }{ 2 } = 42.31 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42.31 * (42.31-40)(42.31-33)(42.31-11.63) } ; ; T = sqrt{ 27987.53 } = 167.29 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 167.29 }{ 40 } = 8.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 167.29 }{ 33 } = 10.14 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 167.29 }{ 11.63 } = 28.77 ; ;

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{ 33**2+11.63**2-40**2 }{ 2 * 33 * 11.63 } ) = 119° 19' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 40**2+11.63**2-33**2 }{ 2 * 40 * 11.63 } ) = 46° ; ; gamma = 180° - alpha - beta = 180° - 119° 19' - 46° = 14° 41' ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 167.29 }{ 42.31 } = 3.95 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 40 }{ 2 * sin 119° 19' } = 22.94 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 11.63**2 - 40**2 } }{ 2 } = 14.564 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.63**2+2 * 40**2 - 33**2 } }{ 2 } = 24.4 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 40**2 - 11.63**2 } }{ 2 } = 36.204 ; ;
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