Triangle calculator SSA

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Triangle has two solutions with side c=156.8898874987 and with side c=56.09106565283

#1 Acute scalene triangle.

Sides: a = 130   b = 90   c = 156.8898874987

Area: T = 5849.205451817
Perimeter: p = 376.8898874987
Semiperimeter: s = 188.4444437493

Angle ∠ A = α = 55.94548867294° = 55°56'42″ = 0.97664224731 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 89.05551132706° = 89°3'18″ = 1.55443049423 rad

Height: ha = 89.98877618181
Height: hb = 129.9822322626
Height: hc = 74.56549367256

Median: ma = 110.1465628816
Median: mb = 136.8655114428
Median: mc = 79.66547364041

Inradius: r = 31.03994119135
Circumradius: R = 78.45551058029

Vertex coordinates: A[156.8898874987; 0] B[0; 0] C[106.4989765758; 74.56549367256]
Centroid: CG[87.79328802481; 24.85549789085]
Coordinates of the circumscribed circle: U[78.44444374934; 1.29437746692]
Coordinates of the inscribed circle: I[98.44444374934; 31.03994119135]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.0555113271° = 124°3'18″ = 0.97664224731 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 90.94548867294° = 90°56'42″ = 1.55443049423 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 130 ; ; b = 90 ; ; beta = 35° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 130**2 + c**2 -2 * 130 * c * cos (35° ) ; ; ; ; c**2 -212.98c +8800 =0 ; ; p=1; q=-212.98; r=8800 ; ; D = q**2 - 4pr = 212.98**2 - 4 * 1 * 8800 = 10160.2808444 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 212.98 ± sqrt{ 10160.28 } }{ 2 } ; ; c_{1,2} = 106.48976576 ± 50.3991092293 ; ; c_{1} = 156.888874989 ; ; c_{2} = 56.0906565307 ; ; ; ; text{ Factored form: } ; ; (c -156.888874989) (c -56.0906565307) = 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 = 130 ; ; b = 90 ; ; c = 156.89 ; ;

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

p = a+b+c = 130+90+156.89 = 376.89 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 376.89 }{ 2 } = 188.44 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 188.44 * (188.44-130)(188.44-90)(188.44-156.89) } ; ; T = sqrt{ 34213193.5 } = 5849.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5849.2 }{ 130 } = 89.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5849.2 }{ 90 } = 129.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5849.2 }{ 156.89 } = 74.56 ; ;

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{ 90**2+156.89**2-130**2 }{ 2 * 90 * 156.89 } ) = 55° 56'42" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 130**2+156.89**2-90**2 }{ 2 * 130 * 156.89 } ) = 35° ; ; gamma = 180° - alpha - beta = 180° - 55° 56'42" - 35° = 89° 3'18" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5849.2 }{ 188.44 } = 31.04 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 130 }{ 2 * sin 55° 56'42" } = 78.46 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 156.89**2 - 130**2 } }{ 2 } = 110.146 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 156.89**2+2 * 130**2 - 90**2 } }{ 2 } = 136.865 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 130**2 - 156.89**2 } }{ 2 } = 79.665 ; ;







#2 Obtuse scalene triangle.

Sides: a = 130   b = 90   c = 56.09106565283

Area: T = 2091.198812747
Perimeter: p = 276.0910656528
Semiperimeter: s = 138.0455328264

Angle ∠ A = α = 124.0555113271° = 124°3'18″ = 2.16551701805 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 20.94548867294° = 20°56'42″ = 0.36655572349 rad

Height: ha = 32.17222788841
Height: hb = 46.47110694993
Height: hc = 74.56549367256

Median: ma = 37.3910919685
Median: mb = 89.4321990221
Median: mc = 108.2298737231

Inradius: r = 15.1498633813
Circumradius: R = 78.45551058029

Vertex coordinates: A[56.09106565283; 0] B[0; 0] C[106.4989765758; 74.56549367256]
Centroid: CG[54.19334740953; 24.85549789085]
Coordinates of the circumscribed circle: U[28.04553282642; 73.27111620565]
Coordinates of the inscribed circle: I[48.04553282642; 15.1498633813]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 55.94548867294° = 55°56'42″ = 2.16551701805 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 159.0555113271° = 159°3'18″ = 0.36655572349 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 130 ; ; b = 90 ; ; beta = 35° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 130**2 + c**2 -2 * 130 * c * cos (35° ) ; ; ; ; c**2 -212.98c +8800 =0 ; ; p=1; q=-212.98; r=8800 ; ; D = q**2 - 4pr = 212.98**2 - 4 * 1 * 8800 = 10160.2808444 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 212.98 ± sqrt{ 10160.28 } }{ 2 } ; ; c_{1,2} = 106.48976576 ± 50.3991092293 ; ; c_{1} = 156.888874989 ; ; c_{2} = 56.0906565307 ; ; ; ; text{ Factored form: } ; ; (c -156.888874989) (c -56.0906565307) = 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 = 130 ; ; b = 90 ; ; c = 56.09 ; ;

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

p = a+b+c = 130+90+56.09 = 276.09 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 276.09 }{ 2 } = 138.05 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 138.05 * (138.05-130)(138.05-90)(138.05-56.09) } ; ; T = sqrt{ 4373109.61 } = 2091.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2091.2 }{ 130 } = 32.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2091.2 }{ 90 } = 46.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2091.2 }{ 56.09 } = 74.56 ; ;

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{ 90**2+56.09**2-130**2 }{ 2 * 90 * 56.09 } ) = 124° 3'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 130**2+56.09**2-90**2 }{ 2 * 130 * 56.09 } ) = 35° ; ; gamma = 180° - alpha - beta = 180° - 124° 3'18" - 35° = 20° 56'42" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2091.2 }{ 138.05 } = 15.15 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 130 }{ 2 * sin 124° 3'18" } = 78.46 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 56.09**2 - 130**2 } }{ 2 } = 37.391 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 56.09**2+2 * 130**2 - 90**2 } }{ 2 } = 89.432 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 130**2 - 56.09**2 } }{ 2 } = 108.229 ; ;
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