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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 130 ; ; b = 90 ; ; beta = 35° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 90**2 = 130**2 + c**2 -2 * 90 * c * cos (35° ) ; ; ; ; c**2 -212.98c +8800 =0 ; ; p=1; q=-212.979531515; 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} = 106.489765758 ± 50.3991092293 ; ; c_{1} = 156.888874987 ; ;
c_{2} = 56.0906565283 ; ; ; ; (c -156.888874987) (c -56.0906565283) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 130**2-90**2-56.09**2 }{ 2 * 90 * 56.09 } ) = 124° 3'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 90**2-130**2-56.09**2 }{ 2 * 130 * 56.09 } ) = 35° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 56.09**2-130**2-90**2 }{ 2 * 90 * 130 } ) = 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 ; ;




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