Triangle calculator SSA

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Triangle has two solutions with side c=122.0411264747 and with side c=41.78991441111

#1 Acute scalene triangle.

Sides: a = 100   b = 70   c = 122.0411264747

Area: T = 35009.99968606
Perimeter: p = 292.0411264747
Semiperimeter: s = 146.0210632373

Angle ∠ A = α = 55.02442676241° = 55°1'27″ = 0.96603546385 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 89.97657323759° = 89°58'33″ = 1.57703727769 rad

Height: ha = 709.9999937212
Height: hb = 100.9999910303
Height: hc = 57.35876436351

Median: ma = 86.00660181062
Median: mb = 105.9344107588
Median: mc = 61.04549213674

Inradius: r = 23.96992133171
Circumradius: R = 61.02106378467

Vertex coordinates: A[122.0411264747; 0] B[0; 0] C[81.91552044289; 57.35876436351]
Centroid: CG[67.98554897252; 19.1199214545]
Coordinates of the circumscribed circle: U[61.02106323734; 0.02658452869]
Coordinates of the inscribed circle: I[76.02106323734; 23.96992133171]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.9765732376° = 124°58'33″ = 0.96603546385 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 90.02442676241° = 90°1'27″ = 1.57703727769 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 = 100 ; ; b = 70 ; ; c = 122.04 ; ;

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

p = a+b+c = 100+70+122.04 = 292.04 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 292.04 }{ 2 } = 146.02 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 146.02 * (146.02-100)(146.02-70)(146.02-122.04) } ; ; T = sqrt{ 12249997.8 } = 3500 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3500 }{ 100 } = 70 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3500 }{ 70 } = 100 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3500 }{ 122.04 } = 57.36 ; ;

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{ 100**2-70**2-122.04**2 }{ 2 * 70 * 122.04 } ) = 55° 1'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 70**2-100**2-122.04**2 }{ 2 * 100 * 122.04 } ) = 35° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 122.04**2-100**2-70**2 }{ 2 * 70 * 100 } ) = 89° 58'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3500 }{ 146.02 } = 23.97 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 55° 1'27" } = 61.02 ; ;





#2 Obtuse scalene triangle.

Sides: a = 100   b = 70   c = 41.78991441111

Area: T = 1198.463341787
Perimeter: p = 211.7899144111
Semiperimeter: s = 105.8954572056

Angle ∠ A = α = 124.9765732376° = 124°58'33″ = 2.1811238015 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 20.02442676241° = 20°1'27″ = 0.34994894003 rad

Height: ha = 23.96992683574
Height: hb = 34.24218119391
Height: hc = 57.35876436351

Median: ma = 28.69108745557
Median: mb = 68.1777461692
Median: mc = 83.74661453359

Inradius: r = 11.31875150964
Circumradius: R = 61.02106378467

Vertex coordinates: A[41.78991441111; 0] B[0; 0] C[81.91552044289; 57.35876436351]
Centroid: CG[41.23547828467; 19.1199214545]
Coordinates of the circumscribed circle: U[20.89545720555; 57.33217983482]
Coordinates of the inscribed circle: I[35.89545720555; 11.31875150964]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 55.02442676241° = 55°1'27″ = 2.1811238015 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 159.9765732376° = 159°58'33″ = 0.34994894003 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 70 ; ; beta = 35° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 70**2 = 100**2 + c**2 -2 * 70 * c * cos (35° ) ; ; ; ; c**2 -163.83c +5100 =0 ; ; p=1; q=-163.830408858; r=5100 ; ; D = q**2 - 4pr = 163.83**2 - 4 * 1 * 5100 = 6440.40286651 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 163.83 ± sqrt{ 6440.4 } }{ 2 } ; ; c_{1,2} = 81.9152044289 ± 40.1260603178 ; ; c_{1} = 122.041264747 ; ;
c_{2} = 41.7891441111 ; ; ; ; (c -122.041264747) (c -41.7891441111) = 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 = 100 ; ; b = 70 ; ; c = 41.79 ; ;

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

p = a+b+c = 100+70+41.79 = 211.79 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 211.79 }{ 2 } = 105.89 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 105.89 * (105.89-100)(105.89-70)(105.89-41.79) } ; ; T = sqrt{ 1436314.56 } = 1198.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1198.46 }{ 100 } = 23.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1198.46 }{ 70 } = 34.24 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1198.46 }{ 41.79 } = 57.36 ; ;

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{ 100**2-70**2-41.79**2 }{ 2 * 70 * 41.79 } ) = 124° 58'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 70**2-100**2-41.79**2 }{ 2 * 100 * 41.79 } ) = 35° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 41.79**2-100**2-70**2 }{ 2 * 70 * 100 } ) = 20° 1'27" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1198.46 }{ 105.89 } = 11.32 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 124° 58'33" } = 61.02 ; ;




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