Triangle calculator SSA

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Triangle has two solutions with side c=86.88799436125 and with side c=12.74217209769

#1 Acute scalene triangle.

Sides: a = 66   b = 57   c = 86.88799436125

Area: T = 1880.946625769
Perimeter: p = 209.8879943612
Semiperimeter: s = 104.9439971806

Angle ∠ A = α = 49.43331142069° = 49°25'59″ = 0.86327706024 rad
Angle ∠ B = β = 41° = 0.71655849933 rad
Angle ∠ C = γ = 89.56768857931° = 89°34'1″ = 1.56332370578 rad

Height: ha = 56.99883714452
Height: hb = 65.99881143049
Height: hc = 43.32998959134

Median: ma = 65.64772566148
Median: mb = 71.69224842718
Median: mc = 43.76660696142

Inradius: r = 17.92440209933
Circumradius: R = 43.44112129711

Vertex coordinates: A[86.88799436125; 0] B[0; 0] C[49.81108322947; 43.32998959134]
Centroid: CG[45.56435919691; 14.43332986378]
Coordinates of the circumscribed circle: U[43.44399718062; 0.32883806848]
Coordinates of the inscribed circle: I[47.94399718062; 17.92440209933]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.5676885793° = 130°34'1″ = 0.86327706024 rad
∠ B' = β' = 139° = 0.71655849933 rad
∠ C' = γ' = 90.43331142069° = 90°25'59″ = 1.56332370578 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 66 ; ; b = 57 ; ; beta = 41° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 57**2 = 66**2 + c**2 -2 * 66 * c * cos (41° ) ; ; ; ; c**2 -99.622c +1107 =0 ; ; p=1; q=-99.622; r=1107 ; ; D = q**2 - 4pr = 99.622**2 - 4 * 1 * 1107 = 5496.47605556 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 99.62 ± sqrt{ 5496.48 } }{ 2 } ; ; c_{1,2} = 49.81083229 ± 37.0691113178 ; ; c_{1} = 86.8799436078 ; ;
c_{2} = 12.7417209722 ; ; ; ; text{ Factored form: } ; ; (c -86.8799436078) (c -12.7417209722) = 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 = 66 ; ; b = 57 ; ; c = 86.88 ; ;

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

p = a+b+c = 66+57+86.88 = 209.88 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 209.88 }{ 2 } = 104.94 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 104.94 * (104.94-66)(104.94-57)(104.94-86.88) } ; ; T = sqrt{ 3537958.82 } = 1880.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1880.95 }{ 66 } = 57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1880.95 }{ 57 } = 66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1880.95 }{ 86.88 } = 43.3 ; ;

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{ 57**2+86.88**2-66**2 }{ 2 * 57 * 86.88 } ) = 49° 25'59" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 66**2+86.88**2-57**2 }{ 2 * 66 * 86.88 } ) = 41° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 66**2+57**2-86.88**2 }{ 2 * 66 * 57 } ) = 89° 34'1" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1880.95 }{ 104.94 } = 17.92 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 66 }{ 2 * sin 49° 25'59" } = 43.44 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 57**2+2 * 86.88**2 - 66**2 } }{ 2 } = 65.647 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 86.88**2+2 * 66**2 - 57**2 } }{ 2 } = 71.692 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 57**2+2 * 66**2 - 86.88**2 } }{ 2 } = 43.766 ; ;







#2 Obtuse scalene triangle.

Sides: a = 66   b = 57   c = 12.74217209769

Area: T = 275.8587596029
Perimeter: p = 135.7421720977
Semiperimeter: s = 67.87108604885

Angle ∠ A = α = 130.5676885793° = 130°34'1″ = 2.27988220512 rad
Angle ∠ B = β = 41° = 0.71655849933 rad
Angle ∠ C = γ = 8.43331142069° = 8°25'59″ = 0.14771856091 rad

Height: ha = 8.35993210918
Height: hb = 9.67992138957
Height: hc = 43.32998959134

Median: ma = 24.83329564637
Median: mb = 38.03884769244
Median: mc = 61.33444286403

Inradius: r = 4.06444481894
Circumradius: R = 43.44112129711

Vertex coordinates: A[12.74217209769; 0] B[0; 0] C[49.81108322947; 43.32998959134]
Centroid: CG[20.85108510905; 14.43332986378]
Coordinates of the circumscribed circle: U[6.37108604885; 42.97215152286]
Coordinates of the inscribed circle: I[10.87108604885; 4.06444481894]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 49.43331142069° = 49°25'59″ = 2.27988220512 rad
∠ B' = β' = 139° = 0.71655849933 rad
∠ C' = γ' = 171.5676885793° = 171°34'1″ = 0.14771856091 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 66 ; ; b = 57 ; ; beta = 41° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 57**2 = 66**2 + c**2 -2 * 66 * c * cos (41° ) ; ; ; ; c**2 -99.622c +1107 =0 ; ; p=1; q=-99.622; r=1107 ; ; D = q**2 - 4pr = 99.622**2 - 4 * 1 * 1107 = 5496.47605556 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 99.62 ± sqrt{ 5496.48 } }{ 2 } ; ; c_{1,2} = 49.81083229 ± 37.0691113178 ; ; c_{1} = 86.8799436078 ; ; : Nr. 1
c_{2} = 12.7417209722 ; ; ; ; text{ Factored form: } ; ; (c -86.8799436078) (c -12.7417209722) = 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 = 66 ; ; b = 57 ; ; c = 12.74 ; ;

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

p = a+b+c = 66+57+12.74 = 135.74 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 135.74 }{ 2 } = 67.87 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 67.87 * (67.87-66)(67.87-57)(67.87-12.74) } ; ; T = sqrt{ 76097.41 } = 275.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 275.86 }{ 66 } = 8.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 275.86 }{ 57 } = 9.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 275.86 }{ 12.74 } = 43.3 ; ;

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{ 57**2+12.74**2-66**2 }{ 2 * 57 * 12.74 } ) = 130° 34'1" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 66**2+12.74**2-57**2 }{ 2 * 66 * 12.74 } ) = 41° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 66**2+57**2-12.74**2 }{ 2 * 66 * 57 } ) = 8° 25'59" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 275.86 }{ 67.87 } = 4.06 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 66 }{ 2 * sin 130° 34'1" } = 43.44 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 57**2+2 * 12.74**2 - 66**2 } }{ 2 } = 24.833 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.74**2+2 * 66**2 - 57**2 } }{ 2 } = 38.038 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 57**2+2 * 66**2 - 12.74**2 } }{ 2 } = 61.334 ; ;
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