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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 57**2 = 66**2 + c**2 -2 * 57 * c * cos (41° ) ; ; ; ; c**2 -99.622c +1107 =0 ; ; p=1; q=-99.6216645894; 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} = 49.8108322947 ± 37.0691113178 ; ; c_{1} = 86.8799436125 ; ;
c_{2} = 12.7417209769 ; ; ; ; (c -86.8799436125) (c -12.7417209769) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 66**2-57**2-12.74**2 }{ 2 * 57 * 12.74 } ) = 130° 34'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 57**2-66**2-12.74**2 }{ 2 * 66 * 12.74 } ) = 41° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.74**2-66**2-57**2 }{ 2 * 57 * 66 } ) = 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 ; ;




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