Triangle calculator SSA

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Triangle has two solutions with side c=454.4365580922 and with side c=36.74988830125

#1 Acute scalene triangle.

Sides: a = 840   b = 830   c = 454.4365580922

Area: T = 182523.1411072
Perimeter: p = 2124.436558092
Semiperimeter: s = 1062.218779046

Angle ∠ A = α = 75.42766930667° = 75°25'36″ = 1.31664441379 rad
Angle ∠ B = β = 73° = 1.2744090354 rad
Angle ∠ C = γ = 31.57333069333° = 31°34'24″ = 0.55110581617 rad

Height: ha = 434.5798907314
Height: hb = 439.8154797764
Height: hc = 803.2965995009

Median: ma = 520.8770280016
Median: mb = 532.7587776671
Median: mc = 803.5066114288

Inradius: r = 171.832212587
Circumradius: R = 433.9622078942

Vertex coordinates: A[454.4365580922; 0] B[0; 0] C[245.5922231967; 803.2965995009]
Centroid: CG[233.3432604296; 267.765533167]
Coordinates of the circumscribed circle: U[227.2187790461; 369.7233087808]
Coordinates of the inscribed circle: I[232.2187790461; 171.832212587]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 104.5733306933° = 104°34'24″ = 1.31664441379 rad
∠ B' = β' = 107° = 1.2744090354 rad
∠ C' = γ' = 148.4276693067° = 148°25'36″ = 0.55110581617 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 = 840 ; ; b = 830 ; ; c = 454.44 ; ;

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

p = a+b+c = 840+830+454.44 = 2124.44 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2124.44 }{ 2 } = 1062.22 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1062.22 * (1062.22-840)(1062.22-830)(1062.22-454.44) } ; ; T = sqrt{ 33314697026.8 } = 182523.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 182523.14 }{ 840 } = 434.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 182523.14 }{ 830 } = 439.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 182523.14 }{ 454.44 } = 803.3 ; ;

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{ 840**2-830**2-454.44**2 }{ 2 * 830 * 454.44 } ) = 75° 25'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 830**2-840**2-454.44**2 }{ 2 * 840 * 454.44 } ) = 73° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 454.44**2-840**2-830**2 }{ 2 * 830 * 840 } ) = 31° 34'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 182523.14 }{ 1062.22 } = 171.83 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 840 }{ 2 * sin 75° 25'36" } = 433.96 ; ;





#2 Obtuse scalene triangle.

Sides: a = 840   b = 830   c = 36.74988830125

Area: T = 14760.11552725
Perimeter: p = 1706.749888301
Semiperimeter: s = 853.3744441506

Angle ∠ A = α = 104.5733306933° = 104°34'24″ = 1.82551485157 rad
Angle ∠ B = β = 73° = 1.2744090354 rad
Angle ∠ C = γ = 2.42766930667° = 2°25'36″ = 0.04223537839 rad

Height: ha = 35.14331316012
Height: hb = 35.56765428253
Height: hc = 803.2965995009

Median: ma = 410.7621780356
Median: mb = 425.735494125
Median: mc = 834.8132781346

Inradius: r = 17.29661768651
Circumradius: R = 433.9622078942

Vertex coordinates: A[36.74988830125; 0] B[0; 0] C[245.5922231967; 803.2965995009]
Centroid: CG[94.11437049932; 267.765533167]
Coordinates of the circumscribed circle: U[18.37444415062; 433.5732907202]
Coordinates of the inscribed circle: I[23.37444415062; 17.29661768651]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 75.42766930667° = 75°25'36″ = 1.82551485157 rad
∠ B' = β' = 107° = 1.2744090354 rad
∠ C' = γ' = 177.5733306933° = 177°34'24″ = 0.04223537839 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 840 ; ; b = 830 ; ; beta = 73° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 830**2 = 840**2 + c**2 -2 * 830 * c * cos (73° ) ; ; ; ; c**2 -491.184c +16700 =0 ; ; p=1; q=-491.184463934; r=16700 ; ; D = q**2 - 4pr = 491.184**2 - 4 * 1 * 16700 = 174462.17761 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 491.18 ± sqrt{ 174462.18 } }{ 2 } ; ; c_{1,2} = 245.592231967 ± 208.843348955 ; ;
c_{1} = 454.435580922 ; ; c_{2} = 36.7488830125 ; ; ; ; (c -454.435580922) (c -36.7488830125) = 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 = 840 ; ; b = 830 ; ; c = 36.75 ; ;

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

p = a+b+c = 840+830+36.75 = 1706.75 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1706.75 }{ 2 } = 853.37 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 853.37 * (853.37-840)(853.37-830)(853.37-36.75) } ; ; T = sqrt{ 217861002.86 } = 14760.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14760.12 }{ 840 } = 35.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14760.12 }{ 830 } = 35.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14760.12 }{ 36.75 } = 803.3 ; ;

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{ 840**2-830**2-36.75**2 }{ 2 * 830 * 36.75 } ) = 104° 34'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 830**2-840**2-36.75**2 }{ 2 * 840 * 36.75 } ) = 73° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 36.75**2-840**2-830**2 }{ 2 * 830 * 840 } ) = 2° 25'36" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14760.12 }{ 853.37 } = 17.3 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 840 }{ 2 * sin 104° 34'24" } = 433.96 ; ;




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