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?

1. Use Law of Cosines

a = 840 ; ; b = 830 ; ; beta = 73° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 830**2 = 840**2 + c**2 -2 * 840 * c * cos (73° ) ; ; ; ; c**2 -491.184c +16700 =0 ; ; p=1; q=-491.184; 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} = 245.59223197 ± 208.843348955 ; ; c_{1} = 454.435580922 ; ; c_{2} = 36.7488830125 ; ; ; ; text{ Factored form: } ; ; (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 = 454.44 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 830**2+454.44**2-840**2 }{ 2 * 830 * 454.44 } ) = 75° 25'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 840**2+454.44**2-830**2 }{ 2 * 840 * 454.44 } ) = 73° ; ;
 gamma = 180° - alpha - beta = 180° - 75° 25'36" - 73° = 31° 34'24" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 830**2+2 * 454.44**2 - 840**2 } }{ 2 } = 520.87 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 454.44**2+2 * 840**2 - 830**2 } }{ 2 } = 532.758 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 830**2+2 * 840**2 - 454.44**2 } }{ 2 } = 803.506 ; ;



#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 840 ; ; b = 830 ; ; beta = 73° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 830**2 = 840**2 + c**2 -2 * 840 * c * cos (73° ) ; ; ; ; c**2 -491.184c +16700 =0 ; ; p=1; q=-491.184; 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 } ; ; : Nr. 1
c_{1,2} = 245.59223197 ± 208.843348955 ; ; c_{1} = 454.435580922 ; ; c_{2} = 36.7488830125 ; ; ; ; text{ Factored form: } ; ; (c -454.435580922) (c -36.7488830125) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 830**2+36.75**2-840**2 }{ 2 * 830 * 36.75 } ) = 104° 34'24" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 840**2+36.75**2-830**2 }{ 2 * 840 * 36.75 } ) = 73° ; ;
 gamma = 180° - alpha - beta = 180° - 104° 34'24" - 73° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 830**2+2 * 36.75**2 - 840**2 } }{ 2 } = 410.762 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 36.75**2+2 * 840**2 - 830**2 } }{ 2 } = 425.735 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 830**2+2 * 840**2 - 36.75**2 } }{ 2 } = 834.813 ; ;
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