Triangle calculator SSA

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Triangle has two solutions with side c=93.19771127024 and with side c=70.81876445452

#1 Acute scalene triangle.

Sides: a = 125   b = 95   c = 93.19771127024

Area: T = 4396.047711285
Perimeter: p = 313.1977112702
Semiperimeter: s = 156.5998556351

Angle ∠ A = α = 83.23656175722° = 83°14'8″ = 1.45327355816 rad
Angle ∠ B = β = 49° = 0.85552113335 rad
Angle ∠ C = γ = 47.76443824278° = 47°45'52″ = 0.83436457385 rad

Height: ha = 70.33767538056
Height: hb = 92.54883602705
Height: hc = 94.33986975278

Median: ma = 70.35498465388
Median: mb = 99.49442255009
Median: mc = 100.765494701

Inradius: r = 28.07220794321
Circumradius: R = 62.93881171841

Vertex coordinates: A[93.19771127024; 0] B[0; 0] C[82.00773786238; 94.33986975278]
Centroid: CG[58.40114971087; 31.44662325093]
Coordinates of the circumscribed circle: U[46.59985563512; 42.30658050468]
Coordinates of the inscribed circle: I[61.59985563512; 28.07220794321]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 96.76443824278° = 96°45'52″ = 1.45327355816 rad
∠ B' = β' = 131° = 0.85552113335 rad
∠ C' = γ' = 132.2365617572° = 132°14'8″ = 0.83436457385 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 125 ; ; b = 95 ; ; beta = 49° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 95**2 = 125**2 + c**2 -2 * 125 * c * cos (49° ) ; ; ; ; c**2 -164.015c +6600 =0 ; ; p=1; q=-164.015; r=6600 ; ; D = q**2 - 4pr = 164.015**2 - 4 * 1 * 6600 = 500.840594998 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 164.01 ± sqrt{ 500.84 } }{ 2 } ; ; c_{1,2} = 82.00737862 ± 11.1897340786 ; ; c_{1} = 93.1971126986 ; ;
c_{2} = 70.8176445414 ; ; ; ; text{ Factored form: } ; ; (c -93.1971126986) (c -70.8176445414) = 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 = 125 ; ; b = 95 ; ; c = 93.2 ; ;

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

p = a+b+c = 125+95+93.2 = 313.2 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 313.2 }{ 2 } = 156.6 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 156.6 * (156.6-125)(156.6-95)(156.6-93.2) } ; ; T = sqrt{ 19325230.22 } = 4396.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4396.05 }{ 125 } = 70.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4396.05 }{ 95 } = 92.55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4396.05 }{ 93.2 } = 94.34 ; ;

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{ 95**2+93.2**2-125**2 }{ 2 * 95 * 93.2 } ) = 83° 14'8" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 125**2+93.2**2-95**2 }{ 2 * 125 * 93.2 } ) = 49° ; ; gamma = 180° - alpha - beta = 180° - 83° 14'8" - 49° = 47° 45'52" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4396.05 }{ 156.6 } = 28.07 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 125 }{ 2 * sin 83° 14'8" } = 62.94 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 95**2+2 * 93.2**2 - 125**2 } }{ 2 } = 70.35 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 93.2**2+2 * 125**2 - 95**2 } }{ 2 } = 99.494 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 95**2+2 * 125**2 - 93.2**2 } }{ 2 } = 100.765 ; ;







#2 Obtuse scalene triangle.

Sides: a = 125   b = 95   c = 70.81876445452

Area: T = 3340.422217419
Perimeter: p = 290.8187644545
Semiperimeter: s = 145.4098822273

Angle ∠ A = α = 96.76443824278° = 96°45'52″ = 1.6898857072 rad
Angle ∠ B = β = 49° = 0.85552113335 rad
Angle ∠ C = γ = 34.23656175722° = 34°14'8″ = 0.59875242481 rad

Height: ha = 53.44767547871
Height: hb = 70.32546773514
Height: hc = 94.33986975278

Median: ma = 55.80216074093
Median: mb = 89.79987716479
Median: mc = 105.2219842735

Inradius: r = 22.97326238201
Circumradius: R = 62.93881171841

Vertex coordinates: A[70.81876445452; 0] B[0; 0] C[82.00773786238; 94.33986975278]
Centroid: CG[50.94216743897; 31.44662325093]
Coordinates of the circumscribed circle: U[35.40988222726; 52.0332892481]
Coordinates of the inscribed circle: I[50.40988222726; 22.97326238201]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 83.23656175722° = 83°14'8″ = 1.6898857072 rad
∠ B' = β' = 131° = 0.85552113335 rad
∠ C' = γ' = 145.7644382428° = 145°45'52″ = 0.59875242481 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 125 ; ; b = 95 ; ; beta = 49° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 95**2 = 125**2 + c**2 -2 * 125 * c * cos (49° ) ; ; ; ; c**2 -164.015c +6600 =0 ; ; p=1; q=-164.015; r=6600 ; ; D = q**2 - 4pr = 164.015**2 - 4 * 1 * 6600 = 500.840594998 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 164.01 ± sqrt{ 500.84 } }{ 2 } ; ; c_{1,2} = 82.00737862 ± 11.1897340786 ; ; c_{1} = 93.1971126986 ; ; : Nr. 1
c_{2} = 70.8176445414 ; ; ; ; text{ Factored form: } ; ; (c -93.1971126986) (c -70.8176445414) = 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 = 125 ; ; b = 95 ; ; c = 70.82 ; ;

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

p = a+b+c = 125+95+70.82 = 290.82 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 290.82 }{ 2 } = 145.41 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 145.41 * (145.41-125)(145.41-95)(145.41-70.82) } ; ; T = sqrt{ 11158420.3 } = 3340.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3340.42 }{ 125 } = 53.45 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3340.42 }{ 95 } = 70.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3340.42 }{ 70.82 } = 94.34 ; ;

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{ 95**2+70.82**2-125**2 }{ 2 * 95 * 70.82 } ) = 96° 45'52" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 125**2+70.82**2-95**2 }{ 2 * 125 * 70.82 } ) = 49° ; ; gamma = 180° - alpha - beta = 180° - 96° 45'52" - 49° = 34° 14'8" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3340.42 }{ 145.41 } = 22.97 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 125 }{ 2 * sin 96° 45'52" } = 62.94 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 95**2+2 * 70.82**2 - 125**2 } }{ 2 } = 55.802 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 70.82**2+2 * 125**2 - 95**2 } }{ 2 } = 89.799 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 95**2+2 * 125**2 - 70.82**2 } }{ 2 } = 105.22 ; ;
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