Triangle calculator SSA

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Triangle has two solutions with side c=45.58217164964 and with side c=17.1021593795

#1 Acute scalene triangle.

Sides: a = 39.4   b = 27.8   c = 45.58217164964

Area: T = 544.1533231782
Perimeter: p = 112.7821716496
Semiperimeter: s = 56.39108582482

Angle ∠ A = α = 59.18773549657° = 59°11'14″ = 1.03330142197 rad
Angle ∠ B = β = 37.3° = 37°18' = 0.6511007811 rad
Angle ∠ C = γ = 83.51326450343° = 83°30'46″ = 1.45875706229 rad

Height: ha = 27.6221991461
Height: hb = 39.14877145167
Height: hc = 23.87659429705

Median: ma = 32.20552237902
Median: mb = 40.27217821728
Median: mc = 25.36109301941

Inradius: r = 9.65496710404
Circumradius: R = 22.93877327914

Vertex coordinates: A[45.58217164964; 0] B[0; 0] C[31.34216551457; 23.87659429705]
Centroid: CG[25.64111238807; 7.95986476568]
Coordinates of the circumscribed circle: U[22.79108582482; 2.59215952466]
Coordinates of the inscribed circle: I[28.59108582482; 9.65496710404]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120.8132645034° = 120°48'46″ = 1.03330142197 rad
∠ B' = β' = 142.7° = 142°42' = 0.6511007811 rad
∠ C' = γ' = 96.48773549657° = 96°29'14″ = 1.45875706229 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 39.4 ; ; b = 27.8 ; ; beta = 37° 18' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 27.8**2 = 39.4**2 + c**2 -2 * 39.4 * c * cos (37° 18') ; ; ; ; c**2 -62.683c +779.52 =0 ; ; p=1; q=-62.683; r=779.52 ; ; D = q**2 - 4pr = 62.683**2 - 4 * 1 * 779.52 = 811.117389084 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 62.68 ± sqrt{ 811.12 } }{ 2 } ; ; c_{1,2} = 31.34165515 ± 14.2400613507 ; ; c_{1} = 45.5817165007 ; ; c_{2} = 17.1015937993 ; ; ; ; text{ Factored form: } ; ; (c -45.5817165007) (c -17.1015937993) = 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 = 39.4 ; ; b = 27.8 ; ; c = 45.58 ; ;

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

p = a+b+c = 39.4+27.8+45.58 = 112.78 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 112.78 }{ 2 } = 56.39 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 56.39 * (56.39-39.4)(56.39-27.8)(56.39-45.58) } ; ; T = sqrt{ 296102.74 } = 544.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 544.15 }{ 39.4 } = 27.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 544.15 }{ 27.8 } = 39.15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 544.15 }{ 45.58 } = 23.88 ; ;

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{ 27.8**2+45.58**2-39.4**2 }{ 2 * 27.8 * 45.58 } ) = 59° 11'14" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 39.4**2+45.58**2-27.8**2 }{ 2 * 39.4 * 45.58 } ) = 37° 18' ; ; gamma = 180° - alpha - beta = 180° - 59° 11'14" - 37° 18' = 83° 30'46" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 544.15 }{ 56.39 } = 9.65 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 39.4 }{ 2 * sin 59° 11'14" } = 22.94 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.8**2+2 * 45.58**2 - 39.4**2 } }{ 2 } = 32.205 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 45.58**2+2 * 39.4**2 - 27.8**2 } }{ 2 } = 40.272 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.8**2+2 * 39.4**2 - 45.58**2 } }{ 2 } = 25.361 ; ;







#2 Obtuse scalene triangle.

Sides: a = 39.4   b = 27.8   c = 17.1021593795

Area: T = 204.1588339077
Perimeter: p = 84.3021593795
Semiperimeter: s = 42.15107968975

Angle ∠ A = α = 120.8132645034° = 120°48'46″ = 2.10985784339 rad
Angle ∠ B = β = 37.3° = 37°18' = 0.6511007811 rad
Angle ∠ C = γ = 21.88773549657° = 21°53'14″ = 0.38220064087 rad

Height: ha = 10.36333674658
Height: hb = 14.68876502933
Height: hc = 23.87659429705

Median: ma = 12.02334044748
Median: mb = 27.00437452063
Median: mc = 33.00773305861

Inradius: r = 4.84435226402
Circumradius: R = 22.93877327914

Vertex coordinates: A[17.1021593795; 0] B[0; 0] C[31.34216551457; 23.87659429705]
Centroid: CG[16.14877496469; 7.95986476568]
Coordinates of the circumscribed circle: U[8.55107968975; 21.28443477238]
Coordinates of the inscribed circle: I[14.35107968975; 4.84435226402]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 59.18773549657° = 59°11'14″ = 2.10985784339 rad
∠ B' = β' = 142.7° = 142°42' = 0.6511007811 rad
∠ C' = γ' = 158.1132645034° = 158°6'46″ = 0.38220064087 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 39.4 ; ; b = 27.8 ; ; beta = 37° 18' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 27.8**2 = 39.4**2 + c**2 -2 * 39.4 * c * cos (37° 18') ; ; ; ; c**2 -62.683c +779.52 =0 ; ; p=1; q=-62.683; r=779.52 ; ; D = q**2 - 4pr = 62.683**2 - 4 * 1 * 779.52 = 811.117389084 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 62.68 ± sqrt{ 811.12 } }{ 2 } ; ; c_{1,2} = 31.34165515 ± 14.2400613507 ; ; c_{1} = 45.5817165007 ; ; c_{2} = 17.1015937993 ; ; ; ; text{ Factored form: } ; ; (c -45.5817165007) (c -17.1015937993) = 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 = 39.4 ; ; b = 27.8 ; ; c = 17.1 ; ;

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

p = a+b+c = 39.4+27.8+17.1 = 84.3 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 84.3 }{ 2 } = 42.15 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42.15 * (42.15-39.4)(42.15-27.8)(42.15-17.1) } ; ; T = sqrt{ 41680.63 } = 204.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 204.16 }{ 39.4 } = 10.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 204.16 }{ 27.8 } = 14.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 204.16 }{ 17.1 } = 23.88 ; ;

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{ 27.8**2+17.1**2-39.4**2 }{ 2 * 27.8 * 17.1 } ) = 120° 48'46" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 39.4**2+17.1**2-27.8**2 }{ 2 * 39.4 * 17.1 } ) = 37° 18' ; ; gamma = 180° - alpha - beta = 180° - 120° 48'46" - 37° 18' = 21° 53'14" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 204.16 }{ 42.15 } = 4.84 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 39.4 }{ 2 * sin 120° 48'46" } = 22.94 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.8**2+2 * 17.1**2 - 39.4**2 } }{ 2 } = 12.023 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.1**2+2 * 39.4**2 - 27.8**2 } }{ 2 } = 27.004 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.8**2+2 * 39.4**2 - 17.1**2 } }{ 2 } = 33.007 ; ;
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