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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 27.8**2 = 39.4**2 + c**2 -2 * 27.8 * c * cos (37° 18') ; ; ; ; c**2 -62.683c +779.52 =0 ; ; p=1; q=-62.6833102914; 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} = 31.3416551457 ± 14.2400613507 ; ;
c_{1} = 45.5817164964 ; ; c_{2} = 17.101593795 ; ; ; ; (c -45.5817164964) (c -17.101593795) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 39.4**2-27.8**2-17.1**2 }{ 2 * 27.8 * 17.1 } ) = 120° 48'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27.8**2-39.4**2-17.1**2 }{ 2 * 39.4 * 17.1 } ) = 37° 18' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 17.1**2-39.4**2-27.8**2 }{ 2 * 27.8 * 39.4 } ) = 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 ; ;




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