Triangle calculator SSA

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Triangle has two solutions with side c=95.06993151523 and with side c=59.08332067214

#1 Acute scalene triangle.

Sides: a = 89   b = 48   c = 95.06993151523

Area: T = 2115.292226214
Perimeter: p = 232.0699315152
Semiperimeter: s = 116.0354657576

Angle ∠ A = α = 67.98546304596° = 67°59'5″ = 1.18765556423 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 82.01553695404° = 82°55″ = 1.43114382357 rad

Height: ha = 47.53546575761
Height: hb = 88.13771775891
Height: hc = 44.5

Median: ma = 60.752226203
Median: mb = 88.90221222568
Median: mc = 53.41330726426

Inradius: r = 18.23298315549
Circumradius: R = 48

Vertex coordinates: A[95.06993151523; 0] B[0; 0] C[77.07662609368; 44.5]
Centroid: CG[57.38218586964; 14.83333333333]
Coordinates of the circumscribed circle: U[47.53546575761; 6.66875579578]
Coordinates of the inscribed circle: I[68.03546575761; 18.23298315549]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.015536954° = 112°55″ = 1.18765556423 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 97.98546304596° = 97°59'5″ = 1.43114382357 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 89 ; ; b = 48 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 48**2 = 89**2 + c**2 -2 * 89 * c * cos (30° ) ; ; ; ; c**2 -154.153c +5617 =0 ; ; p=1; q=-154.153; r=5617 ; ; D = q**2 - 4pr = 154.153**2 - 4 * 1 * 5617 = 1295 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 154.15 ± sqrt{ 1295 } }{ 2 } ; ;
c_{1,2} = 77.07626094 ± 17.9930542154 ; ; c_{1} = 95.0693151523 ; ; c_{2} = 59.0832067214 ; ; ; ; text{ Factored form: } ; ; (c -95.0693151523) (c -59.0832067214) = 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 = 89 ; ; b = 48 ; ; c = 95.07 ; ;

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

p = a+b+c = 89+48+95.07 = 232.07 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 232.07 }{ 2 } = 116.03 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 116.03 * (116.03-89)(116.03-48)(116.03-95.07) } ; ; T = sqrt{ 4474461.35 } = 2115.29 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2115.29 }{ 89 } = 47.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2115.29 }{ 48 } = 88.14 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2115.29 }{ 95.07 } = 44.5 ; ;

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{ 48**2+95.07**2-89**2 }{ 2 * 48 * 95.07 } ) = 67° 59'5" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 89**2+95.07**2-48**2 }{ 2 * 89 * 95.07 } ) = 30° ; ;
 gamma = 180° - alpha - beta = 180° - 67° 59'5" - 30° = 82° 55" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2115.29 }{ 116.03 } = 18.23 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 89 }{ 2 * sin 67° 59'5" } = 48 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 95.07**2 - 89**2 } }{ 2 } = 60.752 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 95.07**2+2 * 89**2 - 48**2 } }{ 2 } = 88.902 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 89**2 - 95.07**2 } }{ 2 } = 53.413 ; ;



#2 Obtuse scalene triangle.

Sides: a = 89   b = 48   c = 59.08332067214

Area: T = 1314.601134955
Perimeter: p = 196.0833206721
Semiperimeter: s = 98.04216033607

Angle ∠ A = α = 112.015536954° = 112°55″ = 1.95550370113 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 37.98546304596° = 37°59'5″ = 0.66329568667 rad

Height: ha = 29.54216033607
Height: hb = 54.77550562313
Height: hc = 44.5

Median: ma = 30.28546934645
Median: mb = 71.62334085913
Median: mc = 65.1143698028

Inradius: r = 13.40986072085
Circumradius: R = 48

Vertex coordinates: A[59.08332067214; 0] B[0; 0] C[77.07662609368; 44.5]
Centroid: CG[45.38664892194; 14.83333333333]
Coordinates of the circumscribed circle: U[29.54216033607; 37.83224420422]
Coordinates of the inscribed circle: I[50.04216033607; 13.40986072085]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 67.98546304596° = 67°59'5″ = 1.95550370113 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 142.015536954° = 142°55″ = 0.66329568667 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 89 ; ; b = 48 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 48**2 = 89**2 + c**2 -2 * 89 * c * cos (30° ) ; ; ; ; c**2 -154.153c +5617 =0 ; ; p=1; q=-154.153; r=5617 ; ; D = q**2 - 4pr = 154.153**2 - 4 * 1 * 5617 = 1295 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 154.15 ± sqrt{ 1295 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 77.07626094 ± 17.9930542154 ; ; c_{1} = 95.0693151523 ; ; c_{2} = 59.0832067214 ; ; ; ; text{ Factored form: } ; ; (c -95.0693151523) (c -59.0832067214) = 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 = 89 ; ; b = 48 ; ; c = 59.08 ; ;

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

p = a+b+c = 89+48+59.08 = 196.08 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 196.08 }{ 2 } = 98.04 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 98.04 * (98.04-89)(98.04-48)(98.04-59.08) } ; ; T = sqrt{ 1728176.71 } = 1314.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1314.6 }{ 89 } = 29.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1314.6 }{ 48 } = 54.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1314.6 }{ 59.08 } = 44.5 ; ;

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{ 48**2+59.08**2-89**2 }{ 2 * 48 * 59.08 } ) = 112° 55" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 89**2+59.08**2-48**2 }{ 2 * 89 * 59.08 } ) = 30° ; ;
 gamma = 180° - alpha - beta = 180° - 112° 55" - 30° = 37° 59'5" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1314.6 }{ 98.04 } = 13.41 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 89 }{ 2 * sin 112° 55" } = 48 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 59.08**2 - 89**2 } }{ 2 } = 30.285 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 59.08**2+2 * 89**2 - 48**2 } }{ 2 } = 71.623 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 89**2 - 59.08**2 } }{ 2 } = 65.114 ; ;
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