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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 89 ; ; b = 48 ; ; beta = 30° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 48**2 = 89**2 + c**2 -2 * 48 * c * cos (30° ) ; ; ; ; c**2 -154.153c +5617 =0 ; ; p=1; q=-154.152521874; 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} = 77.0762609368 ± 17.9930542154 ; ; c_{1} = 95.0693151523 ; ;
c_{2} = 59.0832067214 ; ; ; ; (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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 89**2-48**2-59.08**2 }{ 2 * 48 * 59.08 } ) = 112° 55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 48**2-89**2-59.08**2 }{ 2 * 89 * 59.08 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 59.08**2-89**2-48**2 }{ 2 * 48 * 89 } ) = 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 ; ;




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