Triangle calculator SSA

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Triangle has two solutions with side c=110.9711084646 and with side c=18.45552580209

#1 Obtuse scalene triangle.

Sides: a = 72   b = 56   c = 110.9711084646

Area: T = 1751.275477892
Perimeter: p = 238.9711084646
Semiperimeter: s = 119.4865542323

Angle ∠ A = α = 34.30765196075° = 34°18'23″ = 0.59987617221 rad
Angle ∠ B = β = 26° = 0.45437856055 rad
Angle ∠ C = γ = 119.6933480392° = 119°41'37″ = 2.0899045326 rad

Height: ha = 48.64765216368
Height: hb = 62.54655278187
Height: hc = 31.56327225688

Median: ma = 80.18328586032
Median: mb = 89.24884779354
Median: mc = 32.88439564698

Inradius: r = 14.65767923188
Circumradius: R = 63.87328169157

Vertex coordinates: A[110.9711084646; 0] B[0; 0] C[64.71331713335; 31.56327225688]
Centroid: CG[58.56114186599; 10.52109075229]
Coordinates of the circumscribed circle: U[55.48655423231; -31.64400273998]
Coordinates of the inscribed circle: I[63.48655423231; 14.65767923188]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.6933480392° = 145°41'37″ = 0.59987617221 rad
∠ B' = β' = 154° = 0.45437856055 rad
∠ C' = γ' = 60.30765196075° = 60°18'23″ = 2.0899045326 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 = 72 ; ; b = 56 ; ; c = 110.97 ; ;

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

p = a+b+c = 72+56+110.97 = 238.97 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 238.97 }{ 2 } = 119.49 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 119.49 * (119.49-72)(119.49-56)(119.49-110.97) } ; ; T = sqrt{ 3066963.35 } = 1751.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1751.27 }{ 72 } = 48.65 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1751.27 }{ 56 } = 62.55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1751.27 }{ 110.97 } = 31.56 ; ;

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{ 72**2-56**2-110.97**2 }{ 2 * 56 * 110.97 } ) = 34° 18'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 56**2-72**2-110.97**2 }{ 2 * 72 * 110.97 } ) = 26° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 110.97**2-72**2-56**2 }{ 2 * 56 * 72 } ) = 119° 41'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1751.27 }{ 119.49 } = 14.66 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 72 }{ 2 * sin 34° 18'23" } = 63.87 ; ;





#2 Obtuse scalene triangle.

Sides: a = 72   b = 56   c = 18.45552580209

Area: T = 291.2499094424
Perimeter: p = 146.4555258021
Semiperimeter: s = 73.22876290104

Angle ∠ A = α = 145.6933480392° = 145°41'37″ = 2.54328309315 rad
Angle ∠ B = β = 26° = 0.45437856055 rad
Angle ∠ C = γ = 8.30765196075° = 8°18'23″ = 0.14549761165 rad

Height: ha = 8.09902526229
Height: hb = 10.40217533723
Height: hc = 31.56327225688

Median: ma = 21.03108885763
Median: mb = 44.47880650918
Median: mc = 63.8354558531

Inradius: r = 3.9777311547
Circumradius: R = 63.87328169157

Vertex coordinates: A[18.45552580209; 0] B[0; 0] C[64.71331713335; 31.56327225688]
Centroid: CG[27.72328097848; 10.52109075229]
Coordinates of the circumscribed circle: U[9.22876290104; 63.20327499686]
Coordinates of the inscribed circle: I[17.22876290104; 3.9777311547]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 34.30765196075° = 34°18'23″ = 2.54328309315 rad
∠ B' = β' = 154° = 0.45437856055 rad
∠ C' = γ' = 171.6933480392° = 171°41'37″ = 0.14549761165 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 72 ; ; b = 56 ; ; beta = 26° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 56**2 = 72**2 + c**2 -2 * 56 * c * cos (26° ) ; ; ; ; c**2 -129.426c +2048 =0 ; ; p=1; q=-129.426342667; r=2048 ; ; D = q**2 - 4pr = 129.426**2 - 4 * 1 * 2048 = 8559.17817618 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 129.43 ± sqrt{ 8559.18 } }{ 2 } ; ; c_{1,2} = 64.7131713335 ± 46.2579133127 ; ; c_{1} = 110.971084646 ; ;
c_{2} = 18.4552580209 ; ; ; ; (c -110.971084646) (c -18.4552580209) = 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 = 72 ; ; b = 56 ; ; c = 18.46 ; ;

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

p = a+b+c = 72+56+18.46 = 146.46 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 146.46 }{ 2 } = 73.23 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 73.23 * (73.23-72)(73.23-56)(73.23-18.46) } ; ; T = sqrt{ 84826.04 } = 291.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 291.25 }{ 72 } = 8.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 291.25 }{ 56 } = 10.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 291.25 }{ 18.46 } = 31.56 ; ;

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{ 72**2-56**2-18.46**2 }{ 2 * 56 * 18.46 } ) = 145° 41'37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 56**2-72**2-18.46**2 }{ 2 * 72 * 18.46 } ) = 26° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 18.46**2-72**2-56**2 }{ 2 * 56 * 72 } ) = 8° 18'23" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 291.25 }{ 73.23 } = 3.98 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 72 }{ 2 * sin 145° 41'37" } = 63.87 ; ;




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