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?

1. Use Law of Cosines

a = 72 ; ; b = 56 ; ; beta = 26° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 56**2 = 72**2 + c**2 -2 * 72 * c * cos (26° ) ; ; ; ; c**2 -129.426c +2048 =0 ; ; p=1; q=-129.426; 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} = 64.71317133 ± 46.2579133127 ; ; c_{1} = 110.971084643 ; ;
c_{2} = 18.4552580173 ; ; ; ; text{ Factored form: } ; ; (c -110.971084643) (c -18.4552580173) = 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 = 110.97 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 56**2+110.97**2-72**2 }{ 2 * 56 * 110.97 } ) = 34° 18'23" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 72**2+110.97**2-56**2 }{ 2 * 72 * 110.97 } ) = 26° ; ; gamma = 180° - alpha - beta = 180° - 34° 18'23" - 26° = 119° 41'37" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 110.97**2 - 72**2 } }{ 2 } = 80.183 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 110.97**2+2 * 72**2 - 56**2 } }{ 2 } = 89.248 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 72**2 - 110.97**2 } }{ 2 } = 32.884 ; ;







#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

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How did we calculate this triangle?

1. Use Law of Cosines

a = 72 ; ; b = 56 ; ; beta = 26° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 56**2 = 72**2 + c**2 -2 * 72 * c * cos (26° ) ; ; ; ; c**2 -129.426c +2048 =0 ; ; p=1; q=-129.426; 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} = 64.71317133 ± 46.2579133127 ; ; c_{1} = 110.971084643 ; ; : Nr. 1
c_{2} = 18.4552580173 ; ; ; ; text{ Factored form: } ; ; (c -110.971084643) (c -18.4552580173) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 56**2+18.46**2-72**2 }{ 2 * 56 * 18.46 } ) = 145° 41'37" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 72**2+18.46**2-56**2 }{ 2 * 72 * 18.46 } ) = 26° ; ; gamma = 180° - alpha - beta = 180° - 145° 41'37" - 26° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 18.46**2 - 72**2 } }{ 2 } = 21.031 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.46**2+2 * 72**2 - 56**2 } }{ 2 } = 44.478 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 56**2+2 * 72**2 - 18.46**2 } }{ 2 } = 63.835 ; ;
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