Triangle calculator SSA

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Triangle has two solutions with side c=271.251072158 and with side c=39.3554734018

#1 Obtuse scalene triangle.

Sides: a = 170   b = 135   c = 271.251072158

Area: T = 9377.847667384
Perimeter: p = 576.251072158
Semiperimeter: s = 288.125536079

Angle ∠ A = α = 30.81096070765° = 30°48'35″ = 0.53877290847 rad
Angle ∠ B = β = 24° = 0.41988790205 rad
Angle ∠ C = γ = 125.1990392923° = 125°11'25″ = 2.18549845484 rad

Height: ha = 110.3287607927
Height: hb = 138.9311061835
Height: hc = 69.14552293229

Median: ma = 196.6622088311
Median: mb = 216.061070207
Median: mc = 71.89106218538

Inradius: r = 32.54878001941
Circumradius: R = 165.9555050151

Vertex coordinates: A[271.251072158; 0] B[0; 0] C[155.3032727799; 69.14552293229]
Centroid: CG[142.1844483127; 23.04884097743]
Coordinates of the circumscribed circle: U[135.625536079; -95.63991142851]
Coordinates of the inscribed circle: I[153.125536079; 32.54878001941]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.1990392923° = 149°11'25″ = 0.53877290847 rad
∠ B' = β' = 156° = 0.41988790205 rad
∠ C' = γ' = 54.81096070765° = 54°48'35″ = 2.18549845484 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 = 170 ; ; b = 135 ; ; c = 271.25 ; ;

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

p = a+b+c = 170+135+271.25 = 576.25 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 576.25 }{ 2 } = 288.13 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 288.13 * (288.13-170)(288.13-135)(288.13-271.25) } ; ; T = sqrt{ 87944008.24 } = 9377.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9377.85 }{ 170 } = 110.33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9377.85 }{ 135 } = 138.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9377.85 }{ 271.25 } = 69.15 ; ;

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{ 170**2-135**2-271.25**2 }{ 2 * 135 * 271.25 } ) = 30° 48'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 135**2-170**2-271.25**2 }{ 2 * 170 * 271.25 } ) = 24° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 271.25**2-170**2-135**2 }{ 2 * 135 * 170 } ) = 125° 11'25" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9377.85 }{ 288.13 } = 32.55 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 170 }{ 2 * sin 30° 48'35" } = 165.96 ; ;





#2 Obtuse scalene triangle.

Sides: a = 170   b = 135   c = 39.3554734018

Area: T = 1360.596605431
Perimeter: p = 344.3554734018
Semiperimeter: s = 172.1777367009

Angle ∠ A = α = 149.1990392923° = 149°11'25″ = 2.60438635689 rad
Angle ∠ B = β = 24° = 0.41988790205 rad
Angle ∠ C = γ = 6.81096070765° = 6°48'35″ = 0.11988500643 rad

Height: ha = 16.00770124036
Height: hb = 20.15769785824
Height: hc = 69.14552293229

Median: ma = 51.59435804613
Median: mb = 103.2876724921
Median: mc = 152.2344362834

Inradius: r = 7.90222933034
Circumradius: R = 165.9555050151

Vertex coordinates: A[39.3554734018; 0] B[0; 0] C[155.3032727799; 69.14552293229]
Centroid: CG[64.88658206058; 23.04884097743]
Coordinates of the circumscribed circle: U[19.6777367009; 164.7844343608]
Coordinates of the inscribed circle: I[37.1777367009; 7.90222933034]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 30.81096070765° = 30°48'35″ = 2.60438635689 rad
∠ B' = β' = 156° = 0.41988790205 rad
∠ C' = γ' = 173.1990392923° = 173°11'25″ = 0.11988500643 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 170 ; ; b = 135 ; ; beta = 24° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 135**2 = 170**2 + c**2 -2 * 135 * c * cos (24° ) ; ; ; ; c**2 -310.605c +10675 =0 ; ; p=1; q=-310.605455598; r=10675 ; ; D = q**2 - 4pr = 310.605**2 - 4 * 1 * 10675 = 53775.7490475 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 310.61 ± sqrt{ 53775.75 } }{ 2 } ; ; c_{1,2} = 155.302727799 ± 115.947993781 ; ;
c_{1} = 271.25072158 ; ; c_{2} = 39.354734018 ; ; ; ; (c -271.25072158) (c -39.354734018) = 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 = 170 ; ; b = 135 ; ; c = 39.35 ; ;

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

p = a+b+c = 170+135+39.35 = 344.35 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 344.35 }{ 2 } = 172.18 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 172.18 * (172.18-170)(172.18-135)(172.18-39.35) } ; ; T = sqrt{ 1851221.62 } = 1360.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1360.6 }{ 170 } = 16.01 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1360.6 }{ 135 } = 20.16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1360.6 }{ 39.35 } = 69.15 ; ;

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{ 170**2-135**2-39.35**2 }{ 2 * 135 * 39.35 } ) = 149° 11'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 135**2-170**2-39.35**2 }{ 2 * 170 * 39.35 } ) = 24° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 39.35**2-170**2-135**2 }{ 2 * 135 * 170 } ) = 6° 48'35" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1360.6 }{ 172.18 } = 7.9 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 170 }{ 2 * sin 149° 11'25" } = 165.96 ; ;




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