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?

1. Use Law of Cosines

a = 170 ; ; b = 135 ; ; beta = 24° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 135**2 = 170**2 + c**2 -2 * 170 * c * cos (24° ) ; ; ; ; c**2 -310.605c +10675 =0 ; ; p=1; q=-310.605; 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} = 155.3027278 ± 115.947993781 ; ; c_{1} = 271.250721581 ; ; c_{2} = 39.3547340188 ; ; ; ; text{ Factored form: } ; ; (c -271.250721581) (c -39.3547340188) = 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 = 271.25 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 135**2+271.25**2-170**2 }{ 2 * 135 * 271.25 } ) = 30° 48'35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 170**2+271.25**2-135**2 }{ 2 * 170 * 271.25 } ) = 24° ; ; gamma = 180° - alpha - beta = 180° - 30° 48'35" - 24° = 125° 11'25" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 135**2+2 * 271.25**2 - 170**2 } }{ 2 } = 196.662 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 271.25**2+2 * 170**2 - 135**2 } }{ 2 } = 216.061 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 135**2+2 * 170**2 - 271.25**2 } }{ 2 } = 71.891 ; ;







#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 - 2ac cos beta ; ; 135**2 = 170**2 + c**2 -2 * 170 * c * cos (24° ) ; ; ; ; c**2 -310.605c +10675 =0 ; ; p=1; q=-310.605; 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} = 155.3027278 ± 115.947993781 ; ; c_{1} = 271.250721581 ; ; c_{2} = 39.3547340188 ; ; ; ; text{ Factored form: } ; ; (c -271.250721581) (c -39.3547340188) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 135**2+39.35**2-170**2 }{ 2 * 135 * 39.35 } ) = 149° 11'25" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 170**2+39.35**2-135**2 }{ 2 * 170 * 39.35 } ) = 24° ; ; gamma = 180° - alpha - beta = 180° - 149° 11'25" - 24° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 135**2+2 * 39.35**2 - 170**2 } }{ 2 } = 51.594 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 39.35**2+2 * 170**2 - 135**2 } }{ 2 } = 103.287 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 135**2+2 * 170**2 - 39.35**2 } }{ 2 } = 152.234 ; ;
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