Triangle calculator SSA

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Triangle has two solutions with side c=17.34881178089 and with side c=5.52991300795

#1 Obtuse scalene triangle.

Sides: a = 13.1   b = 8.7   c = 17.34881178089

Area: T = 55.38436332995
Perimeter: p = 39.14881178089
Semiperimeter: s = 19.57440589044

Angle ∠ A = α = 47.21547675553° = 47°12'53″ = 0.82440531494 rad
Angle ∠ B = β = 29.17° = 29°10'12″ = 0.50991125428 rad
Angle ∠ C = γ = 103.6155232445° = 103°36'55″ = 1.80884269614 rad

Height: ha = 8.45655165343
Height: hb = 12.7321869724
Height: hc = 6.38549731607

Median: ma = 12.05990669521
Median: mb = 14.74331711567
Median: mc = 6.95877799708

Inradius: r = 2.82994404124
Circumradius: R = 8.92548613213

Vertex coordinates: A[17.34881178089; 0] B[0; 0] C[11.43986239442; 6.38549731607]
Centroid: CG[9.59655805844; 2.12883243869]
Coordinates of the circumscribed circle: U[8.67440589044; -2.10109168778]
Coordinates of the inscribed circle: I[10.87440589044; 2.82994404124]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.7855232445° = 132°47'7″ = 0.82440531494 rad
∠ B' = β' = 150.83° = 150°49'48″ = 0.50991125428 rad
∠ C' = γ' = 76.38547675553° = 76°23'5″ = 1.80884269614 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 13.1 ; ; b = 8.7 ; ; beta = 29° 10'12" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 8.7**2 = 13.1**2 + c**2 -2 * 13.1 * c * cos (29° 10'12") ; ; ; ; c**2 -22.877c +95.92 =0 ; ; p=1; q=-22.877; r=95.92 ; ; D = q**2 - 4pr = 22.877**2 - 4 * 1 * 95.92 = 139.688470947 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 22.88 ± sqrt{ 139.69 } }{ 2 } ; ; c_{1,2} = 11.43862394 ± 5.90949386468 ; ;
c_{1} = 17.3481178047 ; ; c_{2} = 5.52913007532 ; ; ; ; text{ Factored form: } ; ; (c -17.3481178047) (c -5.52913007532) = 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 = 13.1 ; ; b = 8.7 ; ; c = 17.35 ; ;

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

p = a+b+c = 13.1+8.7+17.35 = 39.15 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 39.15 }{ 2 } = 19.57 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.57 * (19.57-13.1)(19.57-8.7)(19.57-17.35) } ; ; T = sqrt{ 3067.35 } = 55.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 55.38 }{ 13.1 } = 8.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 55.38 }{ 8.7 } = 12.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 55.38 }{ 17.35 } = 6.38 ; ;

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{ 8.7**2+17.35**2-13.1**2 }{ 2 * 8.7 * 17.35 } ) = 47° 12'53" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13.1**2+17.35**2-8.7**2 }{ 2 * 13.1 * 17.35 } ) = 29° 10'12" ; ; gamma = 180° - alpha - beta = 180° - 47° 12'53" - 29° 10'12" = 103° 36'55" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 55.38 }{ 19.57 } = 2.83 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13.1 }{ 2 * sin 47° 12'53" } = 8.92 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.7**2+2 * 17.35**2 - 13.1**2 } }{ 2 } = 12.059 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.35**2+2 * 13.1**2 - 8.7**2 } }{ 2 } = 14.743 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.7**2+2 * 13.1**2 - 17.35**2 } }{ 2 } = 6.958 ; ;





#2 Obtuse scalene triangle.

Sides: a = 13.1   b = 8.7   c = 5.52991300795

Area: T = 17.652167358
Perimeter: p = 27.32991300795
Semiperimeter: s = 13.66545650398

Angle ∠ A = α = 132.7855232445° = 132°47'7″ = 2.31875395042 rad
Angle ∠ B = β = 29.17° = 29°10'12″ = 0.50991125428 rad
Angle ∠ C = γ = 18.04547675553° = 18°2'41″ = 0.31549406066 rad

Height: ha = 2.69549119969
Height: hb = 4.05878559954
Height: hc = 6.38549731607

Median: ma = 3.1988146294
Median: mb = 9.06546643467
Median: mc = 10.77106629388

Inradius: r = 1.29217845192
Circumradius: R = 8.92548613213

Vertex coordinates: A[5.52991300795; 0] B[0; 0] C[11.43986239442; 6.38549731607]
Centroid: CG[5.65659180079; 2.12883243869]
Coordinates of the circumscribed circle: U[2.76545650398; 8.48658900385]
Coordinates of the inscribed circle: I[4.96545650398; 1.29217845192]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 47.21547675553° = 47°12'53″ = 2.31875395042 rad
∠ B' = β' = 150.83° = 150°49'48″ = 0.50991125428 rad
∠ C' = γ' = 161.9555232445° = 161°57'19″ = 0.31549406066 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 13.1 ; ; b = 8.7 ; ; beta = 29° 10'12" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 8.7**2 = 13.1**2 + c**2 -2 * 13.1 * c * cos (29° 10'12") ; ; ; ; c**2 -22.877c +95.92 =0 ; ; p=1; q=-22.877; r=95.92 ; ; D = q**2 - 4pr = 22.877**2 - 4 * 1 * 95.92 = 139.688470947 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 22.88 ± sqrt{ 139.69 } }{ 2 } ; ; c_{1,2} = 11.43862394 ± 5.90949386468 ; ; : Nr. 1
c_{1} = 17.3481178047 ; ; c_{2} = 5.52913007532 ; ; ; ; text{ Factored form: } ; ; (c -17.3481178047) (c -5.52913007532) = 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 = 13.1 ; ; b = 8.7 ; ; c = 5.53 ; ;

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

p = a+b+c = 13.1+8.7+5.53 = 27.33 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.33 }{ 2 } = 13.66 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.66 * (13.66-13.1)(13.66-8.7)(13.66-5.53) } ; ; T = sqrt{ 311.58 } = 17.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.65 }{ 13.1 } = 2.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.65 }{ 8.7 } = 4.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.65 }{ 5.53 } = 6.38 ; ;

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{ 8.7**2+5.53**2-13.1**2 }{ 2 * 8.7 * 5.53 } ) = 132° 47'7" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13.1**2+5.53**2-8.7**2 }{ 2 * 13.1 * 5.53 } ) = 29° 10'12" ; ; gamma = 180° - alpha - beta = 180° - 132° 47'7" - 29° 10'12" = 18° 2'41" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.65 }{ 13.66 } = 1.29 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13.1 }{ 2 * sin 132° 47'7" } = 8.92 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.7**2+2 * 5.53**2 - 13.1**2 } }{ 2 } = 3.198 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.53**2+2 * 13.1**2 - 8.7**2 } }{ 2 } = 9.065 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.7**2+2 * 13.1**2 - 5.53**2 } }{ 2 } = 10.771 ; ;
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