Triangle calculator SSA

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Triangle has two solutions with side c=102.3832524717 and with side c=73.81114245661

#1 Acute scalene triangle.

Sides: a = 131   b = 98   c = 102.3832524717

Area: T = 4963.157671584
Perimeter: p = 331.3832524717
Semiperimeter: s = 165.6911262358

Angle ∠ A = α = 81.61880760505° = 81°37'5″ = 1.42545041562 rad
Angle ∠ B = β = 47.74° = 47°44'24″ = 0.83332201849 rad
Angle ∠ C = γ = 50.64219239495° = 50°38'31″ = 0.88438683125 rad

Height: ha = 75.77333849747
Height: hb = 101.2898912568
Height: hc = 96.95332003548

Median: ma = 75.84774830411
Median: mb = 106.867716373
Median: mc = 103.7439841229

Inradius: r = 29.95442452945
Circumradius: R = 66.20772007578

Vertex coordinates: A[102.3832524717; 0] B[0; 0] C[88.09769746413; 96.95332003548]
Centroid: CG[63.49331664526; 32.31877334516]
Coordinates of the circumscribed circle: U[51.19112623583; 41.98662845503]
Coordinates of the inscribed circle: I[67.69112623583; 29.95442452945]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 98.38219239495° = 98°22'55″ = 1.42545041562 rad
∠ B' = β' = 132.26° = 132°15'36″ = 0.83332201849 rad
∠ C' = γ' = 129.3588076051° = 129°21'29″ = 0.88438683125 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 131 ; ; b = 98 ; ; beta = 47° 44'24" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 98**2 = 131**2 + c**2 -2 * 131 * c * cos (47° 44'24") ; ; ; ; c**2 -176.194c +7557 =0 ; ; p=1; q=-176.194; r=7557 ; ; D = q**2 - 4pr = 176.194**2 - 4 * 1 * 7557 = 816.30776381 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 176.19 ± sqrt{ 816.31 } }{ 2 } ; ; c_{1,2} = 88.09697464 ± 14.2855500752 ; ; c_{1} = 102.382524715 ; ;
c_{2} = 73.8114245648 ; ; ; ; text{ Factored form: } ; ; (c -102.382524715) (c -73.8114245648) = 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 = 131 ; ; b = 98 ; ; c = 102.38 ; ;

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

p = a+b+c = 131+98+102.38 = 331.38 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 331.38 }{ 2 } = 165.69 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 165.69 * (165.69-131)(165.69-98)(165.69-102.38) } ; ; T = sqrt{ 24632924.59 } = 4963.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4963.16 }{ 131 } = 75.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4963.16 }{ 98 } = 101.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4963.16 }{ 102.38 } = 96.95 ; ;

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{ 98**2+102.38**2-131**2 }{ 2 * 98 * 102.38 } ) = 81° 37'5" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 131**2+102.38**2-98**2 }{ 2 * 131 * 102.38 } ) = 47° 44'24" ; ; gamma = 180° - alpha - beta = 180° - 81° 37'5" - 47° 44'24" = 50° 38'31" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4963.16 }{ 165.69 } = 29.95 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 131 }{ 2 * sin 81° 37'5" } = 66.21 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 98**2+2 * 102.38**2 - 131**2 } }{ 2 } = 75.847 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 102.38**2+2 * 131**2 - 98**2 } }{ 2 } = 106.867 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 98**2+2 * 131**2 - 102.38**2 } }{ 2 } = 103.74 ; ;







#2 Obtuse scalene triangle.

Sides: a = 131   b = 98   c = 73.81114245661

Area: T = 3578.127691722
Perimeter: p = 302.8111424566
Semiperimeter: s = 151.4065712283

Angle ∠ A = α = 98.38219239495° = 98°22'55″ = 1.71770884974 rad
Angle ∠ B = β = 47.74° = 47°44'24″ = 0.83332201849 rad
Angle ∠ C = γ = 33.87880760505° = 33°52'41″ = 0.59112839713 rad

Height: ha = 54.62878918659
Height: hb = 73.02329983105
Height: hc = 96.95332003548

Median: ma = 56.88442086895
Median: mb = 94.35986943436
Median: mc = 109.6387896737

Inradius: r = 23.6332707533
Circumradius: R = 66.20772007578

Vertex coordinates: A[73.81114245661; 0] B[0; 0] C[88.09769746413; 96.95332003548]
Centroid: CG[53.96994664025; 32.31877334516]
Coordinates of the circumscribed circle: U[36.9065712283; 54.96769158045]
Coordinates of the inscribed circle: I[53.4065712283; 23.6332707533]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 81.61880760505° = 81°37'5″ = 1.71770884974 rad
∠ B' = β' = 132.26° = 132°15'36″ = 0.83332201849 rad
∠ C' = γ' = 146.1221923949° = 146°7'19″ = 0.59112839713 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 131 ; ; b = 98 ; ; beta = 47° 44'24" ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 98**2 = 131**2 + c**2 -2 * 131 * c * cos (47° 44'24") ; ; ; ; c**2 -176.194c +7557 =0 ; ; p=1; q=-176.194; r=7557 ; ; D = q**2 - 4pr = 176.194**2 - 4 * 1 * 7557 = 816.30776381 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 176.19 ± sqrt{ 816.31 } }{ 2 } ; ; c_{1,2} = 88.09697464 ± 14.2855500752 ; ; c_{1} = 102.382524715 ; ; : Nr. 1
c_{2} = 73.8114245648 ; ; ; ; text{ Factored form: } ; ; (c -102.382524715) (c -73.8114245648) = 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 = 131 ; ; b = 98 ; ; c = 73.81 ; ;

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

p = a+b+c = 131+98+73.81 = 302.81 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 302.81 }{ 2 } = 151.41 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 151.41 * (151.41-131)(151.41-98)(151.41-73.81) } ; ; T = sqrt{ 12802992.24 } = 3578.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3578.13 }{ 131 } = 54.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3578.13 }{ 98 } = 73.02 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3578.13 }{ 73.81 } = 96.95 ; ;

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{ 98**2+73.81**2-131**2 }{ 2 * 98 * 73.81 } ) = 98° 22'55" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 131**2+73.81**2-98**2 }{ 2 * 131 * 73.81 } ) = 47° 44'24" ; ; gamma = 180° - alpha - beta = 180° - 98° 22'55" - 47° 44'24" = 33° 52'41" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3578.13 }{ 151.41 } = 23.63 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 131 }{ 2 * sin 98° 22'55" } = 66.21 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 98**2+2 * 73.81**2 - 131**2 } }{ 2 } = 56.884 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 73.81**2+2 * 131**2 - 98**2 } }{ 2 } = 94.359 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 98**2+2 * 131**2 - 73.81**2 } }{ 2 } = 109.638 ; ;
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