Triangle calculator SSA

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Triangle has two solutions with side c=5.84662287756 and with side c=1.14879553725

#1 Obtuse scalene triangle.

Sides: a = 3.531146   b = 2.4   c = 5.84662287756

Area: T = 1.43766646507
Perimeter: p = 11.77876887756
Semiperimeter: s = 5.88988443878

Angle ∠ A = α = 11.81769192781° = 11°49'1″ = 0.20662441488 rad
Angle ∠ B = β = 8° = 0.14396263402 rad
Angle ∠ C = γ = 160.1833080722° = 160°10'59″ = 2.79657221646 rad

Height: ha = 0.81436377876
Height: hb = 1.19772205423
Height: hc = 0.49114842391

Median: ma = 4.1055044825
Median: mb = 4.67881193138
Median: mc = 0.7565650145

Inradius: r = 0.24439637654
Circumradius: R = 8.62223558412

Vertex coordinates: A[5.84662287756; 0] B[0; 0] C[3.4977092074; 0.49114842391]
Centroid: CG[3.11444402832; 0.16438280797]
Coordinates of the circumscribed circle: U[2.92331143878; -8.11217459605]
Coordinates of the inscribed circle: I[3.48988443878; 0.24439637654]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 168.1833080722° = 168°10'59″ = 0.20662441488 rad
∠ B' = β' = 172° = 0.14396263402 rad
∠ C' = γ' = 19.81769192781° = 19°49'1″ = 2.79657221646 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 3.53 ; ; b = 2.4 ; ; beta = 8° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 2.4**2 = 3.53**2 + c**2 -2 * 3.53 * c * cos (8° ) ; ; ; ; c**2 -6.994c +6.711 =0 ; ; p=1; q=-6.994; r=6.711 ; ; D = q**2 - 4pr = 6.994**2 - 4 * 1 * 6.711 = 22.0737729708 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 6.99 ± sqrt{ 22.07 } }{ 2 } ; ; c_{1,2} = 3.49709207 ± 2.34913670158 ; ; c_{1} = 5.84622877158 ; ;
c_{2} = 1.14795536842 ; ; ; ; (c -5.84622877158) (c -1.14795536842) = 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 = 3.53 ; ; b = 2.4 ; ; c = 5.85 ; ;

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

p = a+b+c = 3.53+2.4+5.85 = 11.78 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.78 }{ 2 } = 5.89 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.89 * (5.89-3.53)(5.89-2.4)(5.89-5.85) } ; ; T = sqrt{ 2.06 } = 1.44 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.44 }{ 3.53 } = 0.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.44 }{ 2.4 } = 1.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.44 }{ 5.85 } = 0.49 ; ;

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{ 3.53**2-2.4**2-5.85**2 }{ 2 * 2.4 * 5.85 } ) = 11° 49'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2.4**2-3.53**2-5.85**2 }{ 2 * 3.53 * 5.85 } ) = 8° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.85**2-3.53**2-2.4**2 }{ 2 * 2.4 * 3.53 } ) = 160° 10'59" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.44 }{ 5.89 } = 0.24 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.53 }{ 2 * sin 11° 49'1" } = 8.62 ; ;





#2 Obtuse scalene triangle.

Sides: a = 3.531146   b = 2.4   c = 1.14879553725

Area: T = 0.28221009864
Perimeter: p = 7.07994153725
Semiperimeter: s = 3.54397076862

Angle ∠ A = α = 168.1833080722° = 168°10'59″ = 2.93553485047 rad
Angle ∠ B = β = 8° = 0.14396263402 rad
Angle ∠ C = γ = 3.81769192781° = 3°49'1″ = 0.06766178087 rad

Height: ha = 0.16597645089
Height: hb = 0.23550841553
Height: hc = 0.49114842391

Median: ma = 0.64989209009
Median: mb = 2.33554883075
Median: mc = 2.96441448145

Inradius: r = 0.0879696125
Circumradius: R = 8.62223558412

Vertex coordinates: A[1.14879553725; 0] B[0; 0] C[3.4977092074; 0.49114842391]
Centroid: CG[1.54883491488; 0.16438280797]
Coordinates of the circumscribed circle: U[0.57439776862; 8.60332301996]
Coordinates of the inscribed circle: I[1.14397076862; 0.0879696125]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 11.81769192781° = 11°49'1″ = 2.93553485047 rad
∠ B' = β' = 172° = 0.14396263402 rad
∠ C' = γ' = 176.1833080722° = 176°10'59″ = 0.06766178087 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 3.53 ; ; b = 2.4 ; ; beta = 8° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 2.4**2 = 3.53**2 + c**2 -2 * 3.53 * c * cos (8° ) ; ; ; ; c**2 -6.994c +6.711 =0 ; ; p=1; q=-6.994; r=6.711 ; ; D = q**2 - 4pr = 6.994**2 - 4 * 1 * 6.711 = 22.0737729708 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 6.99 ± sqrt{ 22.07 } }{ 2 } ; ; c_{1,2} = 3.49709207 ± 2.34913670158 ; ; c_{1} = 5.84622877158 ; ; : Nr. 1
c_{2} = 1.14795536842 ; ; ; ; (c -5.84622877158) (c -1.14795536842) = 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 = 3.53 ; ; b = 2.4 ; ; c = 1.15 ; ;

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

p = a+b+c = 3.53+2.4+1.15 = 7.08 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 7.08 }{ 2 } = 3.54 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.54 * (3.54-3.53)(3.54-2.4)(3.54-1.15) } ; ; T = sqrt{ 0.08 } = 0.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.28 }{ 3.53 } = 0.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.28 }{ 2.4 } = 0.24 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.28 }{ 1.15 } = 0.49 ; ;

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{ 3.53**2-2.4**2-1.15**2 }{ 2 * 2.4 * 1.15 } ) = 168° 10'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2.4**2-3.53**2-1.15**2 }{ 2 * 3.53 * 1.15 } ) = 8° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.15**2-3.53**2-2.4**2 }{ 2 * 2.4 * 3.53 } ) = 3° 49'1" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.28 }{ 3.54 } = 0.08 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.53 }{ 2 * sin 168° 10'59" } = 8.62 ; ;




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