Triangle calculator SSA

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Triangle has two solutions with side c=14.48106911099 and with side c=5.24656059882

#1 Acute scalene triangle.

Sides: a = 12.35   b = 8.75   c = 14.48106911099

Area: T = 53.81332567882
Perimeter: p = 35.58106911099
Semiperimeter: s = 17.7990345555

Angle ∠ A = α = 58.14985238878° = 58°8'55″ = 1.0154883197 rad
Angle ∠ B = β = 37° = 0.64657718232 rad
Angle ∠ C = γ = 84.85114761122° = 84°51'5″ = 1.48109376333 rad

Height: ha = 8.71546974556
Height: hb = 12.33001729802
Height: hc = 7.43224155359

Median: ma = 10.24767474113
Median: mb = 12.72765797648
Median: mc = 7.8821617616

Inradius: r = 3.02548573094
Circumradius: R = 7.27696756174

Vertex coordinates: A[14.48106911099; 0] B[0; 0] C[9.86331485491; 7.43224155359]
Centroid: CG[8.11546132197; 2.47774718453]
Coordinates of the circumscribed circle: U[7.2440345555; 0.6522364796]
Coordinates of the inscribed circle: I[9.0440345555; 3.02548573094]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.8511476112° = 121°51'5″ = 1.0154883197 rad
∠ B' = β' = 143° = 0.64657718232 rad
∠ C' = γ' = 95.14985238878° = 95°8'55″ = 1.48109376333 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 12.35 ; ; b = 8.75 ; ; beta = 37° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 8.75**2 = 12.35**2 + c**2 -2 * 12.35 * c * cos (37° ) ; ; ; ; c**2 -19.726c +75.96 =0 ; ; p=1; q=-19.726; r=75.96 ; ; D = q**2 - 4pr = 19.726**2 - 4 * 1 * 75.96 = 85.2867972052 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 19.73 ± sqrt{ 85.29 } }{ 2 } ; ; c_{1,2} = 9.86314855 ± 4.61754256085 ; ; c_{1} = 14.4806911109 ; ; c_{2} = 5.24560598915 ; ; ; ; text{ Factored form: } ; ; (c -14.4806911109) (c -5.24560598915) = 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 = 12.35 ; ; b = 8.75 ; ; c = 14.48 ; ;

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

p = a+b+c = 12.35+8.75+14.48 = 35.58 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 35.58 }{ 2 } = 17.79 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.79 * (17.79-12.35)(17.79-8.75)(17.79-14.48) } ; ; T = sqrt{ 2895.87 } = 53.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 53.81 }{ 12.35 } = 8.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 53.81 }{ 8.75 } = 12.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 53.81 }{ 14.48 } = 7.43 ; ;

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{ 8.75**2+14.48**2-12.35**2 }{ 2 * 8.75 * 14.48 } ) = 58° 8'55" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.35**2+14.48**2-8.75**2 }{ 2 * 12.35 * 14.48 } ) = 37° ; ; gamma = 180° - alpha - beta = 180° - 58° 8'55" - 37° = 84° 51'5" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 53.81 }{ 17.79 } = 3.02 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.35 }{ 2 * sin 58° 8'55" } = 7.27 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.75**2+2 * 14.48**2 - 12.35**2 } }{ 2 } = 10.247 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.48**2+2 * 12.35**2 - 8.75**2 } }{ 2 } = 12.727 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.75**2+2 * 12.35**2 - 14.48**2 } }{ 2 } = 7.882 ; ;







#2 Obtuse scalene triangle.

Sides: a = 12.35   b = 8.75   c = 5.24656059882

Area: T = 19.49437617211
Perimeter: p = 26.34656059882
Semiperimeter: s = 13.17328029941

Angle ∠ A = α = 121.8511476112° = 121°51'5″ = 2.12767094566 rad
Angle ∠ B = β = 37° = 0.64657718232 rad
Angle ∠ C = γ = 21.14985238878° = 21°8'55″ = 0.36991113738 rad

Height: ha = 3.15768844893
Height: hb = 4.45657169648
Height: hc = 7.43224155359

Median: ma = 3.72994525191
Median: mb = 8.41989557602
Median: mc = 10.37660977469

Inradius: r = 1.48798491809
Circumradius: R = 7.27696756174

Vertex coordinates: A[5.24656059882; 0] B[0; 0] C[9.86331485491; 7.43224155359]
Centroid: CG[5.03662515124; 2.47774718453]
Coordinates of the circumscribed circle: U[2.62328029941; 6.788005074]
Coordinates of the inscribed circle: I[4.42328029941; 1.48798491809]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 58.14985238878° = 58°8'55″ = 2.12767094566 rad
∠ B' = β' = 143° = 0.64657718232 rad
∠ C' = γ' = 158.8511476112° = 158°51'5″ = 0.36991113738 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 12.35 ; ; b = 8.75 ; ; beta = 37° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 8.75**2 = 12.35**2 + c**2 -2 * 12.35 * c * cos (37° ) ; ; ; ; c**2 -19.726c +75.96 =0 ; ; p=1; q=-19.726; r=75.96 ; ; D = q**2 - 4pr = 19.726**2 - 4 * 1 * 75.96 = 85.2867972052 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 19.73 ± sqrt{ 85.29 } }{ 2 } ; ; c_{1,2} = 9.86314855 ± 4.61754256085 ; ; c_{1} = 14.4806911109 ; ; c_{2} = 5.24560598915 ; ; ; ; text{ Factored form: } ; ; (c -14.4806911109) (c -5.24560598915) = 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 = 12.35 ; ; b = 8.75 ; ; c = 5.25 ; ;

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

p = a+b+c = 12.35+8.75+5.25 = 26.35 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.35 }{ 2 } = 13.17 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.17 * (13.17-12.35)(13.17-8.75)(13.17-5.25) } ; ; T = sqrt{ 380.01 } = 19.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 19.49 }{ 12.35 } = 3.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 19.49 }{ 8.75 } = 4.46 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 19.49 }{ 5.25 } = 7.43 ; ;

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{ 8.75**2+5.25**2-12.35**2 }{ 2 * 8.75 * 5.25 } ) = 121° 51'5" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.35**2+5.25**2-8.75**2 }{ 2 * 12.35 * 5.25 } ) = 37° ; ; gamma = 180° - alpha - beta = 180° - 121° 51'5" - 37° = 21° 8'55" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 19.49 }{ 13.17 } = 1.48 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.35 }{ 2 * sin 121° 51'5" } = 7.27 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.75**2+2 * 5.25**2 - 12.35**2 } }{ 2 } = 3.729 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.25**2+2 * 12.35**2 - 8.75**2 } }{ 2 } = 8.419 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.75**2+2 * 12.35**2 - 5.25**2 } }{ 2 } = 10.376 ; ;
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