Triangle calculator SSA

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Triangle has two solutions with side c=42.45503886154 and with side c=15.90109145032

#1 Acute scalene triangle.

Sides: a = 42   b = 33   c = 42.45503886154

Area: T = 641.2611335488
Perimeter: p = 117.4550388615
Semiperimeter: s = 58.72551943077

Angle ∠ A = α = 66.28801639305° = 66°16'49″ = 1.15768070893 rad
Angle ∠ B = β = 46° = 0.80328514559 rad
Angle ∠ C = γ = 67.72198360695° = 67°43'11″ = 1.18219341083 rad

Height: ha = 30.53662540709
Height: hb = 38.86443233629
Height: hc = 30.21222716142

Median: ma = 31.69441279545
Median: mb = 38.86985958943
Median: mc = 31.2410856688

Inradius: r = 10.92196971257
Circumradius: R = 22.93876992518

Vertex coordinates: A[42.45503886154; 0] B[0; 0] C[29.17656515593; 30.21222716142]
Centroid: CG[23.87553467249; 10.07107572047]
Coordinates of the circumscribed circle: U[21.22551943077; 8.69765035253]
Coordinates of the inscribed circle: I[25.72551943077; 10.92196971257]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 113.7219836069° = 113°43'11″ = 1.15768070893 rad
∠ B' = β' = 134° = 0.80328514559 rad
∠ C' = γ' = 112.2880163931° = 112°16'49″ = 1.18219341083 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 42 ; ; b = 33 ; ; beta = 46° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 33**2 = 42**2 + c**2 -2 * 42 * c * cos (46° ) ; ; ; ; c**2 -58.351c +675 =0 ; ; p=1; q=-58.351; r=675 ; ; D = q**2 - 4pr = 58.351**2 - 4 * 1 * 675 = 704.874575634 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 58.35 ± sqrt{ 704.87 } }{ 2 } ; ; c_{1,2} = 29.17565156 ± 13.2747370561 ; ; c_{1} = 42.4503886161 ; ;
c_{2} = 15.9009145039 ; ; ; ; text{ Factored form: } ; ; (c -42.4503886161) (c -15.9009145039) = 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 = 42 ; ; b = 33 ; ; c = 42.45 ; ;

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

p = a+b+c = 42+33+42.45 = 117.45 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 117.45 }{ 2 } = 58.73 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 58.73 * (58.73-42)(58.73-33)(58.73-42.45) } ; ; T = sqrt{ 411216.1 } = 641.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 641.26 }{ 42 } = 30.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 641.26 }{ 33 } = 38.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 641.26 }{ 42.45 } = 30.21 ; ;

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{ 33**2+42.45**2-42**2 }{ 2 * 33 * 42.45 } ) = 66° 16'49" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 42**2+42.45**2-33**2 }{ 2 * 42 * 42.45 } ) = 46° ; ; gamma = 180° - alpha - beta = 180° - 66° 16'49" - 46° = 67° 43'11" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 641.26 }{ 58.73 } = 10.92 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 42 }{ 2 * sin 66° 16'49" } = 22.94 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 42.45**2 - 42**2 } }{ 2 } = 31.694 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 42.45**2+2 * 42**2 - 33**2 } }{ 2 } = 38.869 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 42**2 - 42.45**2 } }{ 2 } = 31.241 ; ;







#2 Obtuse scalene triangle.

Sides: a = 42   b = 33   c = 15.90109145032

Area: T = 240.2011373942
Perimeter: p = 90.90109145032
Semiperimeter: s = 45.45504572516

Angle ∠ A = α = 113.7219836069° = 113°43'11″ = 1.98547855642 rad
Angle ∠ B = β = 46° = 0.80328514559 rad
Angle ∠ C = γ = 20.28801639305° = 20°16'49″ = 0.35439556334 rad

Height: ha = 11.43881606639
Height: hb = 14.55876590268
Height: hc = 30.21222716142

Median: ma = 15.1633098002
Median: mb = 27.1322444435
Median: mc = 36.92327603179

Inradius: r = 5.28549055536
Circumradius: R = 22.93876992518

Vertex coordinates: A[15.90109145032; 0] B[0; 0] C[29.17656515593; 30.21222716142]
Centroid: CG[15.02655220208; 10.07107572047]
Coordinates of the circumscribed circle: U[7.95504572516; 21.51657680889]
Coordinates of the inscribed circle: I[12.45504572516; 5.28549055536]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 66.28801639305° = 66°16'49″ = 1.98547855642 rad
∠ B' = β' = 134° = 0.80328514559 rad
∠ C' = γ' = 159.7219836069° = 159°43'11″ = 0.35439556334 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 42 ; ; b = 33 ; ; beta = 46° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 33**2 = 42**2 + c**2 -2 * 42 * c * cos (46° ) ; ; ; ; c**2 -58.351c +675 =0 ; ; p=1; q=-58.351; r=675 ; ; D = q**2 - 4pr = 58.351**2 - 4 * 1 * 675 = 704.874575634 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 58.35 ± sqrt{ 704.87 } }{ 2 } ; ; c_{1,2} = 29.17565156 ± 13.2747370561 ; ; c_{1} = 42.4503886161 ; ; : Nr. 1
c_{2} = 15.9009145039 ; ; ; ; text{ Factored form: } ; ; (c -42.4503886161) (c -15.9009145039) = 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 = 42 ; ; b = 33 ; ; c = 15.9 ; ;

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

p = a+b+c = 42+33+15.9 = 90.9 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 90.9 }{ 2 } = 45.45 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 45.45 * (45.45-42)(45.45-33)(45.45-15.9) } ; ; T = sqrt{ 57696.7 } = 240.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 240.2 }{ 42 } = 11.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 240.2 }{ 33 } = 14.56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 240.2 }{ 15.9 } = 30.21 ; ;

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{ 33**2+15.9**2-42**2 }{ 2 * 33 * 15.9 } ) = 113° 43'11" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 42**2+15.9**2-33**2 }{ 2 * 42 * 15.9 } ) = 46° ; ; gamma = 180° - alpha - beta = 180° - 113° 43'11" - 46° = 20° 16'49" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 240.2 }{ 45.45 } = 5.28 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 42 }{ 2 * sin 113° 43'11" } = 22.94 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 15.9**2 - 42**2 } }{ 2 } = 15.163 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.9**2+2 * 42**2 - 33**2 } }{ 2 } = 27.132 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 42**2 - 15.9**2 } }{ 2 } = 36.923 ; ;
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