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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

5. 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{ 42**2-33**2-42.45**2 }{ 2 * 33 * 42.45 } ) = 66° 16'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 33**2-42**2-42.45**2 }{ 2 * 42 * 42.45 } ) = 46° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 42.45**2-42**2-33**2 }{ 2 * 33 * 42 } ) = 67° 43'11" ; ;

6. Inradius

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

7. Circumradius

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





#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 - 2bc cos( beta ) ; ; 33**2 = 42**2 + c**2 -2 * 33 * c * cos (46° ) ; ; ; ; c**2 -58.351c +675 =0 ; ; p=1; q=-58.3513031186; 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} = 29.1756515593 ± 13.2747370561 ; ; c_{1} = 42.4503886154 ; ;
c_{2} = 15.9009145032 ; ; ; ; (c -42.4503886154) (c -15.9009145032) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 42**2-33**2-15.9**2 }{ 2 * 33 * 15.9 } ) = 113° 43'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 33**2-42**2-15.9**2 }{ 2 * 42 * 15.9 } ) = 46° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15.9**2-42**2-33**2 }{ 2 * 33 * 42 } ) = 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 ; ;




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