Triangle calculator SSA

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Triangle has two solutions with side c=57.75549166089 and with side c=2.28655197055

#1 Obtuse scalene triangle.

Sides: a = 34   b = 32   c = 57.75549166089

Area: T = 460.9432946302
Perimeter: p = 123.7554916609
Semiperimeter: s = 61.87774583044

Angle ∠ A = α = 29.9221535133° = 29°55'18″ = 0.52222293053 rad
Angle ∠ B = β = 28° = 0.48986921906 rad
Angle ∠ C = γ = 122.0788464867° = 122°4'42″ = 2.13106711577 rad

Height: ha = 27.11442909589
Height: hb = 28.80989341439
Height: hc = 15.96220331347

Median: ma = 43.48435048754
Median: mb = 44.60773446447
Median: mc = 16.00328872981

Inradius: r = 7.44992870091
Circumradius: R = 34.0810871491

Vertex coordinates: A[57.75549166089; 0] B[0; 0] C[30.02202181572; 15.96220331347]
Centroid: CG[29.25883782554; 5.32106777116]
Coordinates of the circumscribed circle: U[28.87774583044; -18.10996741259]
Coordinates of the inscribed circle: I[29.87774583044; 7.44992870091]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0788464867° = 150°4'42″ = 0.52222293053 rad
∠ B' = β' = 152° = 0.48986921906 rad
∠ C' = γ' = 57.9221535133° = 57°55'18″ = 2.13106711577 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 34 ; ; b = 32 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 32**2 = 34**2 + c**2 -2 * 34 * c * cos (28° ) ; ; ; ; c**2 -60.04c +132 =0 ; ; p=1; q=-60.04; r=132 ; ; D = q**2 - 4pr = 60.04**2 - 4 * 1 * 132 = 3076.85399282 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 60.04 ± sqrt{ 3076.85 } }{ 2 } ; ; c_{1,2} = 30.02021816 ± 27.7346984517 ; ; c_{1} = 57.7549166117 ; ;
c_{2} = 2.28551970831 ; ; ; ; (c -57.7549166117) (c -2.28551970831) = 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 = 34 ; ; b = 32 ; ; c = 57.75 ; ;

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

p = a+b+c = 34+32+57.75 = 123.75 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 123.75 }{ 2 } = 61.88 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 61.88 * (61.88-34)(61.88-32)(61.88-57.75) } ; ; T = sqrt{ 212468.4 } = 460.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 460.94 }{ 34 } = 27.11 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 460.94 }{ 32 } = 28.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 460.94 }{ 57.75 } = 15.96 ; ;

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{ 34**2-32**2-57.75**2 }{ 2 * 32 * 57.75 } ) = 29° 55'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32**2-34**2-57.75**2 }{ 2 * 34 * 57.75 } ) = 28° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 57.75**2-34**2-32**2 }{ 2 * 32 * 34 } ) = 122° 4'42" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 460.94 }{ 61.88 } = 7.45 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 34 }{ 2 * sin 29° 55'18" } = 34.08 ; ;





#2 Obtuse scalene triangle.

Sides: a = 34   b = 32   c = 2.28655197055

Area: T = 18.24107706347
Perimeter: p = 68.28655197055
Semiperimeter: s = 34.14327598528

Angle ∠ A = α = 150.0788464867° = 150°4'42″ = 2.61993633483 rad
Angle ∠ B = β = 28° = 0.48986921906 rad
Angle ∠ C = γ = 1.9221535133° = 1°55'18″ = 0.03435371148 rad

Height: ha = 1.07329865079
Height: hb = 1.14400481647
Height: hc = 15.96220331347

Median: ma = 15.02203794946
Median: mb = 18.0176986434
Median: mc = 32.99553648248

Inradius: r = 0.53442500347
Circumradius: R = 34.0810871491

Vertex coordinates: A[2.28655197055; 0] B[0; 0] C[30.02202181572; 15.96220331347]
Centroid: CG[10.76985792876; 5.32106777116]
Coordinates of the circumscribed circle: U[1.14327598528; 34.06217072606]
Coordinates of the inscribed circle: I[2.14327598528; 0.53442500347]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.9221535133° = 29°55'18″ = 2.61993633483 rad
∠ B' = β' = 152° = 0.48986921906 rad
∠ C' = γ' = 178.0788464867° = 178°4'42″ = 0.03435371148 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 34 ; ; b = 32 ; ; beta = 28° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 32**2 = 34**2 + c**2 -2 * 34 * c * cos (28° ) ; ; ; ; c**2 -60.04c +132 =0 ; ; p=1; q=-60.04; r=132 ; ; D = q**2 - 4pr = 60.04**2 - 4 * 1 * 132 = 3076.85399282 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 60.04 ± sqrt{ 3076.85 } }{ 2 } ; ; c_{1,2} = 30.02021816 ± 27.7346984517 ; ; c_{1} = 57.7549166117 ; ; : Nr. 1
c_{2} = 2.28551970831 ; ; ; ; (c -57.7549166117) (c -2.28551970831) = 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 = 34 ; ; b = 32 ; ; c = 2.29 ; ;

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

p = a+b+c = 34+32+2.29 = 68.29 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 68.29 }{ 2 } = 34.14 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.14 * (34.14-34)(34.14-32)(34.14-2.29) } ; ; T = sqrt{ 332.73 } = 18.24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 18.24 }{ 34 } = 1.07 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 18.24 }{ 32 } = 1.14 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 18.24 }{ 2.29 } = 15.96 ; ;

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{ 34**2-32**2-2.29**2 }{ 2 * 32 * 2.29 } ) = 150° 4'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 32**2-34**2-2.29**2 }{ 2 * 34 * 2.29 } ) = 28° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.29**2-34**2-32**2 }{ 2 * 32 * 34 } ) = 1° 55'18" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 18.24 }{ 34.14 } = 0.53 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 34 }{ 2 * sin 150° 4'42" } = 34.08 ; ;




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