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?

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 ; ;

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

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

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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 - 2bc cos( beta ) ; ; 32**2 = 34**2 + c**2 -2 * 32 * c * cos (28° ) ; ; ; ; c**2 -60.04c +132 =0 ; ; p=1; q=-60.0404363144; 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} = 30.0202181572 ± 27.7346984517 ; ; c_{1} = 57.7549166089 ; ;
c_{2} = 2.28551970552 ; ; ; ; (c -57.7549166089) (c -2.28551970552) = 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 = 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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