Triangle calculator SSA

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Triangle has two solutions with side c=12.77882225669 and with side c=2.70546015061

#1 Acute scalene triangle.

Sides: a = 10.5   b = 8.7   c = 12.77882225669

Area: T = 45.32224206938
Perimeter: p = 31.97882225669
Semiperimeter: s = 15.98991112834

Angle ∠ A = α = 54.62437148787° = 54°37'25″ = 0.95333636743 rad
Angle ∠ B = β = 42.5° = 42°30' = 0.74217649321 rad
Angle ∠ C = γ = 82.87662851213° = 82°52'35″ = 1.44664640472 rad

Height: ha = 8.63328420369
Height: hb = 10.41989472859
Height: hc = 7.094369718

Median: ma = 9.58876997233
Median: mb = 10.85655969889
Median: mc = 7.22114442467

Inradius: r = 2.83545803522
Circumradius: R = 6.43988144632

Vertex coordinates: A[12.77882225669; 0] B[0; 0] C[7.74114120365; 7.094369718]
Centroid: CG[6.84398782011; 2.36545657267]
Coordinates of the circumscribed circle: U[6.38991112834; 0.79884915149]
Coordinates of the inscribed circle: I[7.28991112834; 2.83545803522]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.3766285121° = 125°22'35″ = 0.95333636743 rad
∠ B' = β' = 137.5° = 137°30' = 0.74217649321 rad
∠ C' = γ' = 97.12437148787° = 97°7'25″ = 1.44664640472 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 = 10.5 ; ; b = 8.7 ; ; c = 12.78 ; ;

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

p = a+b+c = 10.5+8.7+12.78 = 31.98 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.98 }{ 2 } = 15.99 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.99 * (15.99-10.5)(15.99-8.7)(15.99-12.78) } ; ; T = sqrt{ 2054.12 } = 45.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 45.32 }{ 10.5 } = 8.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 45.32 }{ 8.7 } = 10.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 45.32 }{ 12.78 } = 7.09 ; ;

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{ 10.5**2-8.7**2-12.78**2 }{ 2 * 8.7 * 12.78 } ) = 54° 37'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.7**2-10.5**2-12.78**2 }{ 2 * 10.5 * 12.78 } ) = 42° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.78**2-10.5**2-8.7**2 }{ 2 * 8.7 * 10.5 } ) = 82° 52'35" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 45.32 }{ 15.99 } = 2.83 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.5 }{ 2 * sin 54° 37'25" } = 6.44 ; ;





#2 Obtuse scalene triangle.

Sides: a = 10.5   b = 8.7   c = 2.70546015061

Area: T = 9.59328120385
Perimeter: p = 21.90546015061
Semiperimeter: s = 10.95223007531

Angle ∠ A = α = 125.3766285121° = 125°22'35″ = 2.18882289793 rad
Angle ∠ B = β = 42.5° = 42°30' = 0.74217649321 rad
Angle ∠ C = γ = 12.12437148787° = 12°7'25″ = 0.21215987422 rad

Height: ha = 1.8277202293
Height: hb = 2.20552441468
Height: hc = 7.094369718

Median: ma = 3.73436221894
Median: mb = 6.31334724719
Median: mc = 9.54767943663

Inradius: r = 0.87658718606
Circumradius: R = 6.43988144632

Vertex coordinates: A[2.70546015061; 0] B[0; 0] C[7.74114120365; 7.094369718]
Centroid: CG[3.48220045142; 2.36545657267]
Coordinates of the circumscribed circle: U[1.35223007531; 6.2955205665]
Coordinates of the inscribed circle: I[2.25223007531; 0.87658718606]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 54.62437148787° = 54°37'25″ = 2.18882289793 rad
∠ B' = β' = 137.5° = 137°30' = 0.74217649321 rad
∠ C' = γ' = 167.8766285121° = 167°52'35″ = 0.21215987422 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 10.5 ; ; b = 8.7 ; ; beta = 42° 30' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 8.7**2 = 10.5**2 + c**2 -2 * 8.7 * c * cos (42° 30') ; ; ; ; c**2 -15.483c +34.56 =0 ; ; p=1; q=-15.482824073; r=34.56 ; ; D = q**2 - 4pr = 15.483**2 - 4 * 1 * 34.56 = 101.477841276 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.48 ± sqrt{ 101.48 } }{ 2 } ; ; c_{1,2} = 7.74141203651 ± 5.03681053038 ; ;
c_{1} = 12.7782225669 ; ; c_{2} = 2.70460150612 ; ; ; ; (c -12.7782225669) (c -2.70460150612) = 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 = 10.5 ; ; b = 8.7 ; ; c = 2.7 ; ;

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

p = a+b+c = 10.5+8.7+2.7 = 21.9 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.9 }{ 2 } = 10.95 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.95 * (10.95-10.5)(10.95-8.7)(10.95-2.7) } ; ; T = sqrt{ 92.02 } = 9.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9.59 }{ 10.5 } = 1.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9.59 }{ 8.7 } = 2.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9.59 }{ 2.7 } = 7.09 ; ;

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{ 10.5**2-8.7**2-2.7**2 }{ 2 * 8.7 * 2.7 } ) = 125° 22'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.7**2-10.5**2-2.7**2 }{ 2 * 10.5 * 2.7 } ) = 42° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.7**2-10.5**2-8.7**2 }{ 2 * 8.7 * 10.5 } ) = 12° 7'25" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9.59 }{ 10.95 } = 0.88 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.5 }{ 2 * sin 125° 22'35" } = 6.44 ; ;




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