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?

1. Use Law of Cosines

a = 10.5 ; ; b = 8.7 ; ; beta = 42° 30' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 8.7**2 = 10.5**2 + c**2 -2 * 10.5 * c * cos (42° 30') ; ; ; ; c**2 -15.483c +34.56 =0 ; ; p=1; q=-15.483; 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} = 7.74141204 ± 5.03681053038 ; ; c_{1} = 12.7782225704 ; ;
c_{2} = 2.70460150962 ; ; ; ; text{ Factored form: } ; ; (c -12.7782225704) (c -2.70460150962) = 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 = 12.78 ; ;

2. 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 ; ;

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 8.7**2+12.78**2-10.5**2 }{ 2 * 8.7 * 12.78 } ) = 54° 37'25" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.5**2+12.78**2-8.7**2 }{ 2 * 10.5 * 12.78 } ) = 42° 30' ; ; gamma = 180° - alpha - beta = 180° - 54° 37'25" - 42° 30' = 82° 52'35" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.7**2+2 * 12.78**2 - 10.5**2 } }{ 2 } = 9.588 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.78**2+2 * 10.5**2 - 8.7**2 } }{ 2 } = 10.856 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.7**2+2 * 10.5**2 - 12.78**2 } }{ 2 } = 7.221 ; ;







#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

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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 - 2ac cos beta ; ; 8.7**2 = 10.5**2 + c**2 -2 * 10.5 * c * cos (42° 30') ; ; ; ; c**2 -15.483c +34.56 =0 ; ; p=1; q=-15.483; 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} = 7.74141204 ± 5.03681053038 ; ; c_{1} = 12.7782225704 ; ; : Nr. 1
c_{2} = 2.70460150962 ; ; ; ; text{ Factored form: } ; ; (c -12.7782225704) (c -2.70460150962) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 8.7**2+2.7**2-10.5**2 }{ 2 * 8.7 * 2.7 } ) = 125° 22'35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.5**2+2.7**2-8.7**2 }{ 2 * 10.5 * 2.7 } ) = 42° 30' ; ; gamma = 180° - alpha - beta = 180° - 125° 22'35" - 42° 30' = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.7**2+2 * 2.7**2 - 10.5**2 } }{ 2 } = 3.734 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.7**2+2 * 10.5**2 - 8.7**2 } }{ 2 } = 6.313 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.7**2+2 * 10.5**2 - 2.7**2 } }{ 2 } = 9.547 ; ;
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