Triangle calculator SSA

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Triangle has two solutions with side c=15.21552242202 and with side c=6.03442521853

#1 Obtuse scalene triangle.

Sides: a = 10.95   b = 5.3   c = 15.21552242202

Area: T = 20.15329049721
Perimeter: p = 31.46552242202
Semiperimeter: s = 15.73326121101

Angle ∠ A = α = 29.98880764935° = 29°59'17″ = 0.52333906712 rad
Angle ∠ B = β = 14° = 0.24443460953 rad
Angle ∠ C = γ = 136.0121923506° = 136°43″ = 2.37438558872 rad

Height: ha = 3.68108958853
Height: hb = 7.60548698008
Height: hc = 2.64990447568

Median: ma = 9.99110409385
Median: mb = 12.98876970258
Median: mc = 4.01550327499

Inradius: r = 1.2810963697
Circumradius: R = 10.95439485603

Vertex coordinates: A[15.21552242202; 0] B[0; 0] C[10.62547382027; 2.64990447568]
Centroid: CG[8.61333208076; 0.88330149189]
Coordinates of the circumscribed circle: U[7.60876121101; -7.88111945188]
Coordinates of the inscribed circle: I[10.43326121101; 1.2810963697]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0121923506° = 150°43″ = 0.52333906712 rad
∠ B' = β' = 166° = 0.24443460953 rad
∠ C' = γ' = 43.98880764935° = 43°59'17″ = 2.37438558872 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 = 10.95 ; ; b = 5.3 ; ; c = 15.22 ; ;

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

p = a+b+c = 10.95+5.3+15.22 = 31.47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.47 }{ 2 } = 15.73 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.73 * (15.73-10.95)(15.73-5.3)(15.73-15.22) } ; ; T = sqrt{ 406.14 } = 20.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20.15 }{ 10.95 } = 3.68 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20.15 }{ 5.3 } = 7.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20.15 }{ 15.22 } = 2.65 ; ;

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{ 10.95**2-5.3**2-15.22**2 }{ 2 * 5.3 * 15.22 } ) = 29° 59'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.3**2-10.95**2-15.22**2 }{ 2 * 10.95 * 15.22 } ) = 14° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15.22**2-10.95**2-5.3**2 }{ 2 * 5.3 * 10.95 } ) = 136° 43" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20.15 }{ 15.73 } = 1.28 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.95 }{ 2 * sin 29° 59'17" } = 10.95 ; ;





#2 Obtuse scalene triangle.

Sides: a = 10.95   b = 5.3   c = 6.03442521853

Area: T = 7.99325020564
Perimeter: p = 22.28442521853
Semiperimeter: s = 11.14221260926

Angle ∠ A = α = 150.0121923506° = 150°43″ = 2.61882019824 rad
Angle ∠ B = β = 14° = 0.24443460953 rad
Angle ∠ C = γ = 15.98880764935° = 15°59'17″ = 0.27990445759 rad

Height: ha = 1.46598177272
Height: hb = 3.01660385118
Height: hc = 2.64990447568

Median: ma = 1.50884676721
Median: mb = 8.43441478359
Median: mc = 8.05656315793

Inradius: r = 0.7177322887
Circumradius: R = 10.95439485603

Vertex coordinates: A[6.03442521853; 0] B[0; 0] C[10.62547382027; 2.64990447568]
Centroid: CG[5.5532996796; 0.88330149189]
Coordinates of the circumscribed circle: U[3.01771260926; 10.53302392756]
Coordinates of the inscribed circle: I[5.84221260926; 0.7177322887]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.98880764935° = 29°59'17″ = 2.61882019824 rad
∠ B' = β' = 166° = 0.24443460953 rad
∠ C' = γ' = 164.0121923506° = 164°43″ = 0.27990445759 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 10.95 ; ; b = 5.3 ; ; beta = 14° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 5.3**2 = 10.95**2 + c**2 -2 * 5.3 * c * cos (14° ) ; ; ; ; c**2 -21.249c +91.813 =0 ; ; p=1; q=-21.2494764054; r=91.8125 ; ; D = q**2 - 4pr = 21.249**2 - 4 * 1 * 91.813 = 84.2902475055 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 21.25 ± sqrt{ 84.29 } }{ 2 } ; ; c_{1,2} = 10.6247382027 ± 4.59048601745 ; ;
c_{1} = 15.2152242202 ; ; c_{2} = 6.03425218527 ; ; ; ; (c -15.2152242202) (c -6.03425218527) = 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 = 10.95 ; ; b = 5.3 ; ; c = 6.03 ; ;

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

p = a+b+c = 10.95+5.3+6.03 = 22.28 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.28 }{ 2 } = 11.14 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.14 * (11.14-10.95)(11.14-5.3)(11.14-6.03) } ; ; T = sqrt{ 63.88 } = 7.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.99 }{ 10.95 } = 1.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.99 }{ 5.3 } = 3.02 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.99 }{ 6.03 } = 2.65 ; ;

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{ 10.95**2-5.3**2-6.03**2 }{ 2 * 5.3 * 6.03 } ) = 150° 43" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.3**2-10.95**2-6.03**2 }{ 2 * 10.95 * 6.03 } ) = 14° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.03**2-10.95**2-5.3**2 }{ 2 * 5.3 * 10.95 } ) = 15° 59'17" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.99 }{ 11.14 } = 0.72 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.95 }{ 2 * sin 150° 43" } = 10.95 ; ;




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