Triangle calculator SSA

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Triangle has two solutions with side c=12.79436569493 and with side c=1.36328706842

#1 Obtuse scalene triangle.

Sides: a = 7.81   b = 6.6   c = 12.79436569493

Area: T = 21.11436831041
Perimeter: p = 27.20436569493
Semiperimeter: s = 13.60218284747

Angle ∠ A = α = 30.00765021269° = 30°23″ = 0.52437122591 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 124.9933497873° = 124°59'37″ = 2.18215480815 rad

Height: ha = 5.40768330612
Height: hb = 6.39880857891
Height: hc = 3.30106486242

Median: ma = 9.40105214785
Median: mb = 10.07220841472
Median: mc = 3.37702574777

Inradius: r = 1.55222680016
Circumradius: R = 7.80884652244

Vertex coordinates: A[12.79436569493; 0] B[0; 0] C[7.07882638168; 3.30106486242]
Centroid: CG[6.62439735887; 1.11002162081]
Coordinates of the circumscribed circle: U[6.39768284747; -4.4788025751]
Coordinates of the inscribed circle: I[7.00218284747; 1.55222680016]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.9933497873° = 149°59'37″ = 0.52437122591 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 55.00765021269° = 55°23″ = 2.18215480815 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 = 7.81 ; ; b = 6.6 ; ; c = 12.79 ; ;

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

p = a+b+c = 7.81+6.6+12.79 = 27.2 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.2 }{ 2 } = 13.6 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.6 * (13.6-7.81)(13.6-6.6)(13.6-12.79) } ; ; T = sqrt{ 445.79 } = 21.11 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21.11 }{ 7.81 } = 5.41 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21.11 }{ 6.6 } = 6.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21.11 }{ 12.79 } = 3.3 ; ;

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{ 7.81**2-6.6**2-12.79**2 }{ 2 * 6.6 * 12.79 } ) = 30° 23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.6**2-7.81**2-12.79**2 }{ 2 * 7.81 * 12.79 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.79**2-7.81**2-6.6**2 }{ 2 * 6.6 * 7.81 } ) = 124° 59'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21.11 }{ 13.6 } = 1.55 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.81 }{ 2 * sin 30° 23" } = 7.81 ; ;





#2 Obtuse scalene triangle.

Sides: a = 7.81   b = 6.6   c = 1.36328706842

Area: T = 2.24991786244
Perimeter: p = 15.77328706842
Semiperimeter: s = 7.88664353421

Angle ∠ A = α = 149.9933497873° = 149°59'37″ = 2.61878803945 rad
Angle ∠ B = β = 25° = 0.4366332313 rad
Angle ∠ C = γ = 5.00765021269° = 5°23″ = 0.08773799461 rad

Height: ha = 0.57659740395
Height: hb = 0.68215692801
Height: hc = 3.30106486242

Median: ma = 2.73112420711
Median: mb = 4.53217500208
Median: mc = 7.19881730928

Inradius: r = 0.28551958492
Circumradius: R = 7.80884652244

Vertex coordinates: A[1.36328706842; 0] B[0; 0] C[7.07882638168; 3.30106486242]
Centroid: CG[2.81437115003; 1.11002162081]
Coordinates of the circumscribed circle: U[0.68114353421; 7.77986743752]
Coordinates of the inscribed circle: I[1.28664353421; 0.28551958492]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 30.00765021269° = 30°23″ = 2.61878803945 rad
∠ B' = β' = 155° = 0.4366332313 rad
∠ C' = γ' = 174.9933497873° = 174°59'37″ = 0.08773799461 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 7.81 ; ; b = 6.6 ; ; beta = 25° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 6.6**2 = 7.81**2 + c**2 -2 * 6.6 * c * cos (25° ) ; ; ; ; c**2 -14.157c +17.436 =0 ; ; p=1; q=-14.1565276335; r=17.4361 ; ; D = q**2 - 4pr = 14.157**2 - 4 * 1 * 17.436 = 130.662874638 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 14.16 ± sqrt{ 130.66 } }{ 2 } ; ; c_{1,2} = 7.07826381676 ± 5.71539313255 ; ;
c_{1} = 12.7936569493 ; ; c_{2} = 1.36287068421 ; ; ; ; (c -12.7936569493) (c -1.36287068421) = 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 = 7.81 ; ; b = 6.6 ; ; c = 1.36 ; ;

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

p = a+b+c = 7.81+6.6+1.36 = 15.77 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.77 }{ 2 } = 7.89 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.89 * (7.89-7.81)(7.89-6.6)(7.89-1.36) } ; ; T = sqrt{ 5.06 } = 2.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.25 }{ 7.81 } = 0.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.25 }{ 6.6 } = 0.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.25 }{ 1.36 } = 3.3 ; ;

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{ 7.81**2-6.6**2-1.36**2 }{ 2 * 6.6 * 1.36 } ) = 149° 59'37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.6**2-7.81**2-1.36**2 }{ 2 * 7.81 * 1.36 } ) = 25° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.36**2-7.81**2-6.6**2 }{ 2 * 6.6 * 7.81 } ) = 5° 23" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.25 }{ 7.89 } = 0.29 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.81 }{ 2 * sin 149° 59'37" } = 7.81 ; ;




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