Triangle calculator SSA

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Triangle has two solutions with side c=35.32332293569 and with side c=6.21546073277

#1 Obtuse scalene triangle.

Sides: a = 22.4   b = 16.8   c = 35.32332293569

Area: T = 148.202192372
Perimeter: p = 74.52332293569
Semiperimeter: s = 37.26216146784

Angle ∠ A = α = 29.96553026368° = 29°57'55″ = 0.52329931924 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 128.0354697363° = 128°2'5″ = 2.23546270258 rad

Height: ha = 13.23223146178
Height: hb = 17.64330861571
Height: hc = 8.39111876925

Median: ma = 25.28992322165
Median: mb = 28.35881604851
Median: mc = 8.94880370445

Inradius: r = 3.97773349867
Circumradius: R = 22.42435241655

Vertex coordinates: A[35.32332293569; 0] B[0; 0] C[20.76989183423; 8.39111876925]
Centroid: CG[18.69773825664; 2.79770625642]
Coordinates of the circumscribed circle: U[17.66216146784; -13.81659980801]
Coordinates of the inscribed circle: I[20.46216146784; 3.97773349867]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0354697363° = 150°2'5″ = 0.52329931924 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 51.96553026368° = 51°57'55″ = 2.23546270258 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 = 22.4 ; ; b = 16.8 ; ; c = 35.32 ; ;

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

p = a+b+c = 22.4+16.8+35.32 = 74.52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 74.52 }{ 2 } = 37.26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.26 * (37.26-22.4)(37.26-16.8)(37.26-35.32) } ; ; T = sqrt{ 21963.81 } = 148.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 148.2 }{ 22.4 } = 13.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 148.2 }{ 16.8 } = 17.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 148.2 }{ 35.32 } = 8.39 ; ;

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{ 22.4**2-16.8**2-35.32**2 }{ 2 * 16.8 * 35.32 } ) = 29° 57'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16.8**2-22.4**2-35.32**2 }{ 2 * 22.4 * 35.32 } ) = 22° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 35.32**2-22.4**2-16.8**2 }{ 2 * 16.8 * 22.4 } ) = 128° 2'5" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 148.2 }{ 37.26 } = 3.98 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 22.4 }{ 2 * sin 29° 57'55" } = 22.42 ; ;





#2 Obtuse scalene triangle.

Sides: a = 22.4   b = 16.8   c = 6.21546073277

Area: T = 26.07439682611
Perimeter: p = 45.41546073277
Semiperimeter: s = 22.70773036639

Angle ∠ A = α = 150.0354697363° = 150°2'5″ = 2.61985994612 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 7.96553026368° = 7°57'55″ = 0.13990207569 rad

Height: ha = 2.32880328805
Height: hb = 3.10440438406
Height: hc = 8.39111876925

Median: ma = 5.91552913807
Median: mb = 14.12990718775
Median: mc = 19.55436355684

Inradius: r = 1.14882635124
Circumradius: R = 22.42435241655

Vertex coordinates: A[6.21546073277; 0] B[0; 0] C[20.76989183423; 8.39111876925]
Centroid: CG[8.99545085567; 2.79770625642]
Coordinates of the circumscribed circle: U[3.10773036639; 22.20771857726]
Coordinates of the inscribed circle: I[5.90773036639; 1.14882635124]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.96553026368° = 29°57'55″ = 2.61985994612 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 172.0354697363° = 172°2'5″ = 0.13990207569 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 22.4 ; ; b = 16.8 ; ; beta = 22° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 16.8**2 = 22.4**2 + c**2 -2 * 16.8 * c * cos (22° ) ; ; ; ; c**2 -41.538c +219.52 =0 ; ; p=1; q=-41.5378366846; r=219.52 ; ; D = q**2 - 4pr = 41.538**2 - 4 * 1 * 219.52 = 847.311876436 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 41.54 ± sqrt{ 847.31 } }{ 2 } ; ; c_{1,2} = 20.7689183423 ± 14.5543110146 ; ;
c_{1} = 35.3232293569 ; ; c_{2} = 6.21460732772 ; ; ; ; (c -35.3232293569) (c -6.21460732772) = 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 = 22.4 ; ; b = 16.8 ; ; c = 6.21 ; ;

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

p = a+b+c = 22.4+16.8+6.21 = 45.41 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45.41 }{ 2 } = 22.71 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.71 * (22.71-22.4)(22.71-16.8)(22.71-6.21) } ; ; T = sqrt{ 679.85 } = 26.07 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 26.07 }{ 22.4 } = 2.33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 26.07 }{ 16.8 } = 3.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 26.07 }{ 6.21 } = 8.39 ; ;

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{ 22.4**2-16.8**2-6.21**2 }{ 2 * 16.8 * 6.21 } ) = 150° 2'5" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16.8**2-22.4**2-6.21**2 }{ 2 * 22.4 * 6.21 } ) = 22° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.21**2-22.4**2-16.8**2 }{ 2 * 16.8 * 22.4 } ) = 7° 57'55" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 26.07 }{ 22.71 } = 1.15 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 22.4 }{ 2 * sin 150° 2'5" } = 22.42 ; ;




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