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?

1. Use Law of Cosines

a = 22.4 ; ; b = 16.8 ; ; beta = 22° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 16.8**2 = 22.4**2 + c**2 -2 * 22.4 * c * cos (22° ) ; ; ; ; c**2 -41.538c +219.52 =0 ; ; p=1; q=-41.538; 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} = 20.76891834 ± 14.5543110146 ; ; c_{1} = 35.3232293546 ; ;
c_{2} = 6.21460732543 ; ; ; ; text{ Factored form: } ; ; (c -35.3232293546) (c -6.21460732543) = 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 = 35.32 ; ;

2. 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 ; ;

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.8**2+2 * 35.32**2 - 22.4**2 } }{ 2 } = 25.289 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 35.32**2+2 * 22.4**2 - 16.8**2 } }{ 2 } = 28.358 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.8**2+2 * 22.4**2 - 35.32**2 } }{ 2 } = 8.948 ; ;







#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 - 2ac cos beta ; ; 16.8**2 = 22.4**2 + c**2 -2 * 22.4 * c * cos (22° ) ; ; ; ; c**2 -41.538c +219.52 =0 ; ; p=1; q=-41.538; 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} = 20.76891834 ± 14.5543110146 ; ; c_{1} = 35.3232293546 ; ; : Nr. 1
c_{2} = 6.21460732543 ; ; ; ; text{ Factored form: } ; ; (c -35.3232293546) (c -6.21460732543) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 16.8**2+6.21**2-22.4**2 }{ 2 * 16.8 * 6.21 } ) = 150° 2'5" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 22.4**2+6.21**2-16.8**2 }{ 2 * 22.4 * 6.21 } ) = 22° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 22.4**2+16.8**2-6.21**2 }{ 2 * 22.4 * 16.8 } ) = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.8**2+2 * 6.21**2 - 22.4**2 } }{ 2 } = 5.915 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.21**2+2 * 22.4**2 - 16.8**2 } }{ 2 } = 14.129 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.8**2+2 * 22.4**2 - 6.21**2 } }{ 2 } = 19.554 ; ;
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