Triangle calculator SSA

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Triangle has two solutions with side c=18.10218420219 and with side c=1.12660787701

#1 Obtuse scalene triangle.

Sides: a = 10.79   b = 9.8   c = 18.10218420219

Area: T = 44.33664569294
Perimeter: p = 38.69218420219
Semiperimeter: s = 19.3465921011

Angle ∠ A = α = 29.99902621564° = 29°59'25″ = 0.52334288182 rad
Angle ∠ B = β = 27° = 0.4711238898 rad
Angle ∠ C = γ = 123.0109737844° = 123°35″ = 2.14769249374 rad

Height: ha = 8.21880643057
Height: hb = 9.04882565162
Height: hc = 4.89985574922

Median: ma = 13.51985915425
Median: mb = 14.07326824839
Median: mc = 4.93108091479

Inradius: r = 2.29217728706
Circumradius: R = 10.79331773965

Vertex coordinates: A[18.10218420219; 0] B[0; 0] C[9.6143960396; 4.89985574922]
Centroid: CG[9.2398600806; 1.63328524974]
Coordinates of the circumscribed circle: U[9.0510921011; -5.88799240782]
Coordinates of the inscribed circle: I[9.5465921011; 2.29217728706]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0109737844° = 150°35″ = 0.52334288182 rad
∠ B' = β' = 153° = 0.4711238898 rad
∠ C' = γ' = 56.99902621564° = 56°59'25″ = 2.14769249374 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 10.79 ; ; b = 9.8 ; ; beta = 27° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 9.8**2 = 10.79**2 + c**2 -2 * 10.79 * c * cos (27° ) ; ; ; ; c**2 -19.228c +20.384 =0 ; ; p=1; q=-19.228; r=20.384 ; ; D = q**2 - 4pr = 19.228**2 - 4 * 1 * 20.384 = 288.176537983 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 19.23 ± sqrt{ 288.18 } }{ 2 } ; ; c_{1,2} = 9.6139604 ± 8.48788162592 ; ; c_{1} = 18.1018420259 ; ;
c_{2} = 1.12607877408 ; ; ; ; text{ Factored form: } ; ; (c -18.1018420259) (c -1.12607877408) = 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 = 10.79 ; ; b = 9.8 ; ; c = 18.1 ; ;

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

p = a+b+c = 10.79+9.8+18.1 = 38.69 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38.69 }{ 2 } = 19.35 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.35 * (19.35-10.79)(19.35-9.8)(19.35-18.1) } ; ; T = sqrt{ 1965.72 } = 44.34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 44.34 }{ 10.79 } = 8.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 44.34 }{ 9.8 } = 9.05 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 44.34 }{ 18.1 } = 4.9 ; ;

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{ 9.8**2+18.1**2-10.79**2 }{ 2 * 9.8 * 18.1 } ) = 29° 59'25" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.79**2+18.1**2-9.8**2 }{ 2 * 10.79 * 18.1 } ) = 27° ; ; gamma = 180° - alpha - beta = 180° - 29° 59'25" - 27° = 123° 35" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 44.34 }{ 19.35 } = 2.29 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10.79 }{ 2 * sin 29° 59'25" } = 10.79 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.8**2+2 * 18.1**2 - 10.79**2 } }{ 2 } = 13.519 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 18.1**2+2 * 10.79**2 - 9.8**2 } }{ 2 } = 14.073 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.8**2+2 * 10.79**2 - 18.1**2 } }{ 2 } = 4.931 ; ;







#2 Obtuse scalene triangle.

Sides: a = 10.79   b = 9.8   c = 1.12660787701

Area: T = 2.7588080798
Perimeter: p = 21.71660787701
Semiperimeter: s = 10.8588039385

Angle ∠ A = α = 150.0109737844° = 150°35″ = 2.61881638354 rad
Angle ∠ B = β = 27° = 0.4711238898 rad
Angle ∠ C = γ = 2.99902621564° = 2°59'25″ = 0.05221899201 rad

Height: ha = 0.51112290636
Height: hb = 0.56328736322
Height: hc = 4.89985574922

Median: ma = 4.42113122145
Median: mb = 5.9022209476
Median: mc = 10.29215031288

Inradius: r = 0.25440127826
Circumradius: R = 10.79331773965

Vertex coordinates: A[1.12660787701; 0] B[0; 0] C[9.6143960396; 4.89985574922]
Centroid: CG[3.58800130554; 1.63328524974]
Coordinates of the circumscribed circle: U[0.5633039385; 10.77884815704]
Coordinates of the inscribed circle: I[1.0588039385; 0.25440127826]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.99902621564° = 29°59'25″ = 2.61881638354 rad
∠ B' = β' = 153° = 0.4711238898 rad
∠ C' = γ' = 177.0109737844° = 177°35″ = 0.05221899201 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 10.79 ; ; b = 9.8 ; ; beta = 27° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 9.8**2 = 10.79**2 + c**2 -2 * 10.79 * c * cos (27° ) ; ; ; ; c**2 -19.228c +20.384 =0 ; ; p=1; q=-19.228; r=20.384 ; ; D = q**2 - 4pr = 19.228**2 - 4 * 1 * 20.384 = 288.176537983 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 19.23 ± sqrt{ 288.18 } }{ 2 } ; ; c_{1,2} = 9.6139604 ± 8.48788162592 ; ; c_{1} = 18.1018420259 ; ; : Nr. 1
c_{2} = 1.12607877408 ; ; ; ; text{ Factored form: } ; ; (c -18.1018420259) (c -1.12607877408) = 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 = 10.79 ; ; b = 9.8 ; ; c = 1.13 ; ;

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

p = a+b+c = 10.79+9.8+1.13 = 21.72 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.72 }{ 2 } = 10.86 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.86 * (10.86-10.79)(10.86-9.8)(10.86-1.13) } ; ; T = sqrt{ 7.61 } = 2.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.76 }{ 10.79 } = 0.51 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.76 }{ 9.8 } = 0.56 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.76 }{ 1.13 } = 4.9 ; ;

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{ 9.8**2+1.13**2-10.79**2 }{ 2 * 9.8 * 1.13 } ) = 150° 35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.79**2+1.13**2-9.8**2 }{ 2 * 10.79 * 1.13 } ) = 27° ; ; gamma = 180° - alpha - beta = 180° - 150° 35" - 27° = 2° 59'25" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.76 }{ 10.86 } = 0.25 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10.79 }{ 2 * sin 150° 35" } = 10.79 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.8**2+2 * 1.13**2 - 10.79**2 } }{ 2 } = 4.421 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.13**2+2 * 10.79**2 - 9.8**2 } }{ 2 } = 5.902 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.8**2+2 * 10.79**2 - 1.13**2 } }{ 2 } = 10.292 ; ;
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