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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 9.8**2 = 10.79**2 + c**2 -2 * 9.8 * c * cos (27° ) ; ; ; ; c**2 -19.228c +20.384 =0 ; ; p=1; q=-19.227920792; r=20.3841 ; ; 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} = 9.61396039599 ± 8.48788162592 ; ;
c_{1} = 18.1018420219 ; ; c_{2} = 1.12607877007 ; ; ; ; (c -18.1018420219) (c -1.12607877007) = 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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 10.79**2-9.8**2-1.13**2 }{ 2 * 9.8 * 1.13 } ) = 150° 35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.8**2-10.79**2-1.13**2 }{ 2 * 10.79 * 1.13 } ) = 27° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.13**2-10.79**2-9.8**2 }{ 2 * 9.8 * 10.79 } ) = 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 ; ;




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