Triangle calculator SSA

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Triangle has two solutions with side c=70.6711379089 and with side c=42.46657059008

#1 Acute scalene triangle.

Sides: a = 80   b = 58.3   c = 70.6711379089

Area: T = 1998.888845559
Perimeter: p = 208.9711379089
Semiperimeter: s = 104.4865689545

Angle ∠ A = α = 76.00112254772° = 76°4″ = 1.32664716201 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 58.99987745228° = 58°59'56″ = 1.03297228701 rad

Height: ha = 49.97222113897
Height: hb = 68.57325027645
Height: hc = 56.56985424949

Median: ma = 50.95875010295
Median: mb = 69.62439858897
Median: mc = 60.42221320744

Inradius: r = 19.13107389969
Circumradius: R = 41.22443253432

Vertex coordinates: A[70.6711379089; 0] B[0; 0] C[56.56985424949; 56.56985424949]
Centroid: CG[42.41333071947; 18.85661808316]
Coordinates of the circumscribed circle: U[35.33656895445; 21.23328529504]
Coordinates of the inscribed circle: I[46.18656895445; 19.13107389969]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103.9998774523° = 103°59'56″ = 1.32664716201 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 121.0011225477° = 121°4″ = 1.03297228701 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 = 80 ; ; b = 58.3 ; ; c = 70.67 ; ;

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

p = a+b+c = 80+58.3+70.67 = 208.97 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 208.97 }{ 2 } = 104.49 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 104.49 * (104.49-80)(104.49-58.3)(104.49-70.67) } ; ; T = sqrt{ 3995555.06 } = 1998.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1998.89 }{ 80 } = 49.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1998.89 }{ 58.3 } = 68.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1998.89 }{ 70.67 } = 56.57 ; ;

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{ 80**2-58.3**2-70.67**2 }{ 2 * 58.3 * 70.67 } ) = 76° 4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 58.3**2-80**2-70.67**2 }{ 2 * 80 * 70.67 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 70.67**2-80**2-58.3**2 }{ 2 * 58.3 * 80 } ) = 58° 59'56" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1998.89 }{ 104.49 } = 19.13 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 80 }{ 2 * sin 76° 4" } = 41.22 ; ;





#2 Obtuse scalene triangle.

Sides: a = 80   b = 58.3   c = 42.46657059008

Area: T = 1201.112154441
Perimeter: p = 180.7665705901
Semiperimeter: s = 90.38328529504

Angle ∠ A = α = 103.9998774523° = 103°59'56″ = 1.81551210335 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 31.00112254772° = 31°4″ = 0.54110734567 rad

Height: ha = 30.02877886103
Height: hb = 41.20545126729
Height: hc = 56.56985424949

Median: ma = 31.64403711866
Median: mb = 57.02658326448
Median: mc = 66.698790818

Inradius: r = 13.28991528117
Circumradius: R = 41.22443253432

Vertex coordinates: A[42.46657059008; 0] B[0; 0] C[56.56985424949; 56.56985424949]
Centroid: CG[33.01114161319; 18.85661808316]
Coordinates of the circumscribed circle: U[21.23328529504; 35.33656895445]
Coordinates of the inscribed circle: I[32.08328529504; 13.28991528117]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 76.00112254772° = 76°4″ = 1.81551210335 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 148.9998774523° = 148°59'56″ = 0.54110734567 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 80 ; ; b = 58.3 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 58.3**2 = 80**2 + c**2 -2 * 58.3 * c * cos (45° ) ; ; ; ; c**2 -113.137c +3001.11 =0 ; ; p=1; q=-113.13708499; r=3001.11 ; ; D = q**2 - 4pr = 113.137**2 - 4 * 1 * 3001.11 = 795.56 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 113.14 ± sqrt{ 795.56 } }{ 2 } ; ; c_{1,2} = 56.5685424949 ± 14.1028365941 ; ;
c_{1} = 70.671379089 ; ; c_{2} = 42.4657059008 ; ; ; ; (c -70.671379089) (c -42.4657059008) = 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 = 80 ; ; b = 58.3 ; ; c = 42.47 ; ;

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

p = a+b+c = 80+58.3+42.47 = 180.77 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 180.77 }{ 2 } = 90.38 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 90.38 * (90.38-80)(90.38-58.3)(90.38-42.47) } ; ; T = sqrt{ 1442668.94 } = 1201.11 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1201.11 }{ 80 } = 30.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1201.11 }{ 58.3 } = 41.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1201.11 }{ 42.47 } = 56.57 ; ;

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{ 80**2-58.3**2-42.47**2 }{ 2 * 58.3 * 42.47 } ) = 103° 59'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 58.3**2-80**2-42.47**2 }{ 2 * 80 * 42.47 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 42.47**2-80**2-58.3**2 }{ 2 * 58.3 * 80 } ) = 31° 4" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1201.11 }{ 90.38 } = 13.29 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 80 }{ 2 * sin 103° 59'56" } = 41.22 ; ;




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