Triangle calculator SSA

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Triangle has two solutions with side c=100.8549522814 and with side c=75.4522025827

#1 Acute scalene triangle.

Sides: a = 131.08   b = 97.84   c = 100.8549522814

Area: T = 4891.82875479
Perimeter: p = 329.7769522814
Semiperimeter: s = 164.8854761407

Angle ∠ A = α = 82.5422486563° = 82°32'33″ = 1.44106381633 rad
Angle ∠ B = β = 47.74° = 47°44'24″ = 0.83332201849 rad
Angle ∠ C = γ = 49.7187513437° = 49°43'3″ = 0.86877343054 rad

Height: ha = 74.63988090922
Height: hb = 99.9966474814
Height: hc = 97.01224084161

Median: ma = 74.67436521532
Median: mb = 106.2222078336
Median: mc = 104.0989670175

Inradius: r = 29.66881603937
Circumradius: R = 66.09991073688

Vertex coordinates: A[100.8549522814; 0] B[0; 0] C[88.15107743205; 97.01224084161]
Centroid: CG[633.0000990448; 32.3377469472]
Coordinates of the circumscribed circle: U[50.4254761407; 42.73768158851]
Coordinates of the inscribed circle: I[67.0454761407; 29.66881603937]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 97.4587513437° = 97°27'27″ = 1.44106381633 rad
∠ B' = β' = 132.26° = 132°15'36″ = 0.83332201849 rad
∠ C' = γ' = 130.2822486563° = 130°16'57″ = 0.86877343054 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 = 131.08 ; ; b = 97.84 ; ; c = 100.85 ; ;

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

p = a+b+c = 131.08+97.84+100.85 = 329.77 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 329.77 }{ 2 } = 164.88 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 164.88 * (164.88-131.08)(164.88-97.84)(164.88-100.85) } ; ; T = sqrt{ 23929976.76 } = 4891.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4891.83 }{ 131.08 } = 74.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4891.83 }{ 97.84 } = 100 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4891.83 }{ 100.85 } = 97.01 ; ;

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{ 131.08**2-97.84**2-100.85**2 }{ 2 * 97.84 * 100.85 } ) = 82° 32'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 97.84**2-131.08**2-100.85**2 }{ 2 * 131.08 * 100.85 } ) = 47° 44'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 100.85**2-131.08**2-97.84**2 }{ 2 * 97.84 * 131.08 } ) = 49° 43'3" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4891.83 }{ 164.88 } = 29.67 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 131.08 }{ 2 * sin 82° 32'33" } = 66.1 ; ;





#2 Obtuse scalene triangle.

Sides: a = 131.08   b = 97.84   c = 75.4522025827

Area: T = 3659.891137268
Perimeter: p = 304.3722025827
Semiperimeter: s = 152.1866012914

Angle ∠ A = α = 97.4587513437° = 97°27'27″ = 1.70109544903 rad
Angle ∠ B = β = 47.74° = 47°44'24″ = 0.83332201849 rad
Angle ∠ C = γ = 34.8022486563° = 34°48'9″ = 0.60774179784 rad

Height: ha = 55.84221021159
Height: hb = 74.81438056557
Height: hc = 97.01224084161

Median: ma = 57.77697611272
Median: mb = 95.10216345848
Median: mc = 109.3354642038

Inradius: r = 24.04988025319
Circumradius: R = 66.09991073688

Vertex coordinates: A[75.4522025827; 0] B[0; 0] C[88.15107743205; 97.01224084161]
Centroid: CG[54.53442667158; 32.3377469472]
Coordinates of the circumscribed circle: U[37.72660129135; 54.27655925311]
Coordinates of the inscribed circle: I[54.34660129135; 24.04988025319]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 82.5422486563° = 82°32'33″ = 1.70109544903 rad
∠ B' = β' = 132.26° = 132°15'36″ = 0.83332201849 rad
∠ C' = γ' = 145.1987513437° = 145°11'51″ = 0.60774179784 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 131.08 ; ; b = 97.84 ; ; beta = 47° 44'24" ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 97.84**2 = 131.08**2 + c**2 -2 * 97.84 * c * cos (47° 44'24") ; ; ; ; c**2 -176.302c +7609.301 =0 ; ; p=1; q=-176.301548641; r=7609.3008 ; ; D = q**2 - 4pr = 176.302**2 - 4 * 1 * 7609.301 = 645.032853204 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 176.3 ± sqrt{ 645.03 } }{ 2 } ; ; c_{1,2} = 88.1507743205 ± 12.6987484935 ; ;
c_{1} = 100.849522814 ; ; c_{2} = 75.452025827 ; ; ; ; (c -100.849522814) (c -75.452025827) = 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 = 131.08 ; ; b = 97.84 ; ; c = 75.45 ; ;

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

p = a+b+c = 131.08+97.84+75.45 = 304.37 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 304.37 }{ 2 } = 152.19 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 152.19 * (152.19-131.08)(152.19-97.84)(152.19-75.45) } ; ; T = sqrt{ 13394804.86 } = 3659.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3659.89 }{ 131.08 } = 55.84 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3659.89 }{ 97.84 } = 74.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3659.89 }{ 75.45 } = 97.01 ; ;

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{ 131.08**2-97.84**2-75.45**2 }{ 2 * 97.84 * 75.45 } ) = 97° 27'27" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 97.84**2-131.08**2-75.45**2 }{ 2 * 131.08 * 75.45 } ) = 47° 44'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 75.45**2-131.08**2-97.84**2 }{ 2 * 97.84 * 131.08 } ) = 34° 48'9" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3659.89 }{ 152.19 } = 24.05 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 131.08 }{ 2 * sin 97° 27'27" } = 66.1 ; ;




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