Triangle calculator SSA

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Triangle has two solutions with side c=49.29216864377 and with side c=9.96219233077

#1 Obtuse scalene triangle.

Sides: a = 37.7   b = 30.5   c = 49.29216864377

Area: T = 574.5933102667
Perimeter: p = 117.4921686438
Semiperimeter: s = 58.74658432189

Angle ∠ A = α = 49.85330499496° = 49°51'11″ = 0.87700998638 rad
Angle ∠ B = β = 38.2° = 38°12' = 0.66767157743 rad
Angle ∠ C = γ = 91.94769500504° = 91°56'49″ = 1.60547770155 rad

Height: ha = 30.48223927144
Height: hb = 37.67882362404
Height: hc = 23.3143996505

Median: ma = 36.39655722024
Median: mb = 41.14550808231
Median: mc = 23.84401428694

Inradius: r = 9.78110001727
Circumradius: R = 24.66600791879

Vertex coordinates: A[49.29216864377; 0] B[0; 0] C[29.62768048727; 23.3143996505]
Centroid: CG[26.30661637701; 7.77113321683]
Coordinates of the circumscribed circle: U[24.64658432189; -0.83878052199]
Coordinates of the inscribed circle: I[28.24658432189; 9.78110001727]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.147695005° = 130°8'49″ = 0.87700998638 rad
∠ B' = β' = 141.8° = 141°48' = 0.66767157743 rad
∠ C' = γ' = 88.05330499496° = 88°3'11″ = 1.60547770155 rad


How did we calculate this triangle?

1. Use Law of Cosines

a = 37.7 ; ; b = 30.5 ; ; beta = 38° 12' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 30.5**2 = 37.7**2 + c**2 -2 * 37.7 * c * cos (38° 12') ; ; ; ; c**2 -59.254c +491.04 =0 ; ; p=1; q=-59.254; r=491.04 ; ; D = q**2 - 4pr = 59.254**2 - 4 * 1 * 491.04 = 1546.83026786 ; ; D>0 ; ;
 ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 59.25 ± sqrt{ 1546.83 } }{ 2 } ; ; c_{1,2} = 29.62680487 ± 19.664881565 ; ; c_{1} = 49.2916864377 ; ; c_{2} = 9.9619233077 ; ; ; ; text{ Factored form: } ; ; (c -49.2916864377) (c -9.9619233077) = 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 = 37.7 ; ; b = 30.5 ; ; c = 49.29 ; ;

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

p = a+b+c = 37.7+30.5+49.29 = 117.49 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 117.49 }{ 2 } = 58.75 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 58.75 * (58.75-37.7)(58.75-30.5)(58.75-49.29) } ; ; T = sqrt{ 330157.23 } = 574.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 574.59 }{ 37.7 } = 30.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 574.59 }{ 30.5 } = 37.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 574.59 }{ 49.29 } = 23.31 ; ;

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{ 30.5**2+49.29**2-37.7**2 }{ 2 * 30.5 * 49.29 } ) = 49° 51'11" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37.7**2+49.29**2-30.5**2 }{ 2 * 37.7 * 49.29 } ) = 38° 12' ; ;
 gamma = 180° - alpha - beta = 180° - 49° 51'11" - 38° 12' = 91° 56'49" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 574.59 }{ 58.75 } = 9.78 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37.7 }{ 2 * sin 49° 51'11" } = 24.66 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 30.5**2+2 * 49.29**2 - 37.7**2 } }{ 2 } = 36.396 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 49.29**2+2 * 37.7**2 - 30.5**2 } }{ 2 } = 41.145 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 30.5**2+2 * 37.7**2 - 49.29**2 } }{ 2 } = 23.84 ; ;



#2 Obtuse scalene triangle.

Sides: a = 37.7   b = 30.5   c = 9.96219233077

Area: T = 116.1266122589
Perimeter: p = 78.16219233077
Semiperimeter: s = 39.08109616539

Angle ∠ A = α = 130.147695005° = 130°8'49″ = 2.27114927898 rad
Angle ∠ B = β = 38.2° = 38°12' = 0.66767157743 rad
Angle ∠ C = γ = 11.65330499496° = 11°39'11″ = 0.20333840895 rad

Height: ha = 6.16105370074
Height: hb = 7.61548277108
Height: hc = 23.3143996505

Median: ma = 12.62662606497
Median: mb = 22.97217752469
Median: mc = 33.92658017002

Inradius: r = 2.97114243886
Circumradius: R = 24.66600791879

Vertex coordinates: A[9.96219233077; 0] B[0; 0] C[29.62768048727; 23.3143996505]
Centroid: CG[13.19662427268; 7.77113321683]
Coordinates of the circumscribed circle: U[4.98109616539; 24.15218017249]
Coordinates of the inscribed circle: I[8.58109616539; 2.97114243886]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 49.85330499496° = 49°51'11″ = 2.27114927898 rad
∠ B' = β' = 141.8° = 141°48' = 0.66767157743 rad
∠ C' = γ' = 168.347695005° = 168°20'49″ = 0.20333840895 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 37.7 ; ; b = 30.5 ; ; beta = 38° 12' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 30.5**2 = 37.7**2 + c**2 -2 * 37.7 * c * cos (38° 12') ; ; ; ; c**2 -59.254c +491.04 =0 ; ; p=1; q=-59.254; r=491.04 ; ; D = q**2 - 4pr = 59.254**2 - 4 * 1 * 491.04 = 1546.83026786 ; ; D>0 ; ; : Nr. 1
 ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 59.25 ± sqrt{ 1546.83 } }{ 2 } ; ; c_{1,2} = 29.62680487 ± 19.664881565 ; ; c_{1} = 49.2916864377 ; ; c_{2} = 9.9619233077 ; ; ; ; text{ Factored form: } ; ; (c -49.2916864377) (c -9.9619233077) = 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 = 37.7 ; ; b = 30.5 ; ; c = 9.96 ; ;

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

p = a+b+c = 37.7+30.5+9.96 = 78.16 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 78.16 }{ 2 } = 39.08 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.08 * (39.08-37.7)(39.08-30.5)(39.08-9.96) } ; ; T = sqrt{ 13485.28 } = 116.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 116.13 }{ 37.7 } = 6.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 116.13 }{ 30.5 } = 7.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 116.13 }{ 9.96 } = 23.31 ; ;

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{ 30.5**2+9.96**2-37.7**2 }{ 2 * 30.5 * 9.96 } ) = 130° 8'49" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37.7**2+9.96**2-30.5**2 }{ 2 * 37.7 * 9.96 } ) = 38° 12' ; ;
 gamma = 180° - alpha - beta = 180° - 130° 8'49" - 38° 12' = 11° 39'11" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 116.13 }{ 39.08 } = 2.97 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37.7 }{ 2 * sin 130° 8'49" } = 24.66 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 30.5**2+2 * 9.96**2 - 37.7**2 } }{ 2 } = 12.626 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.96**2+2 * 37.7**2 - 30.5**2 } }{ 2 } = 22.972 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 30.5**2+2 * 37.7**2 - 9.96**2 } }{ 2 } = 33.926 ; ;
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