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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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

6. Inradius

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

7. Circumradius

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





#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

Calculate another triangle

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 - 2bc cos( beta ) ; ; 30.5**2 = 37.7**2 + c**2 -2 * 30.5 * c * cos (38° 12') ; ; ; ; c**2 -59.254c +491.04 =0 ; ; p=1; q=-59.2536097454; 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} = 29.6268048727 ± 19.664881565 ; ;
c_{1} = 49.2916864377 ; ; c_{2} = 9.9619233077 ; ; ; ; (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 = 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 37.7**2-30.5**2-9.96**2 }{ 2 * 30.5 * 9.96 } ) = 130° 8'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 30.5**2-37.7**2-9.96**2 }{ 2 * 37.7 * 9.96 } ) = 38° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 9.96**2-37.7**2-30.5**2 }{ 2 * 30.5 * 37.7 } ) = 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 ; ;




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