Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=113.0219732148 and with side c=10.61876149704

#1 Obtuse scalene triangle.

Sides: a = 65   b = 55   c = 113.0219732148

Area: T = 1135.063308284
Perimeter: p = 233.0219732148
Semiperimeter: s = 116.5109866074

Angle ∠ A = α = 21.42200086505° = 21°25'12″ = 0.37438496768 rad
Angle ∠ B = β = 18° = 0.31441592654 rad
Angle ∠ C = γ = 140.5879991349° = 140°34'48″ = 2.45435837115 rad

Height: ha = 34.92550179334
Height: hb = 41.27550211941
Height: hc = 20.08661046344

Median: ma = 82.72223061055
Median: mb = 87.99442039421
Median: mc = 20.77658281736

Inradius: r = 9.74222057126
Circumradius: R = 88.99218693812

Vertex coordinates: A[113.0219732148; 0] B[0; 0] C[61.81986735592; 20.08661046344]
Centroid: CG[58.27994685691; 6.69553682115]
Coordinates of the circumscribed circle: U[56.5109866074; -68.74772752352]
Coordinates of the inscribed circle: I[61.5109866074; 9.74222057126]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.5879991349° = 158°34'48″ = 0.37438496768 rad
∠ B' = β' = 162° = 0.31441592654 rad
∠ C' = γ' = 39.42200086505° = 39°25'12″ = 2.45435837115 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 = 65 ; ; b = 55 ; ; c = 113.02 ; ;

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

p = a+b+c = 65+55+113.02 = 233.02 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 233.02 }{ 2 } = 116.51 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 116.51 * (116.51-65)(116.51-55)(116.51-113.02) } ; ; T = sqrt{ 1288368.2 } = 1135.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1135.06 }{ 65 } = 34.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1135.06 }{ 55 } = 41.28 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1135.06 }{ 113.02 } = 20.09 ; ;

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{ 65**2-55**2-113.02**2 }{ 2 * 55 * 113.02 } ) = 21° 25'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 55**2-65**2-113.02**2 }{ 2 * 65 * 113.02 } ) = 18° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 113.02**2-65**2-55**2 }{ 2 * 55 * 65 } ) = 140° 34'48" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1135.06 }{ 116.51 } = 9.74 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 65 }{ 2 * sin 21° 25'12" } = 88.99 ; ;





#2 Obtuse scalene triangle.

Sides: a = 65   b = 55   c = 10.61876149704

Area: T = 106.6333262631
Perimeter: p = 130.618761497
Semiperimeter: s = 65.30988074852

Angle ∠ A = α = 158.5879991349° = 158°34'48″ = 2.76877429768 rad
Angle ∠ B = β = 18° = 0.31441592654 rad
Angle ∠ C = γ = 3.42200086505° = 3°25'12″ = 0.06596904114 rad

Height: ha = 3.28110234656
Height: hb = 3.87875731866
Height: hc = 20.08661046344

Median: ma = 22.64110440093
Median: mb = 37.58547957801
Median: mc = 59.97334654917

Inradius: r = 1.6332754704
Circumradius: R = 88.99218693812

Vertex coordinates: A[10.61876149704; 0] B[0; 0] C[61.81986735592; 20.08661046344]
Centroid: CG[24.14554295098; 6.69553682115]
Coordinates of the circumscribed circle: U[5.30988074852; 88.83333798696]
Coordinates of the inscribed circle: I[10.30988074852; 1.6332754704]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 21.42200086505° = 21°25'12″ = 2.76877429768 rad
∠ B' = β' = 162° = 0.31441592654 rad
∠ C' = γ' = 176.5879991349° = 176°34'48″ = 0.06596904114 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 65 ; ; b = 55 ; ; beta = 18° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 55**2 = 65**2 + c**2 -2 * 55 * c * cos (18° ) ; ; ; ; c**2 -123.637c +1200 =0 ; ; p=1; q=-123.637347118; r=1200 ; ; D = q**2 - 4pr = 123.637**2 - 4 * 1 * 1200 = 10486.1936025 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 123.64 ± sqrt{ 10486.19 } }{ 2 } ; ; c_{1,2} = 61.8186735592 ± 51.2010585888 ; ; c_{1} = 113.019732148 ; ;
c_{2} = 10.6176149704 ; ; ; ; (c -113.019732148) (c -10.6176149704) = 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 = 65 ; ; b = 55 ; ; c = 10.62 ; ;

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

p = a+b+c = 65+55+10.62 = 130.62 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 130.62 }{ 2 } = 65.31 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 65.31 * (65.31-65)(65.31-55)(65.31-10.62) } ; ; T = sqrt{ 11370.65 } = 106.63 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 106.63 }{ 65 } = 3.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 106.63 }{ 55 } = 3.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 106.63 }{ 10.62 } = 20.09 ; ;

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{ 65**2-55**2-10.62**2 }{ 2 * 55 * 10.62 } ) = 158° 34'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 55**2-65**2-10.62**2 }{ 2 * 65 * 10.62 } ) = 18° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.62**2-65**2-55**2 }{ 2 * 55 * 65 } ) = 3° 25'12" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 106.63 }{ 65.31 } = 1.63 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 65 }{ 2 * sin 158° 34'48" } = 88.99 ; ;




Look also our friend's collection of math examples and problems:

See more informations about triangles or more information about solving triangles.