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?

1. Use Law of Cosines

a = 65 ; ; b = 55 ; ; beta = 18° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 55**2 = 65**2 + c**2 -2 * 65 * c * cos (18° ) ; ; ; ; c**2 -123.637c +1200 =0 ; ; p=1; q=-123.637; 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} = 61.81867356 ± 51.2010585888 ; ; c_{1} = 113.019732149 ; ; c_{2} = 10.6176149712 ; ; ; ; text{ Factored form: } ; ; (c -113.019732149) (c -10.6176149712) = 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 = 113.02 ; ;

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

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

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ 55**2+113.02**2-65**2 }{ 2 * 55 * 113.02 } ) = 21° 25'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 65**2+113.02**2-55**2 }{ 2 * 65 * 113.02 } ) = 18° ; ; gamma = 180° - alpha - beta = 180° - 21° 25'12" - 18° = 140° 34'48" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 55**2+2 * 113.02**2 - 65**2 } }{ 2 } = 82.722 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 113.02**2+2 * 65**2 - 55**2 } }{ 2 } = 87.994 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 55**2+2 * 65**2 - 113.02**2 } }{ 2 } = 20.776 ; ;







#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 - 2ac cos beta ; ; 55**2 = 65**2 + c**2 -2 * 65 * c * cos (18° ) ; ; ; ; c**2 -123.637c +1200 =0 ; ; p=1; q=-123.637; 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} = 61.81867356 ± 51.2010585888 ; ; c_{1} = 113.019732149 ; ; c_{2} = 10.6176149712 ; ; ; ; text{ Factored form: } ; ; (c -113.019732149) (c -10.6176149712) = 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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 55**2+10.62**2-65**2 }{ 2 * 55 * 10.62 } ) = 158° 34'48" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 65**2+10.62**2-55**2 }{ 2 * 65 * 10.62 } ) = 18° ; ; gamma = 180° - alpha - beta = 180° - 158° 34'48" - 18° = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 55**2+2 * 10.62**2 - 65**2 } }{ 2 } = 22.641 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.62**2+2 * 65**2 - 55**2 } }{ 2 } = 37.585 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 55**2+2 * 65**2 - 10.62**2 } }{ 2 } = 59.973 ; ;
Calculate another triangle

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

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