Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=74.49548974278 and with side c=25.50551025722

#1 Acute scalene triangle.

Sides: a = 100   b = 90   c = 74.49548974278

Area: T = 3225.724368124
Perimeter: p = 264.4954897428
Semiperimeter: s = 132.2477448714

Angle ∠ A = α = 74.20768309517° = 74°12'25″ = 1.29551535276 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 45.79331690483° = 45°47'35″ = 0.79992415748 rad

Height: ha = 64.51444736248
Height: hb = 71.6832748472
Height: hc = 86.60325403784

Median: ma = 65.76327924543
Median: mb = 75.82770721536
Median: mc = 87.53664356386

Inradius: r = 24.39215758876
Circumradius: R = 51.96215242271

Vertex coordinates: A[74.49548974278; 0] B[0; 0] C[50; 86.60325403784]
Centroid: CG[41.49882991426; 28.86875134595]
Coordinates of the circumscribed circle: U[37.24774487139; 36.23302023774]
Coordinates of the inscribed circle: I[42.24774487139; 24.39215758876]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105.7933169048° = 105°47'35″ = 1.29551535276 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 134.2076830952° = 134°12'25″ = 0.79992415748 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 90 ; ; beta = 60° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 100**2 + c**2 -2 * 100 * c * cos (60° ) ; ; ; ; c**2 -100c +1900 =0 ; ; p=1; q=-100; r=1900 ; ; D = q**2 - 4pr = 100**2 - 4 * 1 * 1900 = 2400 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 100 ± sqrt{ 2400 } }{ 2 } = 50 ± sqrt{ 600 } ; ; c_{1,2} = 50 ± 24.4948974278 ; ; c_{1} = 74.4948974278 ; ; c_{2} = 25.5051025722 ; ; ; ; text{ Factored form: } ; ; (c -74.4948974278) (c -25.5051025722) = 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 = 100 ; ; b = 90 ; ; c = 74.49 ; ;

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

p = a+b+c = 100+90+74.49 = 264.49 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 264.49 }{ 2 } = 132.25 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 132.25 * (132.25-100)(132.25-90)(132.25-74.49) } ; ; T = sqrt{ 10405293.27 } = 3225.72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3225.72 }{ 100 } = 64.51 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3225.72 }{ 90 } = 71.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3225.72 }{ 74.49 } = 86.6 ; ;

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{ 90**2+74.49**2-100**2 }{ 2 * 90 * 74.49 } ) = 74° 12'25" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+74.49**2-90**2 }{ 2 * 100 * 74.49 } ) = 60° ; ; gamma = 180° - alpha - beta = 180° - 74° 12'25" - 60° = 45° 47'35" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3225.72 }{ 132.25 } = 24.39 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 74° 12'25" } = 51.96 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 74.49**2 - 100**2 } }{ 2 } = 65.763 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 74.49**2+2 * 100**2 - 90**2 } }{ 2 } = 75.827 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 100**2 - 74.49**2 } }{ 2 } = 87.536 ; ;







#2 Obtuse scalene triangle.

Sides: a = 100   b = 90   c = 25.50551025722

Area: T = 1104.403333768
Perimeter: p = 215.5055102572
Semiperimeter: s = 107.7532551286

Angle ∠ A = α = 105.7933169048° = 105°47'35″ = 1.8466439126 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 14.20768309517° = 14°12'25″ = 0.24879559764 rad

Height: ha = 22.08880667536
Height: hb = 24.54222963929
Height: hc = 86.60325403784

Median: ma = 43.30442160604
Median: mb = 57.4487847032
Median: mc = 94.27328616077

Inradius: r = 10.24994402638
Circumradius: R = 51.96215242271

Vertex coordinates: A[25.50551025722; 0] B[0; 0] C[50; 86.60325403784]
Centroid: CG[25.16883675241; 28.86875134595]
Coordinates of the circumscribed circle: U[12.75325512861; 50.37223380011]
Coordinates of the inscribed circle: I[17.75325512861; 10.24994402638]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 74.20768309517° = 74°12'25″ = 1.8466439126 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 165.7933169048° = 165°47'35″ = 0.24879559764 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 100 ; ; b = 90 ; ; beta = 60° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 90**2 = 100**2 + c**2 -2 * 100 * c * cos (60° ) ; ; ; ; c**2 -100c +1900 =0 ; ; p=1; q=-100; r=1900 ; ; D = q**2 - 4pr = 100**2 - 4 * 1 * 1900 = 2400 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 100 ± sqrt{ 2400 } }{ 2 } = 50 ± sqrt{ 600 } ; ; c_{1,2} = 50 ± 24.4948974278 ; ; c_{1} = 74.4948974278 ; ; c_{2} = 25.5051025722 ; ; ; ; text{ Factored form: } ; ; (c -74.4948974278) (c -25.5051025722) = 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 = 100 ; ; b = 90 ; ; c = 25.51 ; ;

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

p = a+b+c = 100+90+25.51 = 215.51 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 215.51 }{ 2 } = 107.75 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 107.75 * (107.75-100)(107.75-90)(107.75-25.51) } ; ; T = sqrt{ 1219706.73 } = 1104.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1104.4 }{ 100 } = 22.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1104.4 }{ 90 } = 24.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1104.4 }{ 25.51 } = 86.6 ; ;

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{ 90**2+25.51**2-100**2 }{ 2 * 90 * 25.51 } ) = 105° 47'35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+25.51**2-90**2 }{ 2 * 100 * 25.51 } ) = 60° ; ; gamma = 180° - alpha - beta = 180° - 105° 47'35" - 60° = 14° 12'25" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1104.4 }{ 107.75 } = 10.25 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 105° 47'35" } = 51.96 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 25.51**2 - 100**2 } }{ 2 } = 43.304 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 25.51**2+2 * 100**2 - 90**2 } }{ 2 } = 57.448 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 100**2 - 25.51**2 } }{ 2 } = 94.273 ; ;
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.