Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=94.51550067534 and with side c=46.55334537968

#1 Acute scalene triangle.

Sides: a = 120   b = 100   c = 94.51550067534

Area: T = 4587.855480122
Perimeter: p = 314.5155006753
Semiperimeter: s = 157.2587503377

Angle ∠ A = α = 76.12548052335° = 76°7'29″ = 1.32986284938 rad
Angle ∠ B = β = 54° = 0.94224777961 rad
Angle ∠ C = γ = 49.87551947665° = 49°52'31″ = 0.87704863637 rad

Height: ha = 76.46442466869
Height: hb = 91.75770960243
Height: hc = 97.0822039325

Median: ma = 76.59333629683
Median: mb = 95.74220662551
Median: mc = 99.83435032672

Inradius: r = 29.17441551449
Circumradius: R = 61.8033398875

Vertex coordinates: A[94.51550067534; 0] B[0; 0] C[70.53442302751; 97.0822039325]
Centroid: CG[55.01664123428; 32.3610679775]
Coordinates of the circumscribed circle: U[47.25875033767; 39.82994926795]
Coordinates of the inscribed circle: I[57.25875033767; 29.17441551449]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103.8755194767° = 103°52'31″ = 1.32986284938 rad
∠ B' = β' = 126° = 0.94224777961 rad
∠ C' = γ' = 130.1254805233° = 130°7'29″ = 0.87704863637 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 = 120 ; ; b = 100 ; ; c = 94.52 ; ;

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

p = a+b+c = 120+100+94.52 = 314.52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 314.52 }{ 2 } = 157.26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 157.26 * (157.26-120)(157.26-100)(157.26-94.52) } ; ; T = sqrt{ 21048411.68 } = 4587.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4587.85 }{ 120 } = 76.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4587.85 }{ 100 } = 91.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4587.85 }{ 94.52 } = 97.08 ; ;

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{ 120**2-100**2-94.52**2 }{ 2 * 100 * 94.52 } ) = 76° 7'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 100**2-120**2-94.52**2 }{ 2 * 120 * 94.52 } ) = 54° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 94.52**2-120**2-100**2 }{ 2 * 100 * 120 } ) = 49° 52'31" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4587.85 }{ 157.26 } = 29.17 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 120 }{ 2 * sin 76° 7'29" } = 61.8 ; ;





#2 Obtuse scalene triangle.

Sides: a = 120   b = 100   c = 46.55334537968

Area: T = 2259.752211611
Perimeter: p = 266.5533453797
Semiperimeter: s = 133.2776726898

Angle ∠ A = α = 103.8755194767° = 103°52'31″ = 1.81329641598 rad
Angle ∠ B = β = 54° = 0.94224777961 rad
Angle ∠ C = γ = 22.12548052335° = 22°7'29″ = 0.38661506977 rad

Height: ha = 37.66325352685
Height: hb = 45.19550423222
Height: hc = 97.0822039325

Median: ma = 49.83658508526
Median: mb = 76.05500626575
Median: mc = 107.9733116955

Inradius: r = 16.95553392306
Circumradius: R = 61.8033398875

Vertex coordinates: A[46.55334537968; 0] B[0; 0] C[70.53442302751; 97.0822039325]
Centroid: CG[39.0299228024; 32.3610679775]
Coordinates of the circumscribed circle: U[23.27767268984; 57.25325466455]
Coordinates of the inscribed circle: I[33.27767268984; 16.95553392306]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 76.12548052335° = 76°7'29″ = 1.81329641598 rad
∠ B' = β' = 126° = 0.94224777961 rad
∠ C' = γ' = 157.8755194767° = 157°52'31″ = 0.38661506977 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 120 ; ; b = 100 ; ; beta = 54° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 100**2 = 120**2 + c**2 -2 * 100 * c * cos (54° ) ; ; ; ; c**2 -141.068c +4400 =0 ; ; p=1; q=-141.06846055; r=4400 ; ; D = q**2 - 4pr = 141.068**2 - 4 * 1 * 4400 = 2300.310562 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 141.07 ± sqrt{ 2300.31 } }{ 2 } ; ; c_{1,2} = 70.5342302751 ± 23.9807764783 ; ; c_{1} = 94.5150067534 ; ;
c_{2} = 46.5534537968 ; ; ; ; (c -94.5150067534) (c -46.5534537968) = 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 = 120 ; ; b = 100 ; ; c = 46.55 ; ;

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

p = a+b+c = 120+100+46.55 = 266.55 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 266.55 }{ 2 } = 133.28 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 133.28 * (133.28-120)(133.28-100)(133.28-46.55) } ; ; T = sqrt{ 5106479.63 } = 2259.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2259.75 }{ 120 } = 37.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2259.75 }{ 100 } = 45.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2259.75 }{ 46.55 } = 97.08 ; ;

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{ 120**2-100**2-46.55**2 }{ 2 * 100 * 46.55 } ) = 103° 52'31" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 100**2-120**2-46.55**2 }{ 2 * 120 * 46.55 } ) = 54° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 46.55**2-120**2-100**2 }{ 2 * 100 * 120 } ) = 22° 7'29" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2259.75 }{ 133.28 } = 16.96 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 120 }{ 2 * sin 103° 52'31" } = 61.8 ; ;




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

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