Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=70.6711379089 and with side c=42.46657059008

#1 Acute scalene triangle.

Sides: a = 80   b = 58.3   c = 70.6711379089

Area: T = 1998.888845559
Perimeter: p = 208.9711379089
Semiperimeter: s = 104.4865689545

Angle ∠ A = α = 76.00112254772° = 76°4″ = 1.32664716201 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 58.99987745228° = 58°59'56″ = 1.03297228701 rad

Height: ha = 49.97222113897
Height: hb = 68.57325027645
Height: hc = 56.56985424949

Median: ma = 50.95875010295
Median: mb = 69.62439858897
Median: mc = 60.42221320744

Inradius: r = 19.13107389969
Circumradius: R = 41.22443253432

Vertex coordinates: A[70.6711379089; 0] B[0; 0] C[56.56985424949; 56.56985424949]
Centroid: CG[42.41333071947; 18.85661808316]
Coordinates of the circumscribed circle: U[35.33656895445; 21.23328529504]
Coordinates of the inscribed circle: I[46.18656895445; 19.13107389969]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103.9998774523° = 103°59'56″ = 1.32664716201 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 121.0011225477° = 121°4″ = 1.03297228701 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 80 ; ; b = 58.3 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 58.3**2 = 80**2 + c**2 -2 * 80 * c * cos (45° ) ; ; ; ; c**2 -113.137c +3001.11 =0 ; ; p=1; q=-113.137; r=3001.11 ; ; D = q**2 - 4pr = 113.137**2 - 4 * 1 * 3001.11 = 795.56 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 113.14 ± sqrt{ 795.56 } }{ 2 } ; ; c_{1,2} = 56.56854249 ± 14.1028365941 ; ; c_{1} = 70.6713790841 ; ;
c_{2} = 42.4657058959 ; ; ; ; text{ Factored form: } ; ; (c -70.6713790841) (c -42.4657058959) = 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 = 80 ; ; b = 58.3 ; ; c = 70.67 ; ;

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

p = a+b+c = 80+58.3+70.67 = 208.97 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 208.97 }{ 2 } = 104.49 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 104.49 * (104.49-80)(104.49-58.3)(104.49-70.67) } ; ; T = sqrt{ 3995555.06 } = 1998.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1998.89 }{ 80 } = 49.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1998.89 }{ 58.3 } = 68.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1998.89 }{ 70.67 } = 56.57 ; ;

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{ 58.3**2+70.67**2-80**2 }{ 2 * 58.3 * 70.67 } ) = 76° 4" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 80**2+70.67**2-58.3**2 }{ 2 * 80 * 70.67 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 76° 4" - 45° = 58° 59'56" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1998.89 }{ 104.49 } = 19.13 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 80 }{ 2 * sin 76° 4" } = 41.22 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 58.3**2+2 * 70.67**2 - 80**2 } }{ 2 } = 50.958 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 70.67**2+2 * 80**2 - 58.3**2 } }{ 2 } = 69.624 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 58.3**2+2 * 80**2 - 70.67**2 } }{ 2 } = 60.422 ; ;







#2 Obtuse scalene triangle.

Sides: a = 80   b = 58.3   c = 42.46657059008

Area: T = 1201.112154441
Perimeter: p = 180.7665705901
Semiperimeter: s = 90.38328529504

Angle ∠ A = α = 103.9998774523° = 103°59'56″ = 1.81551210335 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 31.00112254772° = 31°4″ = 0.54110734567 rad

Height: ha = 30.02877886103
Height: hb = 41.20545126729
Height: hc = 56.56985424949

Median: ma = 31.64403711866
Median: mb = 57.02658326448
Median: mc = 66.698790818

Inradius: r = 13.28991528117
Circumradius: R = 41.22443253432

Vertex coordinates: A[42.46657059008; 0] B[0; 0] C[56.56985424949; 56.56985424949]
Centroid: CG[33.01114161319; 18.85661808316]
Coordinates of the circumscribed circle: U[21.23328529504; 35.33656895445]
Coordinates of the inscribed circle: I[32.08328529504; 13.28991528117]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 76.00112254772° = 76°4″ = 1.81551210335 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 148.9998774523° = 148°59'56″ = 0.54110734567 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 80 ; ; b = 58.3 ; ; beta = 45° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 58.3**2 = 80**2 + c**2 -2 * 80 * c * cos (45° ) ; ; ; ; c**2 -113.137c +3001.11 =0 ; ; p=1; q=-113.137; r=3001.11 ; ; D = q**2 - 4pr = 113.137**2 - 4 * 1 * 3001.11 = 795.56 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 113.14 ± sqrt{ 795.56 } }{ 2 } ; ; c_{1,2} = 56.56854249 ± 14.1028365941 ; ; c_{1} = 70.6713790841 ; ; : Nr. 1
c_{2} = 42.4657058959 ; ; ; ; text{ Factored form: } ; ; (c -70.6713790841) (c -42.4657058959) = 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 = 80 ; ; b = 58.3 ; ; c = 42.47 ; ;

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

p = a+b+c = 80+58.3+42.47 = 180.77 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 180.77 }{ 2 } = 90.38 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 90.38 * (90.38-80)(90.38-58.3)(90.38-42.47) } ; ; T = sqrt{ 1442668.94 } = 1201.11 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1201.11 }{ 80 } = 30.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1201.11 }{ 58.3 } = 41.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1201.11 }{ 42.47 } = 56.57 ; ;

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{ 58.3**2+42.47**2-80**2 }{ 2 * 58.3 * 42.47 } ) = 103° 59'56" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 80**2+42.47**2-58.3**2 }{ 2 * 80 * 42.47 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 103° 59'56" - 45° = 31° 4" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1201.11 }{ 90.38 } = 13.29 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 80 }{ 2 * sin 103° 59'56" } = 41.22 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 58.3**2+2 * 42.47**2 - 80**2 } }{ 2 } = 31.64 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 42.47**2+2 * 80**2 - 58.3**2 } }{ 2 } = 57.026 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 58.3**2+2 * 80**2 - 42.47**2 } }{ 2 } = 66.698 ; ;
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.