Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=115.3698860353 and with side c=56.65330689478

#1 Acute scalene triangle.

Sides: a = 105   b = 67   c = 115.3698860353

Area: T = 3474.075513882
Perimeter: p = 287.3698860353
Semiperimeter: s = 143.6844430177

Angle ∠ A = α = 64.01223407832° = 64°44″ = 1.11772261086 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 80.98876592168° = 80°59'16″ = 1.41435013068 rad

Height: ha = 66.17328597871
Height: hb = 103.7043735487
Height: hc = 60.22655258169

Median: ma = 78.37988043387
Median: mb = 105.0966322341
Median: mc = 66.5554537901

Inradius: r = 24.17985079605
Circumradius: R = 58.40554676533

Vertex coordinates: A[115.3698860353; 0] B[0; 0] C[86.01109646503; 60.22655258169]
Centroid: CG[67.12766083344; 20.07551752723]
Coordinates of the circumscribed circle: U[57.68444301765; 9.14990527935]
Coordinates of the inscribed circle: I[76.68444301765; 24.17985079605]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115.9887659217° = 115°59'16″ = 1.11772261086 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 99.01223407832° = 99°44″ = 1.41435013068 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 = 105 ; ; b = 67 ; ; c = 115.37 ; ;

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

p = a+b+c = 105+67+115.37 = 287.37 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 287.37 }{ 2 } = 143.68 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 143.68 * (143.68-105)(143.68-67)(143.68-115.37) } ; ; T = sqrt{ 12069198.07 } = 3474.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3474.08 }{ 105 } = 66.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3474.08 }{ 67 } = 103.7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3474.08 }{ 115.37 } = 60.23 ; ;

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{ 105**2-67**2-115.37**2 }{ 2 * 67 * 115.37 } ) = 64° 44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 67**2-105**2-115.37**2 }{ 2 * 105 * 115.37 } ) = 35° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 115.37**2-105**2-67**2 }{ 2 * 67 * 105 } ) = 80° 59'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3474.08 }{ 143.68 } = 24.18 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 105 }{ 2 * sin 64° 44" } = 58.41 ; ;





#2 Obtuse scalene triangle.

Sides: a = 105   b = 67   c = 56.65330689478

Area: T = 1705.988043326
Perimeter: p = 228.6533068948
Semiperimeter: s = 114.3276534474

Angle ∠ A = α = 115.9887659217° = 115°59'16″ = 2.0244366545 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 29.01223407832° = 29°44″ = 0.50663608704 rad

Height: ha = 32.49548653954
Height: hb = 50.92547890525
Height: hc = 60.22655258169

Median: ma = 33.06110815098
Median: mb = 77.42876120683
Median: mc = 83.39442890413

Inradius: r = 14.92219990015
Circumradius: R = 58.40554676533

Vertex coordinates: A[56.65330689478; 0] B[0; 0] C[86.01109646503; 60.22655258169]
Centroid: CG[47.5554677866; 20.07551752723]
Coordinates of the circumscribed circle: U[28.32765344739; 51.07664730233]
Coordinates of the inscribed circle: I[47.32765344739; 14.92219990015]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 64.01223407832° = 64°44″ = 2.0244366545 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 150.9887659217° = 150°59'16″ = 0.50663608704 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 105 ; ; b = 67 ; ; beta = 35° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 67**2 = 105**2 + c**2 -2 * 67 * c * cos (35° ) ; ; ; ; c**2 -172.022c +6536 =0 ; ; p=1; q=-172.021929301; r=6536 ; ; D = q**2 - 4pr = 172.022**2 - 4 * 1 * 6536 = 3447.54416033 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 172.02 ± sqrt{ 3447.54 } }{ 2 } ; ; c_{1,2} = 86.0109646503 ± 29.3578957026 ; ; c_{1} = 115.368860353 ; ;
c_{2} = 56.6530689478 ; ; ; ; (c -115.368860353) (c -56.6530689478) = 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 = 105 ; ; b = 67 ; ; c = 56.65 ; ;

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

p = a+b+c = 105+67+56.65 = 228.65 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 228.65 }{ 2 } = 114.33 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 114.33 * (114.33-105)(114.33-67)(114.33-56.65) } ; ; T = sqrt{ 2910369.24 } = 1705.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1705.98 }{ 105 } = 32.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1705.98 }{ 67 } = 50.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1705.98 }{ 56.65 } = 60.23 ; ;

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{ 105**2-67**2-56.65**2 }{ 2 * 67 * 56.65 } ) = 115° 59'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 67**2-105**2-56.65**2 }{ 2 * 105 * 56.65 } ) = 35° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 56.65**2-105**2-67**2 }{ 2 * 67 * 105 } ) = 29° 44" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1705.98 }{ 114.33 } = 14.92 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 105 }{ 2 * sin 115° 59'16" } = 58.41 ; ;




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

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