Triangle calculator SSA

Please enter two sides and a non-included angle
°


Triangle has two solutions with side c=45.39663754419 and with side c=38.85877277994

#1 Acute scalene triangle.

Sides: a = 70   b = 56   c = 45.39663754419

Area: T = 1268.931051094
Perimeter: p = 171.3966375442
Semiperimeter: s = 85.6988187721

Angle ∠ A = α = 86.65331245162° = 86°39'11″ = 1.51223823299 rad
Angle ∠ B = β = 53° = 0.92550245036 rad
Angle ∠ C = γ = 40.34768754838° = 40°20'49″ = 0.70441858201 rad

Height: ha = 36.25551574554
Height: hb = 45.31989468192
Height: hc = 55.90444857033

Median: ma = 37.06596202305
Median: mb = 51.92770204386
Median: mc = 59.18443921502

Inradius: r = 14.80769701902
Circumradius: R = 35.06597984284

Vertex coordinates: A[45.39663754419; 0] B[0; 0] C[42.12770516206; 55.90444857033]
Centroid: CG[29.17444756875; 18.63548285678]
Coordinates of the circumscribed circle: U[22.6988187721; 26.72204367483]
Coordinates of the inscribed circle: I[29.6988187721; 14.80769701902]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 93.34768754838° = 93°20'49″ = 1.51223823299 rad
∠ B' = β' = 127° = 0.92550245036 rad
∠ C' = γ' = 139.6533124516° = 139°39'11″ = 0.70441858201 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 = 70 ; ; b = 56 ; ; c = 45.4 ; ;

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

p = a+b+c = 70+56+45.4 = 171.4 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 171.4 }{ 2 } = 85.7 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 85.7 * (85.7-70)(85.7-56)(85.7-45.4) } ; ; T = sqrt{ 1610184.64 } = 1268.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1268.93 }{ 70 } = 36.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1268.93 }{ 56 } = 45.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1268.93 }{ 45.4 } = 55.9 ; ;

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{ 70**2-56**2-45.4**2 }{ 2 * 56 * 45.4 } ) = 86° 39'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 56**2-70**2-45.4**2 }{ 2 * 70 * 45.4 } ) = 53° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 45.4**2-70**2-56**2 }{ 2 * 56 * 70 } ) = 40° 20'49" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1268.93 }{ 85.7 } = 14.81 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 70 }{ 2 * sin 86° 39'11" } = 35.06 ; ;





#2 Obtuse scalene triangle.

Sides: a = 70   b = 56   c = 38.85877277994

Area: T = 1086.161064411
Perimeter: p = 164.8587727799
Semiperimeter: s = 82.42988638997

Angle ∠ A = α = 93.34768754838° = 93°20'49″ = 1.62992103236 rad
Angle ∠ B = β = 53° = 0.92550245036 rad
Angle ∠ C = γ = 33.65331245162° = 33°39'11″ = 0.58773578264 rad

Height: ha = 31.03331612603
Height: hb = 38.79114515754
Height: hc = 55.90444857033

Median: ma = 33.13655021822
Median: mb = 49.20332672174
Median: mc = 60.33767155849

Inradius: r = 13.17769454621
Circumradius: R = 35.06597984284

Vertex coordinates: A[38.85877277994; 0] B[0; 0] C[42.12770516206; 55.90444857033]
Centroid: CG[26.99549264733; 18.63548285678]
Coordinates of the circumscribed circle: U[19.42988638997; 29.1844048955]
Coordinates of the inscribed circle: I[26.42988638997; 13.17769454621]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 86.65331245162° = 86°39'11″ = 1.62992103236 rad
∠ B' = β' = 127° = 0.92550245036 rad
∠ C' = γ' = 146.3476875484° = 146°20'49″ = 0.58773578264 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 70 ; ; b = 56 ; ; beta = 53° ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 56**2 = 70**2 + c**2 -2 * 56 * c * cos (53° ) ; ; ; ; c**2 -84.254c +1764 =0 ; ; p=1; q=-84.2541032413; r=1764 ; ; D = q**2 - 4pr = 84.254**2 - 4 * 1 * 1764 = 42.7539129934 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 84.25 ± sqrt{ 42.75 } }{ 2 } ; ; c_{1,2} = 42.1270516206 ± 3.26932382127 ; ; c_{1} = 45.3963754419 ; ;
c_{2} = 38.8577277994 ; ; ; ; (c -45.3963754419) (c -38.8577277994) = 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 = 70 ; ; b = 56 ; ; c = 38.86 ; ;

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

p = a+b+c = 70+56+38.86 = 164.86 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 164.86 }{ 2 } = 82.43 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 82.43 * (82.43-70)(82.43-56)(82.43-38.86) } ; ; T = sqrt{ 1179744.94 } = 1086.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1086.16 }{ 70 } = 31.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1086.16 }{ 56 } = 38.79 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1086.16 }{ 38.86 } = 55.9 ; ;

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{ 70**2-56**2-38.86**2 }{ 2 * 56 * 38.86 } ) = 93° 20'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 56**2-70**2-38.86**2 }{ 2 * 70 * 38.86 } ) = 53° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 38.86**2-70**2-56**2 }{ 2 * 56 * 70 } ) = 33° 39'11" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1086.16 }{ 82.43 } = 13.18 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 70 }{ 2 * sin 93° 20'49" } = 35.06 ; ;




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

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