Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, c and angle α.

Triangle has two solutions: a=52; b=15.87663575253; c=62 and a=52; b=71.80548833418; c=62.

#1 Obtuse scalene triangle.

Sides: a = 52   b = 15.87663575253   c = 62

Area: T = 348.0154682068
Perimeter: p = 129.8766357525
Semiperimeter: s = 64.93881787627

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 12.46877469516° = 12°28'4″ = 0.21876032346 rad
Angle ∠ C = γ = 122.5322253048° = 122°31'56″ = 2.13985912556 rad

Height: ha = 13.38551800795
Height: hb = 43.84106204336
Height: hc = 11.22662800667

Median: ma = 37.04109147314
Median: mb = 56.6665556716
Median: mc = 22.73882797093

Inradius: r = 5.35991691159
Circumradius: R = 36.77695526217

Vertex coordinates: A[62; 0] B[0; 0] C[50.77437199333; 11.22662800667]
Centroid: CG[37.59112399778; 3.74220933556]
Coordinates of the circumscribed circle: U[31; -19.77437199333]
Coordinates of the inscribed circle: I[49.06218212373; 5.35991691159]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 167.5322253048° = 167°31'56″ = 0.21876032346 rad
∠ C' = γ' = 57.46877469516° = 57°28'4″ = 2.13985912556 rad


How did we calculate this triangle?

1. Input data entered: side a, c and angle α.

a = 52 ; ; c = 62 ; ; alpha = 45° ; ;

2. From angle α, side c and side a we calculate side b - by using the law of cosines and quadratic equation:

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 52**2 = 62**2 + b**2 - 2 * 62 * b * cos 45° ; ; ; ; ; ; b**2 -87.681b +1140 =0 ; ; p=1; q=-87.681; r=1140 ; ; D = q**2 - 4pr = 87.681**2 - 4 * 1 * 1140 = 3128 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 87.68 ± sqrt{ 3128 } }{ 2 } ; ;
b_{1,2} = 43.84062043 ± 27.9642629082 ; ; b_{1} = 71.8048833418 ; ; b_{2} = 15.8763575253 ; ; ; ; text{ Factored form: } ; ; (b -71.8048833418) (b -15.8763575253) = 0 ; ; ; ; b > 0 ; ; ; ; b = 71.805 ; ;
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 = 52 ; ; b = 15.88 ; ; c = 62 ; ;

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

p = a+b+c = 52+15.88+62 = 129.88 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 129.88 }{ 2 } = 64.94 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 64.94 * (64.94-52)(64.94-15.88)(64.94-62) } ; ; T = sqrt{ 121114.22 } = 348.01 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 348.01 }{ 52 } = 13.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 348.01 }{ 15.88 } = 43.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 348.01 }{ 62 } = 11.23 ; ;

7. 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{ 15.88**2+62**2-52**2 }{ 2 * 15.88 * 62 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 52**2+62**2-15.88**2 }{ 2 * 52 * 62 } ) = 12° 28'4" ; ;
 gamma = 180° - alpha - beta = 180° - 45° - 12° 28'4" = 122° 31'56" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 348.01 }{ 64.94 } = 5.36 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 52 }{ 2 * sin 45° } = 36.77 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.88**2+2 * 62**2 - 52**2 } }{ 2 } = 37.041 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 52**2 - 15.88**2 } }{ 2 } = 56.666 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.88**2+2 * 52**2 - 62**2 } }{ 2 } = 22.738 ; ;



#2 Acute scalene triangle.

Sides: a = 52   b = 71.80548833418   c = 62

Area: T = 1573.985531793
Perimeter: p = 185.8054883342
Semiperimeter: s = 92.90224416709

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 77.53222530484° = 77°31'56″ = 1.35331930922 rad
Angle ∠ C = γ = 57.46877469516° = 57°28'4″ = 1.0033001398 rad

Height: ha = 60.53878968435
Height: hb = 43.84106204336
Height: hc = 50.77437199333

Median: ma = 61.83882619085
Median: mb = 44.55435035891
Median: mc = 54.48882614502

Inradius: r = 16.94223460743
Circumradius: R = 36.77695526217

Vertex coordinates: A[62; 0] B[0; 0] C[11.22662800667; 50.77437199333]
Centroid: CG[24.40987600222; 16.92545733111]
Coordinates of the circumscribed circle: U[31; 19.77437199333]
Coordinates of the inscribed circle: I[21.09875583291; 16.94223460743]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 102.4687746952° = 102°28'4″ = 1.35331930922 rad
∠ C' = γ' = 122.5322253048° = 122°31'56″ = 1.0033001398 rad

Calculate another triangle

How did we calculate this triangle?

1. Input data entered: side a, c and angle α.

a = 52 ; ; c = 62 ; ; alpha = 45° ; ; : Nr. 1

2. From angle α, side c and side a we calculate side b - by using the law of cosines and quadratic equation:

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 52**2 = 62**2 + b**2 - 2 * 62 * b * cos 45° ; ; ; ; ; ; b**2 -87.681b +1140 =0 ; ; p=1; q=-87.681; r=1140 ; ; D = q**2 - 4pr = 87.681**2 - 4 * 1 * 1140 = 3128 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 87.68 ± sqrt{ 3128 } }{ 2 } ; ; : Nr. 1
b_{1,2} = 43.84062043 ± 27.9642629082 ; ; b_{1} = 71.8048833418 ; ; b_{2} = 15.8763575253 ; ; ; ; text{ Factored form: } ; ; (b -71.8048833418) (b -15.8763575253) = 0 ; ; ; ; b > 0 ; ; ; ; b = 71.805 ; ; : 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 = 52 ; ; b = 71.8 ; ; c = 62 ; ;

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

p = a+b+c = 52+71.8+62 = 185.8 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 185.8 }{ 2 } = 92.9 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 92.9 * (92.9-52)(92.9-71.8)(92.9-62) } ; ; T = sqrt{ 2477429.78 } = 1573.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1573.99 }{ 52 } = 60.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1573.99 }{ 71.8 } = 43.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1573.99 }{ 62 } = 50.77 ; ;

7. 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{ 71.8**2+62**2-52**2 }{ 2 * 71.8 * 62 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 52**2+62**2-71.8**2 }{ 2 * 52 * 62 } ) = 77° 31'56" ; ;
 gamma = 180° - alpha - beta = 180° - 45° - 77° 31'56" = 57° 28'4" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1573.99 }{ 92.9 } = 16.94 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 52 }{ 2 * sin 45° } = 36.77 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 71.8**2+2 * 62**2 - 52**2 } }{ 2 } = 61.838 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 62**2+2 * 52**2 - 71.8**2 } }{ 2 } = 44.554 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 71.8**2+2 * 52**2 - 62**2 } }{ 2 } = 54.488 ; ;
Calculate another triangle


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

See more information about triangles or more details on solving triangles.