Triangle calculator

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

You have entered side b, c and angle β.

Triangle has two solutions: a=71.64404339892; b=35; c=105 and a=136.7944257855; b=35; c=105.

#1 Obtuse scalene triangle.

Sides: a = 71.64404339892   b = 35   c = 105

Area: T = 458.3665564205
Perimeter: p = 211.6440433989
Semiperimeter: s = 105.8220216995

Angle ∠ A = α = 14.44550067074° = 14°26'42″ = 0.25221129275 rad
Angle ∠ B = β = 7° = 0.12221730476 rad
Angle ∠ C = γ = 158.5554993293° = 158°33'18″ = 2.76773066784 rad

Height: ha = 12.79662810575
Height: hb = 26.19223179546
Height: hc = 8.73107726515

Median: ma = 69.5843849092
Median: mb = 88.16113628019
Median: mc = 20.55330019969

Inradius: r = 4.3321550031
Circumradius: R = 143.5966408342

Vertex coordinates: A[105; 0] B[0; 0] C[71.10664370579; 8.73107726515]
Centroid: CG[58.7022145686; 2.91102575505]
Coordinates of the circumscribed circle: U[52.5; -133.6555072813]
Coordinates of the inscribed circle: I[70.82202169946; 4.3321550031]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.5554993293° = 165°33'18″ = 0.25221129275 rad
∠ B' = β' = 173° = 0.12221730476 rad
∠ C' = γ' = 21.44550067074° = 21°26'42″ = 2.76773066784 rad




How did we calculate this triangle?

1. Input data entered: side b, c and angle β.

b = 35 ; ; c = 105 ; ; beta = 7° ; ;

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 35**2 = 105**2 + a**2 - 2 * 105 * a * cos 7° ; ; ; ; ; ; a**2 -208.435a +9800 =0 ; ; p=1; q=-208.435; r=9800 ; ; D = q**2 - 4pr = 208.435**2 - 4 * 1 * 9800 = 4245.02076439 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 208.43 ± sqrt{ 4245.02 } }{ 2 } ; ; a_{1,2} = 104.21734592 ± 32.5769119331 ; ; a_{1} = 136.794257853 ; ; a_{2} = 71.6404339869 ; ;
 ; ; text{ Factored form: } ; ; (a -136.794257853) (a -71.6404339869) = 0 ; ; ; ; a > 0 ; ; ; ; a = 136.794 ; ;
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 = 71.64 ; ; b = 35 ; ; c = 105 ; ;

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

p = a+b+c = 71.64+35+105 = 211.64 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 211.64 }{ 2 } = 105.82 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 105.82 * (105.82-71.64)(105.82-35)(105.82-105) } ; ; T = sqrt{ 210098.99 } = 458.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 458.37 }{ 71.64 } = 12.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 458.37 }{ 35 } = 26.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 458.37 }{ 105 } = 8.73 ; ;

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{ 35**2+105**2-71.64**2 }{ 2 * 35 * 105 } ) = 14° 26'42" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 71.64**2+105**2-35**2 }{ 2 * 71.64 * 105 } ) = 7° ; ; gamma = 180° - alpha - beta = 180° - 14° 26'42" - 7° = 158° 33'18" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 458.37 }{ 105.82 } = 4.33 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 71.64 }{ 2 * sin 14° 26'42" } = 143.6 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 35**2+2 * 105**2 - 71.64**2 } }{ 2 } = 69.584 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 105**2+2 * 71.64**2 - 35**2 } }{ 2 } = 88.161 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 35**2+2 * 71.64**2 - 105**2 } }{ 2 } = 20.553 ; ;







#2 Obtuse scalene triangle.

Sides: a = 136.7944257855   b = 35   c = 105

Area: T = 875.2298885288
Perimeter: p = 276.7944257855
Semiperimeter: s = 138.3977128928

Angle ∠ A = α = 151.5554993293° = 151°33'18″ = 2.64551336308 rad
Angle ∠ B = β = 7° = 0.12221730476 rad
Angle ∠ C = γ = 21.44550067074° = 21°26'42″ = 0.37442859752 rad

Height: ha = 12.79662810575
Height: hb = 50.01330791593
Height: hc = 16.67110263864

Median: ma = 38.03772548227
Median: mb = 120.6765533938
Median: mc = 84.92769361929

Inradius: r = 6.32440393213
Circumradius: R = 143.5966408342

Vertex coordinates: A[105; 0] B[0; 0] C[135.7754614201; 16.67110263864]
Centroid: CG[80.25882047337; 5.55770087955]
Coordinates of the circumscribed circle: U[52.5; 133.6555072813]
Coordinates of the inscribed circle: I[103.3977128928; 6.32440393213]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 28.44550067074° = 28°26'42″ = 2.64551336308 rad
∠ B' = β' = 173° = 0.12221730476 rad
∠ C' = γ' = 158.5554993293° = 158°33'18″ = 0.37442859752 rad

Calculate another triangle

How did we calculate this triangle?

1. Input data entered: side b, c and angle β.

b = 35 ; ; c = 105 ; ; beta = 7° ; ; : Nr. 1

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 35**2 = 105**2 + a**2 - 2 * 105 * a * cos 7° ; ; ; ; ; ; a**2 -208.435a +9800 =0 ; ; p=1; q=-208.435; r=9800 ; ; D = q**2 - 4pr = 208.435**2 - 4 * 1 * 9800 = 4245.02076439 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 208.43 ± sqrt{ 4245.02 } }{ 2 } ; ; a_{1,2} = 104.21734592 ± 32.5769119331 ; ; a_{1} = 136.794257853 ; ; a_{2} = 71.6404339869 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (a -136.794257853) (a -71.6404339869) = 0 ; ; ; ; a > 0 ; ; ; ; a = 136.794 ; ; : 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 = 136.79 ; ; b = 35 ; ; c = 105 ; ;

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

p = a+b+c = 136.79+35+105 = 276.79 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 276.79 }{ 2 } = 138.4 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 138.4 * (138.4-136.79)(138.4-35)(138.4-105) } ; ; T = sqrt{ 766025.6 } = 875.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 875.23 }{ 136.79 } = 12.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 875.23 }{ 35 } = 50.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 875.23 }{ 105 } = 16.67 ; ;

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{ 35**2+105**2-136.79**2 }{ 2 * 35 * 105 } ) = 151° 33'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 136.79**2+105**2-35**2 }{ 2 * 136.79 * 105 } ) = 7° ; ; gamma = 180° - alpha - beta = 180° - 151° 33'18" - 7° = 21° 26'42" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 875.23 }{ 138.4 } = 6.32 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 136.79 }{ 2 * sin 151° 33'18" } = 143.6 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 35**2+2 * 105**2 - 136.79**2 } }{ 2 } = 38.037 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 105**2+2 * 136.79**2 - 35**2 } }{ 2 } = 120.676 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 35**2+2 * 136.79**2 - 105**2 } }{ 2 } = 84.927 ; ;
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.