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=1.84767624318; b=6.2; c=7.5 and a=9.6443904215; b=6.2; c=7.5.

#1 Obtuse scalene triangle.

Sides: a = 1.84767624318   b = 6.2   c = 7.5

Area: T = 4.45215350344
Perimeter: p = 15.54767624318
Semiperimeter: s = 7.77333812159

Angle ∠ A = α = 11.0388226456° = 11°2'18″ = 0.19326533952 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 128.9621773544° = 128°57'42″ = 2.25108075576 rad

Height: ha = 4.82109070726
Height: hb = 1.43659790434
Height: hc = 1.18770760092

Median: ma = 6.81985311564
Median: mb = 4.49766949796
Median: mc = 2.62196880997

Inradius: r = 0.57326639297
Circumradius: R = 4.82327438633

Vertex coordinates: A[7.5; 0] B[0; 0] C[1.41547020986; 1.18770760092]
Centroid: CG[2.97215673662; 0.39656920031]
Coordinates of the circumscribed circle: U[3.75; -3.03325498134]
Coordinates of the inscribed circle: I[1.57333812159; 0.57326639297]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 168.9621773544° = 168°57'42″ = 0.19326533952 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 51.0388226456° = 51°2'18″ = 2.25108075576 rad




How did we calculate this triangle?

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

b = 6.2 ; ; c = 7.5 ; ; beta = 40° ; ;

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 6.2**2 = 7.5**2 + a**2 - 2 * 7.5 * a * cos(40° ) ; ; ; ; ; ; a**2 -11.491a +17.81 =0 ; ; a=1; b=-11.491; c=17.81 ; ; D = b**2 - 4ac = 11.491**2 - 4 * 1 * 17.81 = 60.7954199875 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 11.49 ± sqrt{ 60.8 } }{ 2 } ; ; a_{1,2} = 5.74533332 ± 3.89857089161 ; ; a_{1} = 9.64390421161 ; ; a_{2} = 1.84676242839 ; ; ; ;
 text{ Factored form: } ; ; (a -9.64390421161) (a -1.84676242839) = 0 ; ; ; ; a > 0 ; ; ; ; a = 9.644 ; ;


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 = 1.85 ; ; b = 6.2 ; ; c = 7.5 ; ;

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

p = a+b+c = 1.85+6.2+7.5 = 15.55 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.55 }{ 2 } = 7.77 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.77 * (7.77-1.85)(7.77-6.2)(7.77-7.5) } ; ; T = sqrt{ 19.82 } = 4.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.45 }{ 1.85 } = 4.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.45 }{ 6.2 } = 1.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.45 }{ 7.5 } = 1.19 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 1.85**2-6.2**2-7.5**2 }{ 2 * 6.2 * 7.5 } ) = 11° 2'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.2**2-1.85**2-7.5**2 }{ 2 * 1.85 * 7.5 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.5**2-1.85**2-6.2**2 }{ 2 * 6.2 * 1.85 } ) = 128° 57'42" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.45 }{ 7.77 } = 0.57 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.85 }{ 2 * sin 11° 2'18" } = 4.82 ; ;





#2 Acute scalene triangle.

Sides: a = 9.6443904215   b = 6.2   c = 7.5

Area: T = 23.2466183019
Perimeter: p = 23.3443904215
Semiperimeter: s = 11.67219521075

Angle ∠ A = α = 88.9621773544° = 88°57'42″ = 1.55326758568 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 51.0388226456° = 51°2'18″ = 0.8910785096 rad

Height: ha = 4.82109070726
Height: hb = 7.49987687158
Height: hc = 6.19989821384

Median: ma = 4.90985413183
Median: mb = 8.06333395224
Median: mc = 7.18774852524

Inradius: r = 1.99216276905
Circumradius: R = 4.82327438633

Vertex coordinates: A[7.5; 0] B[0; 0] C[7.38876592339; 6.19989821384]
Centroid: CG[4.9632553078; 2.06663273795]
Coordinates of the circumscribed circle: U[3.75; 3.03325498134]
Coordinates of the inscribed circle: I[5.47219521075; 1.99216276905]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 91.0388226456° = 91°2'18″ = 1.55326758568 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 128.9621773544° = 128°57'42″ = 0.8910785096 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 6.2 ; ; c = 7.5 ; ; beta = 40° ; ; : Nr. 1

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 6.2**2 = 7.5**2 + a**2 - 2 * 7.5 * a * cos(40° ) ; ; ; ; ; ; a**2 -11.491a +17.81 =0 ; ; a=1; b=-11.491; c=17.81 ; ; D = b**2 - 4ac = 11.491**2 - 4 * 1 * 17.81 = 60.7954199875 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 11.49 ± sqrt{ 60.8 } }{ 2 } ; ; a_{1,2} = 5.74533332 ± 3.89857089161 ; ; a_{1} = 9.64390421161 ; ; a_{2} = 1.84676242839 ; ; ; ; : Nr. 1
 text{ Factored form: } ; ; (a -9.64390421161) (a -1.84676242839) = 0 ; ; ; ; a > 0 ; ; ; ; a = 9.644 ; ; : 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 = 9.64 ; ; b = 6.2 ; ; c = 7.5 ; ;

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

p = a+b+c = 9.64+6.2+7.5 = 23.34 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.34 }{ 2 } = 11.67 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.67 * (11.67-9.64)(11.67-6.2)(11.67-7.5) } ; ; T = sqrt{ 540.39 } = 23.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 23.25 }{ 9.64 } = 4.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 23.25 }{ 6.2 } = 7.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 23.25 }{ 7.5 } = 6.2 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 9.64**2-6.2**2-7.5**2 }{ 2 * 6.2 * 7.5 } ) = 88° 57'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.2**2-9.64**2-7.5**2 }{ 2 * 9.64 * 7.5 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.5**2-9.64**2-6.2**2 }{ 2 * 6.2 * 9.64 } ) = 51° 2'18" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 23.25 }{ 11.67 } = 1.99 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.64 }{ 2 * sin 88° 57'42" } = 4.82 ; ;

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

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