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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. 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?

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 ; ;

1. 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 ; ;

2. Semiperimeter of the triangle

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

3. 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 ; ;

4. 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 ; ;

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{ 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" ; ;

6. Inradius

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

7. Circumradius

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




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