Triangle calculator - result

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

You have entered side c, angle α and angle γ.

Right scalene triangle.

Sides: a = 4.33301270189   b = 2.5   c = 5

Area: T = 5.41326587737
Perimeter: p = 11.83301270189
Semiperimeter: s = 5.91550635095

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 2.5
Height: hb = 4.33301270189
Height: hc = 2.16550635095

Median: ma = 3.30771891388
Median: mb = 4.50769390943
Median: mc = 2.5

Inradius: r = 0.91550635095
Circumradius: R = 2.5

Vertex coordinates: A[5; 0] B[0; 0] C[3.75; 2.16550635095]
Centroid: CG[2.91766666667; 0.72216878365]
Coordinates of the circumscribed circle: U[2.5; 0]
Coordinates of the inscribed circle: I[3.41550635095; 0.91550635095]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

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

c = 5 ; ; alpha = 60° ; ; gamma = 90° ; ;

2. From angle α, angle γ and side c we calculate side a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 5 * fraction{ sin 60° }{ sin 90° } = 4.33 ; ;

3. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 4.33**2+5**2 - 2 * 4.33 * 5 * cos 30° } ; ; b = 2.5 ; ;


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 = 4.33 ; ; b = 2.5 ; ; c = 5 ; ;

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

p = a+b+c = 4.33+2.5+5 = 11.83 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.83 }{ 2 } = 5.92 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.92 * (5.92-4.33)(5.92-2.5)(5.92-5) } ; ; T = sqrt{ 29.3 } = 5.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5.41 }{ 4.33 } = 2.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5.41 }{ 2.5 } = 4.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5.41 }{ 5 } = 2.17 ; ;

8. 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{ 2.5**2+5**2-4.33**2 }{ 2 * 2.5 * 5 } ) = 60° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.33**2+5**2-2.5**2 }{ 2 * 4.33 * 5 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 60° - 30° = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5.41 }{ 5.92 } = 0.92 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4.33 }{ 2 * sin 60° } = 2.5 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.5**2+2 * 5**2 - 4.33**2 } }{ 2 } = 3.307 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5**2+2 * 4.33**2 - 2.5**2 } }{ 2 } = 4.507 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.5**2+2 * 4.33**2 - 5**2 } }{ 2 } = 2.5 ; ;
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