Triangle calculator - result

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

You have entered side a and ratio of angles α:β:γ = 1:4:5.

Right scalene triangle.

Sides: a = 2   b = 6.15553670744   c = 6.4722135955

Area: T = 6.15553670744
Perimeter: p = 14.62875030294
Semiperimeter: s = 7.31437515147

Angle ∠ A = α = 18° = 0.31441592654 rad
Angle ∠ B = β = 72° = 1.25766370614 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 6.15553670744
Height: hb = 2
Height: hc = 1.90221130326

Median: ma = 6.23660679775
Median: mb = 3.67704408393
Median: mc = 3.23660679775

Inradius: r = 0.84216155597
Circumradius: R = 3.23660679775

Vertex coordinates: A[6.4722135955; 0] B[0; 0] C[0.61880339887; 1.90221130326]
Centroid: CG[2.36333899812; 0.63440376775]
Coordinates of the circumscribed circle: U[3.23660679775; -0]
Coordinates of the inscribed circle: I[1.15883844403; 0.84216155597]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162° = 0.31441592654 rad
∠ B' = β' = 108° = 1.25766370614 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: side a and ratio of angles α:β:γ.

a = 2 ; ; alpha : beta : gamma = 1:4:5 ; ;

2. From side a and angle β we calculate height hc:

h_c = a * sin beta = 2 * sin 72° = 1.902 ; ;

3. From side a and angle γ we calculate height hb:

h_b = a * sin gamma = 2 * sin 90° = 2 ; ;

4. Calculation of the third side c of the triangle using a Law of Cosines

c**2 = b**2+a**2 - 2ba cos gamma ; ; c = sqrt{ b**2+a**2 - 2ba cos gamma } ; ; c = sqrt{ 6.16**2+2**2 - 2 * 6.16 * 2 * cos 90° } ; ; c = 6.47 ; ;


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 = 2 ; ; b = 6.16 ; ; c = 6.47 ; ;

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

p = a+b+c = 2+6.16+6.47 = 14.63 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 14.63 }{ 2 } = 7.31 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.31 * (7.31-2)(7.31-6.16)(7.31-6.47) } ; ; T = sqrt{ 37.89 } = 6.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6.16 }{ 2 } = 6.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6.16 }{ 6.16 } = 2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6.16 }{ 6.47 } = 1.9 ; ;

9. 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{ 6.16**2+6.47**2-2**2 }{ 2 * 6.16 * 6.47 } ) = 18° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2**2+6.47**2-6.16**2 }{ 2 * 2 * 6.47 } ) = 72° ; ; gamma = 180° - alpha - beta = 180° - 18° - 72° = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6.16 }{ 7.31 } = 0.84 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 2 }{ 2 * sin 18° } = 3.24 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.16**2+2 * 6.47**2 - 2**2 } }{ 2 } = 6.236 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.47**2+2 * 2**2 - 6.16**2 } }{ 2 } = 3.67 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.16**2+2 * 2**2 - 6.47**2 } }{ 2 } = 3.236 ; ;
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