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 = 1   b = 3.07876835372   c = 3.23660679775

Area: T = 1.53988417686
Perimeter: p = 7.31437515147
Semiperimeter: s = 3.65768757573

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

Height: ha = 3.07876835372
Height: hb = 1
Height: hc = 0.95110565163

Median: ma = 3.11880339887
Median: mb = 1.83552204197
Median: mc = 1.61880339887

Inradius: r = 0.42108077798
Circumradius: R = 1.61880339887

Vertex coordinates: A[3.23660679775; 0] B[0; 0] C[0.30990169944; 0.95110565163]
Centroid: CG[1.18216949906; 0.31770188388]
Coordinates of the circumscribed circle: U[1.61880339887; -0]
Coordinates of the inscribed circle: I[0.57991922202; 0.42108077798]

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

2. 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{ 3.08**2+1**2 - 2 * 3.08 * 1 * cos(90° ) } ; ; c = 3.24 ; ;


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 ; ; b = 3.08 ; ; c = 3.24 ; ;

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

p = a+b+c = 1+3.08+3.24 = 7.31 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 7.31 }{ 2 } = 3.66 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.66 * (3.66-1)(3.66-3.08)(3.66-3.24) } ; ; T = sqrt{ 2.37 } = 1.54 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.54 }{ 1 } = 3.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.54 }{ 3.08 } = 1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.54 }{ 3.24 } = 0.95 ; ;

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{ 3.08**2+3.24**2-1**2 }{ 2 * 3.08 * 3.24 } ) = 18° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1**2+3.24**2-3.08**2 }{ 2 * 1 * 3.24 } ) = 72° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 1**2+3.08**2-3.24**2 }{ 2 * 1 * 3.08 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.54 }{ 3.66 } = 0.42 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1 }{ 2 * sin 18° } = 1.62 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.08**2+2 * 3.24**2 - 1**2 } }{ 2 } = 3.118 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.24**2+2 * 1**2 - 3.08**2 } }{ 2 } = 1.835 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.08**2+2 * 1**2 - 3.24**2 } }{ 2 } = 1.618 ; ;
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