Triangle calculator

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

You have entered height hc, angle α, angle β and angle γ.

Acute isosceles triangle.

Sides: a = 27.58444479741   b = 27.58444479741   c = 23.31553829077

Area: T = 291.4422286347
Perimeter: p = 78.48442788559
Semiperimeter: s = 39.24221394279

Angle ∠ A = α = 65° = 1.13444640138 rad
Angle ∠ B = β = 65° = 1.13444640138 rad
Angle ∠ C = γ = 50° = 0.8732664626 rad

Height: ha = 21.1310913087
Height: hb = 21.1310913087
Height: hc = 25

Median: ma = 21.49548594454
Median: mb = 21.49548594454
Median: mc = 25

Inradius: r = 7.42767685349
Circumradius: R = 15.21880354007

Vertex coordinates: A[23.31553829077; 0] B[0; 0] C[11.65876914539; 25]
Centroid: CG[11.65876914539; 8.33333333333]
Coordinates of the circumscribed circle: U[11.65876914539; 9.78219645993]
Coordinates of the inscribed circle: I[11.65876914539; 7.42767685349]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115° = 1.13444640138 rad
∠ B' = β' = 115° = 1.13444640138 rad
∠ C' = γ' = 130° = 0.8732664626 rad

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How did we calculate this triangle?

1. Input data entered: angle α, angle β, angle γ and height hc.

 alpha = 65° ; ; beta = 65° ; ; gamma = 50° ; ; hc = 25 ; ;

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 27.584**2 = 27.584**2 + c**2 - 2 * 27.584 * c * cos(65° ) ; ; ; ; ; ; c**2 -23.315c =0 ; ; a=1; b=-23.315; c=0 ; ; D = b**2 - 4ac = 23.315**2 - 4 * 1 * 0 = 543.607080135 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 23.32 ± sqrt{ 543.61 } }{ 2 } ; ; c_{1,2} = 11.65769145 ± 11.6576914539 ; ; c_{1} = 23.3153829039 ; ; c_{2} = -3.87496434939E-9 ; ;
 ; ; text{ Factored form: } ; ; (c -23.3153829039) (c +3.87496434939E-9) = 0 ; ; ; ; c > 0 ; ; ; ; c = 23.315 ; ;


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 = 27.58 ; ; b = 27.58 ; ; c = 23.32 ; ;

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

p = a+b+c = 27.58+27.58+23.32 = 78.48 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 78.48 }{ 2 } = 39.24 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.24 * (39.24-27.58)(39.24-27.58)(39.24-23.32) } ; ; T = sqrt{ 84938.61 } = 291.44 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 291.44 }{ 27.58 } = 21.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 291.44 }{ 27.58 } = 21.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 291.44 }{ 23.32 } = 25 ; ;

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{ 27.58**2+23.32**2-27.58**2 }{ 2 * 27.58 * 23.32 } ) = 65° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 27.58**2+23.32**2-27.58**2 }{ 2 * 27.58 * 23.32 } ) = 65° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 27.58**2+27.58**2-23.32**2 }{ 2 * 27.58 * 27.58 } ) = 50° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 291.44 }{ 39.24 } = 7.43 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 27.58 }{ 2 * sin 65° } = 15.22 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.58**2+2 * 23.32**2 - 27.58**2 } }{ 2 } = 21.495 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.32**2+2 * 27.58**2 - 27.58**2 } }{ 2 } = 21.495 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.58**2+2 * 27.58**2 - 23.32**2 } }{ 2 } = 25 ; ;
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