Triangle calculator

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

You have entered height ha, height hb and height hc.

Obtuse scalene triangle.

Sides: a = 6.69215220599   b = 5.0198641545   c = 4.0154913236

Area: T = 10.03772830899
Perimeter: p = 15.72550768409
Semiperimeter: s = 7.86325384204

Angle ∠ A = α = 94.94399211322° = 94°56'24″ = 1.65770142153 rad
Angle ∠ B = β = 48.35496321995° = 48°20'59″ = 0.8443860274 rad
Angle ∠ C = γ = 36.71104466683° = 36°42'38″ = 0.64107181642 rad

Height: ha = 3
Height: hb = 4
Height: hc = 5

Median: ma = 3.07655534063
Median: mb = 4.91443979434
Median: mc = 5.56334281828

Inradius: r = 1.27765957447
Circumradius: R = 3.35882350609

Vertex coordinates: A[4.0154913236; 0] B[0; 0] C[4.44770740357; 5]
Centroid: CG[2.82106624239; 1.66766666667]
Coordinates of the circumscribed circle: U[2.0077456618; 2.69221851072]
Coordinates of the inscribed circle: I[2.84438968755; 1.27765957447]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 85.06600788678° = 85°3'36″ = 1.65770142153 rad
∠ B' = β' = 131.65503678° = 131°39'1″ = 0.8443860274 rad
∠ C' = γ' = 143.2989553332° = 143°17'22″ = 0.64107181642 rad

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How did we calculate this triangle?

1. Input data entered: height ha, height hb and height hc.

h_a = 3 ; ; h_b = 4 ; ; h_c = 5 ; ;

2. From height ha, height hb and height hc we calculate area T:

h = fraction{ fraction{ 1 }{ h_a } + fraction{ 1 }{ h_b } + fraction{ 1 }{ h_c } }{ 2 } = (1 / 3 + 1 / 4 + 1 / 5) / 2 = 0.392 ; ; ; ; S_1 = 4 * sqrt{ h * (h - 1 / h_a) * (h - 1 / h_b) * (h - 1 / h_c) } ; ; ; ; S_1 = 4 * sqrt{ 0.392 * (0.392 - 1 / 3) * (0.392 - 1 / 4) * (0.392 - 1 / 5) } = 0.1 ; ; ; ; T = fraction{ 1 }{ S_1 } = fraction{ 1 }{ 0.1 } = 10.037 ; ;

3. From area T and height ha we calculate side a:

T = fraction{ a h_a }{ 2 } ; ; ; ; a = fraction{ 2 T }{ h_a } = fraction{ 2 * 10.037 }{ 3 } = 6.692 ; ;

4. From area T and height hb we calculate side b:

T = fraction{ b h_b }{ 2 } ; ; ; ; b = fraction{ 2 T }{ h_b } = fraction{ 2 * 10.037 }{ 4 } = 5.019 ; ;

5. From area T and height hc we calculate side c:

T = fraction{ c h_c }{ 2 } ; ; ; ; c = fraction{ 2 T }{ h_c } = fraction{ 2 * 10.037 }{ 5 } = 4.015 ; ;
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 = 6.69 ; ; b = 5.02 ; ; c = 4.01 ; ;

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

p = a+b+c = 6.69+5.02+4.01 = 15.73 ; ;

7. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.73 }{ 2 } = 7.86 ; ;

8. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.86 * (7.86-6.69)(7.86-5.02)(7.86-4.01) } ; ; T = sqrt{ 100.75 } = 10.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10.04 }{ 6.69 } = 3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10.04 }{ 5.02 } = 4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10.04 }{ 4.01 } = 5 ; ;

10. 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{ 5.02**2+4.01**2-6.69**2 }{ 2 * 5.02 * 4.01 } ) = 94° 56'24" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.69**2+4.01**2-5.02**2 }{ 2 * 6.69 * 4.01 } ) = 48° 20'59" ; ; gamma = 180° - alpha - beta = 180° - 94° 56'24" - 48° 20'59" = 36° 42'38" ; ;

11. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10.04 }{ 7.86 } = 1.28 ; ;

12. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.69 }{ 2 * sin 94° 56'24" } = 3.36 ; ;

13. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.02**2+2 * 4.01**2 - 6.69**2 } }{ 2 } = 3.076 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.01**2+2 * 6.69**2 - 5.02**2 } }{ 2 } = 4.914 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.02**2+2 * 6.69**2 - 4.01**2 } }{ 2 } = 5.563 ; ;
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