Triangle calculator

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

You have entered circumradius R and aspect ratio a:b:c = 6:7:10.

Obtuse scalene triangle.

Sides: a = 41.32549319419   b = 48.21224205988   c = 68.87548865698

Area: T = 980.1765729496
Perimeter: p = 158.412223911
Semiperimeter: s = 79.20661195552

Angle ∠ A = α = 36.18222872212° = 36°10'56″ = 0.63215000429 rad
Angle ∠ B = β = 43.53111521674° = 43°31'52″ = 0.76597619325 rad
Angle ∠ C = γ = 100.2876560611° = 100°17'12″ = 1.75503306782 rad

Height: ha = 47.43875
Height: hb = 40.66107142857
Height: hc = 28.46325

Median: ma = 55.74218716047
Median: mb = 51.42660695076
Median: mc = 28.81224322125

Inradius: r = 12.375
Circumradius: R = 35

Vertex coordinates: A[68.87548865698; 0] B[0; 0] C[29.96105756579; 28.46325]
Centroid: CG[32.94551540759; 9.48875]
Coordinates of the circumscribed circle: U[34.43774432849; -6.25]
Coordinates of the inscribed circle: I[30.99436989564; 12.375]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.8187712779° = 143°49'4″ = 0.63215000429 rad
∠ B' = β' = 136.4698847833° = 136°28'8″ = 0.76597619325 rad
∠ C' = γ' = 79.71334393885° = 79°42'48″ = 1.75503306782 rad

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How did we calculate this triangle?

1. Input data entered: aspect ratio a:b:c and circumradius R.

a:b:c = 6:7:10R = 35 ; ;

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{ 48.21**2+41.32**2 - 2 * 48.21 * 41.32 * cos(100° 17'12") } ; ; c = 68.87 ; ;


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 = 41.32 ; ; b = 48.21 ; ; c = 68.87 ; ;

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

p = a+b+c = 41.32+48.21+68.87 = 158.41 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 158.41 }{ 2 } = 79.21 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 79.21 * (79.21-41.32)(79.21-48.21)(79.21-68.87) } ; ; T = sqrt{ 960744.46 } = 980.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 980.18 }{ 41.32 } = 47.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 980.18 }{ 48.21 } = 40.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 980.18 }{ 68.87 } = 28.46 ; ;

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{ 48.21**2+68.87**2-41.32**2 }{ 2 * 48.21 * 68.87 } ) = 36° 10'56" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 41.32**2+68.87**2-48.21**2 }{ 2 * 41.32 * 68.87 } ) = 43° 31'52" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 41.32**2+48.21**2-68.87**2 }{ 2 * 41.32 * 48.21 } ) = 100° 17'12" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 980.18 }{ 79.21 } = 12.38 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 41.32 }{ 2 * sin 36° 10'56" } = 35 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 48.21**2+2 * 68.87**2 - 41.32**2 } }{ 2 } = 55.742 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 68.87**2+2 * 41.32**2 - 48.21**2 } }{ 2 } = 51.426 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 48.21**2+2 * 41.32**2 - 68.87**2 } }{ 2 } = 28.812 ; ;
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