13 27 30 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 27   c = 30

Area: T = 175.4999287748
Perimeter: p = 70
Semiperimeter: s = 35

Angle ∠ A = α = 25.67991768138° = 25°40'45″ = 0.44881861846 rad
Angle ∠ B = β = 64.15875871263° = 64°9'27″ = 1.12197611355 rad
Angle ∠ C = γ = 90.16332360599° = 90°9'48″ = 1.57436453335 rad

Height: ha = 276.9998904227
Height: hb = 132.9999472406
Height: hc = 11.76999525165

Median: ma = 27.78993864632
Median: mb = 18.76883243791
Median: mc = 14.96766295471

Inradius: r = 5.01442653642
Circumradius: R = 155.0000608765

Vertex coordinates: A[30; 0] B[0; 0] C[5.66766666667; 11.76999525165]
Centroid: CG[11.88988888889; 3.98999841722]
Coordinates of the circumscribed circle: U[15; -0.04327352162]
Coordinates of the inscribed circle: I[8; 5.01442653642]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.3210823186° = 154°19'15″ = 0.44881861846 rad
∠ B' = β' = 115.8422412874° = 115°50'33″ = 1.12197611355 rad
∠ C' = γ' = 89.83767639401° = 89°50'12″ = 1.57436453335 rad

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How did we calculate this triangle?

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 = 13 ; ; b = 27 ; ; c = 30 ; ;

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

p = a+b+c = 13+27+30 = 70 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 70 }{ 2 } = 35 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35 * (35-13)(35-27)(35-30) } ; ; T = sqrt{ 30800 } = 175.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 175.5 }{ 13 } = 27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 175.5 }{ 27 } = 13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 175.5 }{ 30 } = 11.7 ; ;

5. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 13**2-27**2-30**2 }{ 2 * 27 * 30 } ) = 25° 40'45" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-13**2-30**2 }{ 2 * 13 * 30 } ) = 64° 9'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-13**2-27**2 }{ 2 * 27 * 13 } ) = 90° 9'48" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 175.5 }{ 35 } = 5.01 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 25° 40'45" } = 15 ; ;




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