5 24 27 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 24   c = 27

Area: T = 50.75443101618
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 9.01224514993° = 9°45″ = 0.15772969523 rad
Angle ∠ B = β = 48.75765958652° = 48°45'24″ = 0.85109631299 rad
Angle ∠ C = γ = 122.2310952636° = 122°13'51″ = 2.13333325713 rad

Height: ha = 20.30217240647
Height: hb = 4.23295258468
Height: hc = 3.76595785305

Median: ma = 25.42114476378
Median: mb = 15.26443375225
Median: mc = 10.87442815855

Inradius: r = 1.81326539343
Circumradius: R = 15.95992357263

Vertex coordinates: A[27; 0] B[0; 0] C[3.29662962963; 3.76595785305]
Centroid: CG[10.09987654321; 1.25331928435]
Coordinates of the circumscribed circle: U[13.5; -8.51215923874]
Coordinates of the inscribed circle: I[4; 1.81326539343]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170.9887548501° = 170°59'15″ = 0.15772969523 rad
∠ B' = β' = 131.2433404135° = 131°14'36″ = 0.85109631299 rad
∠ C' = γ' = 57.76990473645° = 57°46'9″ = 2.13333325713 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 = 5 ; ; b = 24 ; ; c = 27 ; ;

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

p = a+b+c = 5+24+27 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-5)(28-24)(28-27) } ; ; T = sqrt{ 2576 } = 50.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 50.75 }{ 5 } = 20.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 50.75 }{ 24 } = 4.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 50.75 }{ 27 } = 3.76 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 24**2+27**2-5**2 }{ 2 * 24 * 27 } ) = 9° 45" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5**2+27**2-24**2 }{ 2 * 5 * 27 } ) = 48° 45'24" ; ; gamma = 180° - alpha - beta = 180° - 9° 45" - 48° 45'24" = 122° 13'51" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 50.75 }{ 28 } = 1.81 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5 }{ 2 * sin 9° 45" } = 15.96 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 27**2 - 5**2 } }{ 2 } = 25.421 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 5**2 - 24**2 } }{ 2 } = 15.264 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 5**2 - 27**2 } }{ 2 } = 10.874 ; ;
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