5 27 28 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 27   c = 28

Area: T = 67.0822039325
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 10.22221793906° = 10°13'20″ = 0.17884106871 rad
Angle ∠ B = β = 73.3988450401° = 73°23'54″ = 1.28110446254 rad
Angle ∠ C = γ = 96.37993702084° = 96°22'46″ = 1.68221373411 rad

Height: ha = 26.833281573
Height: hb = 4.969903995
Height: hc = 4.79215742375

Median: ma = 27.39106918496
Median: mb = 14.90880515159
Median: mc = 13.45436240471

Inradius: r = 2.23660679775
Circumradius: R = 14.08772282582

Vertex coordinates: A[28; 0] B[0; 0] C[1.42985714286; 4.79215742375]
Centroid: CG[9.81095238095; 1.59771914125]
Coordinates of the circumscribed circle: U[14; -1.56552475842]
Coordinates of the inscribed circle: I[3; 2.23660679775]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 169.7787820609° = 169°46'40″ = 0.17884106871 rad
∠ B' = β' = 106.6021549599° = 106°36'6″ = 1.28110446254 rad
∠ C' = γ' = 83.62106297916° = 83°37'14″ = 1.68221373411 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 = 27 ; ; c = 28 ; ;

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

p = a+b+c = 5+27+28 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-5)(30-27)(30-28) } ; ; T = sqrt{ 4500 } = 67.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 67.08 }{ 5 } = 26.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 67.08 }{ 27 } = 4.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 67.08 }{ 28 } = 4.79 ; ;

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{ 27**2+28**2-5**2 }{ 2 * 27 * 28 } ) = 10° 13'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5**2+28**2-27**2 }{ 2 * 5 * 28 } ) = 73° 23'54" ; ; gamma = 180° - alpha - beta = 180° - 10° 13'20" - 73° 23'54" = 96° 22'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 67.08 }{ 30 } = 2.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5 }{ 2 * sin 10° 13'20" } = 14.09 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 28**2 - 5**2 } }{ 2 } = 27.391 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 5**2 - 27**2 } }{ 2 } = 14.908 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 5**2 - 28**2 } }{ 2 } = 13.454 ; ;
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