19 28 30 triangle

Acute scalene triangle.

Sides: a = 19   b = 28   c = 30

Area: T = 258.8522153748
Perimeter: p = 77
Semiperimeter: s = 38.5

Angle ∠ A = α = 38.04875074536° = 38°2'51″ = 0.66440542772 rad
Angle ∠ B = β = 65.26550579885° = 65°15'54″ = 1.13990901484 rad
Angle ∠ C = γ = 76.68774345579° = 76°41'15″ = 1.33884482279 rad

Height: ha = 27.24875951314
Height: hb = 18.48994395534
Height: hc = 17.25768102499

Median: ma = 27.41880597417
Median: mb = 20.84546635857
Median: mc = 18.64113518823

Inradius: r = 6.72334325649
Circumradius: R = 15.4144204372

Vertex coordinates: A[30; 0] B[0; 0] C[7.95; 17.25768102499]
Centroid: CG[12.65; 5.75222700833]
Coordinates of the circumscribed circle: U[15; 3.54993233751]
Coordinates of the inscribed circle: I[10.5; 6.72334325649]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.9522492546° = 141°57'9″ = 0.66440542772 rad
∠ B' = β' = 114.7354942011° = 114°44'6″ = 1.13990901484 rad
∠ C' = γ' = 103.3132565442° = 103°18'45″ = 1.33884482279 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 = 19 ; ; b = 28 ; ; c = 30 ; ;

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

p = a+b+c = 19+28+30 = 77 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 77 }{ 2 } = 38.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.5 * (38.5-19)(38.5-28)(38.5-30) } ; ; T = sqrt{ 67004.44 } = 258.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 258.85 }{ 19 } = 27.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 258.85 }{ 28 } = 18.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 258.85 }{ 30 } = 17.26 ; ;

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{ 28**2+30**2-19**2 }{ 2 * 28 * 30 } ) = 38° 2'51" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 19**2+30**2-28**2 }{ 2 * 19 * 30 } ) = 65° 15'54" ; ; gamma = 180° - alpha - beta = 180° - 38° 2'51" - 65° 15'54" = 76° 41'15" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 258.85 }{ 38.5 } = 6.72 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 19 }{ 2 * sin 38° 2'51" } = 15.41 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 30**2 - 19**2 } }{ 2 } = 27.418 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 30**2+2 * 19**2 - 28**2 } }{ 2 } = 20.845 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 19**2 - 30**2 } }{ 2 } = 18.641 ; ;
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