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: h_a = 27.24875951314
Height: h_b = 18.48994395534
Height: h_c = 17.25768102499

Median: m_a = 27.41880597417
Median: m_b = 20.84546635857
Median: m_c = 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

Calculate another triangle

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

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

6. Inradius

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

7. Circumradius

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

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