19 27 30 triangle

Acute scalene triangle.

Sides: a = 19   b = 27   c = 30

Area: T = 252.0633484067
Perimeter: p = 76
Semiperimeter: s = 38

Angle ∠ A = α = 38.49900685704° = 38°29'24″ = 0.67217784259 rad
Angle ∠ B = β = 62.18218607153° = 62°10'55″ = 1.08552782045 rad
Angle ∠ C = γ = 79.32880707142° = 79°19'41″ = 1.38545360232 rad

Height: ha = 26.53329983228
Height: hb = 18.67113691901
Height: hc = 16.80442322711

Median: ma = 26.91218932816
Median: mb = 21.17219153597
Median: mc = 17.889854382

Inradius: r = 6.63332495807
Circumradius: R = 15.26440118192

Vertex coordinates: A[30; 0] B[0; 0] C[8.86766666667; 16.80442322711]
Centroid: CG[12.95655555556; 5.6011410757]
Coordinates of the circumscribed circle: U[15; 2.82766688554]
Coordinates of the inscribed circle: I[11; 6.63332495807]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.510993143° = 141°30'36″ = 0.67217784259 rad
∠ B' = β' = 117.8188139285° = 117°49'5″ = 1.08552782045 rad
∠ C' = γ' = 100.6721929286° = 100°40'19″ = 1.38545360232 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 = 27 ; ; c = 30 ; ;

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

p = a+b+c = 19+27+30 = 76 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76 }{ 2 } = 38 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38 * (38-19)(38-27)(38-30) } ; ; T = sqrt{ 63536 } = 252.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 252.06 }{ 19 } = 26.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 252.06 }{ 27 } = 18.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 252.06 }{ 30 } = 16.8 ; ;

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**2-27**2-30**2 }{ 2 * 27 * 30 } ) = 38° 29'24" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-19**2-30**2 }{ 2 * 19 * 30 } ) = 62° 10'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-19**2-27**2 }{ 2 * 27 * 19 } ) = 79° 19'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 252.06 }{ 38 } = 6.63 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 38° 29'24" } = 15.26 ; ;




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