19 21 28 triangle

Acute scalene triangle.

Sides: a = 19   b = 21   c = 28

Area: T = 199.4499241663
Perimeter: p = 68
Semiperimeter: s = 34

Angle ∠ A = α = 42.71986445144° = 42°43'7″ = 0.74655809988 rad
Angle ∠ B = β = 48.57438511071° = 48°34'26″ = 0.84877736322 rad
Angle ∠ C = γ = 88.70875043785° = 88°42'27″ = 1.54882380226 rad

Height: ha = 20.99546570172
Height: hb = 18.99551658727
Height: hc = 14.24663744045

Median: ma = 22.85327897641
Median: mb = 21.5
Median: mc = 14.31878210633

Inradius: r = 5.86661541666
Circumradius: R = 14.00435628951

Vertex coordinates: A[28; 0] B[0; 0] C[12.57114285714; 14.24663744045]
Centroid: CG[13.52438095238; 4.74987914682]
Coordinates of the circumscribed circle: U[14; 0.31658698397]
Coordinates of the inscribed circle: I[13; 5.86661541666]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.2811355486° = 137°16'53″ = 0.74655809988 rad
∠ B' = β' = 131.4266148893° = 131°25'34″ = 0.84877736322 rad
∠ C' = γ' = 91.29224956215° = 91°17'33″ = 1.54882380226 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 = 21 ; ; c = 28 ; ;

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

p = a+b+c = 19+21+28 = 68 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 68 }{ 2 } = 34 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34 * (34-19)(34-21)(34-28) } ; ; T = sqrt{ 39780 } = 199.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 199.45 }{ 19 } = 20.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 199.45 }{ 21 } = 19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 199.45 }{ 28 } = 14.25 ; ;

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**2-21**2-28**2 }{ 2 * 21 * 28 } ) = 42° 43'7" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-19**2-28**2 }{ 2 * 19 * 28 } ) = 48° 34'26" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-19**2-21**2 }{ 2 * 21 * 19 } ) = 88° 42'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 199.45 }{ 34 } = 5.87 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 42° 43'7" } = 14 ; ;




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