21 28 30 triangle

Acute scalene triangle.

Sides: a = 21   b = 28   c = 30

Area: T = 282.5549884976
Perimeter: p = 79
Semiperimeter: s = 39.5

Angle ∠ A = α = 42.2798724472° = 42°16'43″ = 0.73879029456 rad
Angle ∠ B = β = 63.76443856588° = 63°45'52″ = 1.11328984753 rad
Angle ∠ C = γ = 73.95768898692° = 73°57'25″ = 1.29107912328 rad

Height: ha = 26.91095128549
Height: hb = 20.18221346411
Height: hc = 18.83766589984

Median: ma = 27.05108779894
Median: mb = 21.78330209108
Median: mc = 19.6855019685

Inradius: r = 7.1533161645
Circumradius: R = 15.60878633703

Vertex coordinates: A[30; 0] B[0; 0] C[9.28333333333; 18.83766589984]
Centroid: CG[13.09444444444; 6.27988863328]
Coordinates of the circumscribed circle: U[15; 4.31333976151]
Coordinates of the inscribed circle: I[11.5; 7.1533161645]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.7211275528° = 137°43'17″ = 0.73879029456 rad
∠ B' = β' = 116.2365614341° = 116°14'8″ = 1.11328984753 rad
∠ C' = γ' = 106.0433110131° = 106°2'35″ = 1.29107912328 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 = 21 ; ; b = 28 ; ; c = 30 ; ;

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

p = a+b+c = 21+28+30 = 79 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 79 }{ 2 } = 39.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.5 * (39.5-21)(39.5-28)(39.5-30) } ; ; T = sqrt{ 79834.44 } = 282.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 282.55 }{ 21 } = 26.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 282.55 }{ 28 } = 20.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 282.55 }{ 30 } = 18.84 ; ;

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{ 21**2-28**2-30**2 }{ 2 * 28 * 30 } ) = 42° 16'43" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28**2-21**2-30**2 }{ 2 * 21 * 30 } ) = 63° 45'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-21**2-28**2 }{ 2 * 28 * 21 } ) = 73° 57'25" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 282.55 }{ 39.5 } = 7.15 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 42° 16'43" } = 15.61 ; ;




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