6 21 24 triangle

Obtuse scalene triangle.

Sides: a = 6   b = 21   c = 24

Area: T = 57.93547693531
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 13.2911177243° = 13°17'28″ = 0.23219748044 rad
Angle ∠ B = β = 53.57664263577° = 53°34'35″ = 0.93550850414 rad
Angle ∠ C = γ = 113.1322396399° = 113°7'57″ = 1.97545328078 rad

Height: ha = 19.31215897844
Height: hb = 5.51875970813
Height: hc = 4.82878974461

Median: ma = 22.34994966386
Median: mb = 13.99110685796
Median: mc = 9.72111110476

Inradius: r = 2.27219517393
Circumradius: R = 13.0499158708

Vertex coordinates: A[24; 0] B[0; 0] C[3.56325; 4.82878974461]
Centroid: CG[9.18875; 1.60992991487]
Coordinates of the circumscribed circle: U[12; -5.12664552067]
Coordinates of the inscribed circle: I[4.5; 2.27219517393]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 166.7098822757° = 166°42'32″ = 0.23219748044 rad
∠ B' = β' = 126.4243573642° = 126°25'25″ = 0.93550850414 rad
∠ C' = γ' = 66.86876036007° = 66°52'3″ = 1.97545328078 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 = 6 ; ; b = 21 ; ; c = 24 ; ;

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

p = a+b+c = 6+21+24 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-6)(25.5-21)(25.5-24) } ; ; T = sqrt{ 3356.44 } = 57.93 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 57.93 }{ 6 } = 19.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 57.93 }{ 21 } = 5.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 57.93 }{ 24 } = 4.83 ; ;

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{ 6**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 13° 17'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-6**2-24**2 }{ 2 * 6 * 24 } ) = 53° 34'35" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-6**2-21**2 }{ 2 * 21 * 6 } ) = 113° 7'57" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 57.93 }{ 25.5 } = 2.27 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6 }{ 2 * sin 13° 17'28" } = 13.05 ; ;




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