8 16 21 triangle

Obtuse scalene triangle.

Sides: a = 8 b = 16 c = 21

Area: T = 56.43998005316
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 19.61659069161° = 19°36'57″ = 0.34223621615 rad
Angle ∠ B = β = 42.17772350071° = 42°10'38″ = 0.73661316203 rad
Angle ∠ C = γ = 118.2076858077° = 118°12'25″ = 2.06330988719 rad

Height: h_a = 14.10999501329
Height: h_b = 7.05499750664
Height: h_c = 5.37114095744

Median: m_a = 18.23545825288
Median: m_b = 13.73295302177
Median: m_c = 7.05333679898

Inradius: r = 2.50766578014
Circumradius: R = 11.91549357563

Vertex coordinates: A[21; 0] B[0; 0] C[5.92985714286; 5.37114095744]
Centroid: CG[8.97661904762; 1.79904698581]
Coordinates of the circumscribed circle: U[10.5; -5.63216688535]
Coordinates of the inscribed circle: I[6.5; 2.50766578014]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.3844093084° = 160°23'3″ = 0.34223621615 rad
∠ B' = β' = 137.8232764993° = 137°49'22″ = 0.73661316203 rad
∠ C' = γ' = 61.79331419232° = 61°47'35″ = 2.06330988719 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.

$a = 8 ; ; b = 16 ; ; c = 21 ; ;$

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

$p = a+b+c = 8+16+21 = 45 ; ;$

2. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;$

3. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-8)(22.5-16)(22.5-21) } ; ; T = sqrt{ 3180.94 } = 56.4 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 56.4 }{ 8 } = 14.1 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 56.4 }{ 16 } = 7.05 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 56.4 }{ 21 } = 5.37 ; ;$

5. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 8**2-16**2-21**2 }{ 2 * 16 * 21 } ) = 19° 36'57" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-8**2-21**2 }{ 2 * 8 * 21 } ) = 42° 10'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-8**2-16**2 }{ 2 * 16 * 8 } ) = 118° 12'25" ; ;$

6. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 56.4 }{ 22.5 } = 2.51 ; ;$

7. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 19° 36'57" } = 11.91 ; ;$

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