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: ha = 14.10999501329
Height: hb = 7.05499750664
Height: hc = 5.37114095744

Median: ma = 18.23545825288
Median: mb = 13.73295302177
Median: mc = 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

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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 = 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 16**2+21**2-8**2 }{ 2 * 16 * 21 } ) = 19° 36'57" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8**2+21**2-16**2 }{ 2 * 8 * 21 } ) = 42° 10'38" ; ; gamma = 180° - alpha - beta = 180° - 19° 36'57" - 42° 10'38" = 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 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16**2+2 * 21**2 - 8**2 } }{ 2 } = 18.235 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 8**2 - 16**2 } }{ 2 } = 13.73 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16**2+2 * 8**2 - 21**2 } }{ 2 } = 7.053 ; ;
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