9 16 20 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 16   c = 20

Area: T = 70.2566227482
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 26.04765623605° = 26°2'48″ = 0.4554598272 rad
Angle ∠ B = β = 51.31878125465° = 51°19'4″ = 0.89656647939 rad
Angle ∠ C = γ = 102.6365625093° = 102°38'8″ = 1.79113295877 rad

Height: ha = 15.6122494996
Height: hb = 8.78220284352
Height: hc = 7.02656227482

Median: ma = 17.54328047928
Median: mb = 13.28553302556
Median: mc = 8.27664726786

Inradius: r = 3.12224989992
Circumradius: R = 10.24882018435

Vertex coordinates: A[20; 0] B[0; 0] C[5.625; 7.02656227482]
Centroid: CG[8.54216666667; 2.34218742494]
Coordinates of the circumscribed circle: U[10; -2.24217941533]
Coordinates of the inscribed circle: I[6.5; 3.12224989992]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.953343764° = 153°57'12″ = 0.4554598272 rad
∠ B' = β' = 128.6822187453° = 128°40'56″ = 0.89656647939 rad
∠ C' = γ' = 77.3644374907° = 77°21'52″ = 1.79113295877 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 = 9 ; ; b = 16 ; ; c = 20 ; ;

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

p = a+b+c = 9+16+20 = 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 * (22.5-9)(22.5-16)(22.5-20) } ; ; T = sqrt{ 4935.94 } = 70.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 70.26 }{ 9 } = 15.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 70.26 }{ 16 } = 8.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 70.26 }{ 20 } = 7.03 ; ;

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{ 9**2-16**2-20**2 }{ 2 * 16 * 20 } ) = 26° 2'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-9**2-20**2 }{ 2 * 9 * 20 } ) = 51° 19'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-9**2-16**2 }{ 2 * 16 * 9 } ) = 102° 38'8" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 70.26 }{ 22.5 } = 3.12 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 26° 2'48" } = 10.25 ; ;




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