11 13 16 triangle

Acute scalene triangle.

Sides: a = 11   b = 13   c = 16

Area: T = 70.99329573972
Perimeter: p = 40
Semiperimeter: s = 20

Angle ∠ A = α = 43.04990798002° = 43°2'57″ = 0.75113481825 rad
Angle ∠ B = β = 53.77884533802° = 53°46'42″ = 0.93986110781 rad
Angle ∠ C = γ = 83.17224668196° = 83°10'21″ = 1.4521633393 rad

Height: ha = 12.90878104359
Height: hb = 10.92219934457
Height: hc = 8.87441196746

Median: ma = 13.5
Median: mb = 12.09333866224
Median: mc = 9

Inradius: r = 3.55496478699
Circumradius: R = 8.05771372284

Vertex coordinates: A[16; 0] B[0; 0] C[6.5; 8.87441196746]
Centroid: CG[7.5; 2.95880398915]
Coordinates of the circumscribed circle: U[8; 0.95878414887]
Coordinates of the inscribed circle: I[7; 3.55496478699]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.95109202° = 136°57'3″ = 0.75113481825 rad
∠ B' = β' = 126.222154662° = 126°13'18″ = 0.93986110781 rad
∠ C' = γ' = 96.82875331804° = 96°49'39″ = 1.4521633393 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 = 11 ; ; b = 13 ; ; c = 16 ; ;

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

p = a+b+c = 11+13+16 = 40 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40 }{ 2 } = 20 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20 * (20-11)(20-13)(20-16) } ; ; T = sqrt{ 5040 } = 70.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 70.99 }{ 11 } = 12.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 70.99 }{ 13 } = 10.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 70.99 }{ 16 } = 8.87 ; ;

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{ 11**2-13**2-16**2 }{ 2 * 13 * 16 } ) = 43° 2'57" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-11**2-16**2 }{ 2 * 11 * 16 } ) = 53° 46'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16**2-11**2-13**2 }{ 2 * 13 * 11 } ) = 83° 10'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 70.99 }{ 20 } = 3.55 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11 }{ 2 * sin 43° 2'57" } = 8.06 ; ;




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