10 13 16 triangle

Acute scalene triangle.

Sides: a = 10   b = 13   c = 16

Area: T = 64.91986991552
Perimeter: p = 39
Semiperimeter: s = 19.5

Angle ∠ A = α = 38.62548328731° = 38°37'29″ = 0.67441305067 rad
Angle ∠ B = β = 54.24111511095° = 54°14'28″ = 0.94766866769 rad
Angle ∠ C = γ = 87.13440160174° = 87°8'2″ = 1.521077547 rad

Height: ha = 12.9843739831
Height: hb = 9.98774921777
Height: hc = 8.11548373944

Median: ma = 13.69330639376
Median: mb = 11.65111801977
Median: mc = 8.39664278119

Inradius: r = 3.32991640592
Circumradius: R = 8.01100187891

Vertex coordinates: A[16; 0] B[0; 0] C[5.844375; 8.11548373944]
Centroid: CG[7.281125; 2.70549457981]
Coordinates of the circumscribed circle: U[8; 0.40105009395]
Coordinates of the inscribed circle: I[6.5; 3.32991640592]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.3755167127° = 141°22'31″ = 0.67441305067 rad
∠ B' = β' = 125.759884889° = 125°45'32″ = 0.94766866769 rad
∠ C' = γ' = 92.86659839826° = 92°51'58″ = 1.521077547 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 = 10 ; ; b = 13 ; ; c = 16 ; ;

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

p = a+b+c = 10+13+16 = 39 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 39 }{ 2 } = 19.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.5 * (19.5-10)(19.5-13)(19.5-16) } ; ; T = sqrt{ 4214.44 } = 64.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 64.92 }{ 10 } = 12.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 64.92 }{ 13 } = 9.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 64.92 }{ 16 } = 8.11 ; ;

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{ 10**2-13**2-16**2 }{ 2 * 13 * 16 } ) = 38° 37'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-10**2-16**2 }{ 2 * 10 * 16 } ) = 54° 14'28" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16**2-10**2-13**2 }{ 2 * 13 * 10 } ) = 87° 8'2" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 64.92 }{ 19.5 } = 3.33 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 38° 37'29" } = 8.01 ; ;




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