4 13 16 triangle

Obtuse scalene triangle.

Sides: a = 4   b = 13   c = 16

Area: T = 18.9988355192
Perimeter: p = 33
Semiperimeter: s = 16.5

Angle ∠ A = α = 10.52656968806° = 10°31'33″ = 0.18437080666 rad
Angle ∠ B = β = 36.42199137286° = 36°25'12″ = 0.63656474079 rad
Angle ∠ C = γ = 133.0544389391° = 133°3'16″ = 2.32222371791 rad

Height: ha = 9.4999177596
Height: hb = 2.92328238757
Height: hc = 2.3754794399

Median: ma = 14.44395290782
Median: mb = 9.68224583655
Median: mc = 5.3398539126

Inradius: r = 1.15114154662
Circumradius: R = 10.94883162041

Vertex coordinates: A[16; 0] B[0; 0] C[3.219875; 2.3754794399]
Centroid: CG[6.406625; 0.7921598133]
Coordinates of the circumscribed circle: U[8; -7.47443312547]
Coordinates of the inscribed circle: I[3.5; 1.15114154662]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 169.4744303119° = 169°28'27″ = 0.18437080666 rad
∠ B' = β' = 143.5880086271° = 143°34'48″ = 0.63656474079 rad
∠ C' = γ' = 46.94656106092° = 46°56'44″ = 2.32222371791 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 = 4 ; ; b = 13 ; ; c = 16 ; ;

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

p = a+b+c = 4+13+16 = 33 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33 }{ 2 } = 16.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.5 * (16.5-4)(16.5-13)(16.5-16) } ; ; T = sqrt{ 360.94 } = 19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 19 }{ 4 } = 9.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 19 }{ 13 } = 2.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 19 }{ 16 } = 2.37 ; ;

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{ 4**2-13**2-16**2 }{ 2 * 13 * 16 } ) = 10° 31'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-4**2-16**2 }{ 2 * 4 * 16 } ) = 36° 25'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16**2-4**2-13**2 }{ 2 * 13 * 4 } ) = 133° 3'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 19 }{ 16.5 } = 1.15 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4 }{ 2 * sin 10° 31'33" } = 10.95 ; ;




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