13 16 17 triangle

Acute scalene triangle.

Sides: a = 13   b = 16   c = 17

Area: T = 98.28552990024
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 46.27766909947° = 46°16'36″ = 0.80876806248 rad
Angle ∠ B = β = 62.80554338089° = 62°48'20″ = 1.09661616081 rad
Angle ∠ C = γ = 70.91878751964° = 70°55'4″ = 1.23877504207 rad

Height: ha = 15.12108152311
Height: hb = 12.28656623753
Height: hc = 11.56329763532

Median: ma = 15.17439909055
Median: mb = 12.84552325787
Median: mc = 11.84327192823

Inradius: r = 4.27332738697
Circumradius: R = 8.99442240495

Vertex coordinates: A[17; 0] B[0; 0] C[5.94111764706; 11.56329763532]
Centroid: CG[7.64770588235; 3.85443254511]
Coordinates of the circumscribed circle: U[8.5; 2.94404194008]
Coordinates of the inscribed circle: I[7; 4.27332738697]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.7233309005° = 133°43'24″ = 0.80876806248 rad
∠ B' = β' = 117.1954566191° = 117°11'40″ = 1.09661616081 rad
∠ C' = γ' = 109.0822124804° = 109°4'56″ = 1.23877504207 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 = 13 ; ; b = 16 ; ; c = 17 ; ;

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

p = a+b+c = 13+16+17 = 46 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46 }{ 2 } = 23 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23 * (23-13)(23-16)(23-17) } ; ; T = sqrt{ 9660 } = 98.29 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 98.29 }{ 13 } = 15.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 98.29 }{ 16 } = 12.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 98.29 }{ 17 } = 11.56 ; ;

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{ 13**2-16**2-17**2 }{ 2 * 16 * 17 } ) = 46° 16'36" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-13**2-17**2 }{ 2 * 13 * 17 } ) = 62° 48'20" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 17**2-13**2-16**2 }{ 2 * 16 * 13 } ) = 70° 55'4" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 98.29 }{ 23 } = 4.27 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 46° 16'36" } = 8.99 ; ;




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