12 13 17 triangle

Acute scalene triangle.

Sides: a = 12   b = 13   c = 17

Area: T = 77.76988883809
Perimeter: p = 42
Semiperimeter: s = 21

Angle ∠ A = α = 44.73219408557° = 44°43'55″ = 0.78107196487 rad
Angle ∠ B = β = 49.687978493° = 49°40'47″ = 0.86770758187 rad
Angle ∠ C = γ = 85.58882742142° = 85°35'18″ = 1.49437971861 rad

Height: ha = 12.96114813968
Height: hb = 11.96444443663
Height: hc = 9.1499280986

Median: ma = 13.89224439894
Median: mb = 13.22003787824
Median: mc = 9.17987798753

Inradius: r = 3.70332803991
Circumradius: R = 8.52552600854

Vertex coordinates: A[17; 0] B[0; 0] C[7.76547058824; 9.1499280986]
Centroid: CG[8.25549019608; 3.05497603287]
Coordinates of the circumscribed circle: U[8.5; 0.65657892373]
Coordinates of the inscribed circle: I[8; 3.70332803991]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.2688059144° = 135°16'5″ = 0.78107196487 rad
∠ B' = β' = 130.322021507° = 130°19'13″ = 0.86770758187 rad
∠ C' = γ' = 94.41217257858° = 94°24'42″ = 1.49437971861 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 = 12 ; ; b = 13 ; ; c = 17 ; ;

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

p = a+b+c = 12+13+17 = 42 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 42 }{ 2 } = 21 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21 * (21-12)(21-13)(21-17) } ; ; T = sqrt{ 6048 } = 77.77 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 77.77 }{ 12 } = 12.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 77.77 }{ 13 } = 11.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 77.77 }{ 17 } = 9.15 ; ;

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{ 12**2-13**2-17**2 }{ 2 * 13 * 17 } ) = 44° 43'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-12**2-17**2 }{ 2 * 12 * 17 } ) = 49° 40'47" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 17**2-12**2-13**2 }{ 2 * 13 * 12 } ) = 85° 35'18" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 77.77 }{ 21 } = 3.7 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 44° 43'55" } = 8.53 ; ;




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