8 17 20 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 17   c = 20

Area: T = 66.97771416231
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 23.20325730572° = 23°12'9″ = 0.40549612948 rad
Angle ∠ B = β = 56.84771120714° = 56°50'50″ = 0.99221692759 rad
Angle ∠ C = γ = 99.95503148714° = 99°57'1″ = 1.74444620829 rad

Height: ha = 16.74442854058
Height: hb = 7.88796637204
Height: hc = 6.69877141623

Median: ma = 18.12545689604
Median: mb = 12.63992246598
Median: mc = 8.74664278423

Inradius: r = 2.97767618499
Circumradius: R = 10.1532717532

Vertex coordinates: A[20; 0] B[0; 0] C[4.375; 6.69877141623]
Centroid: CG[8.125; 2.23325713874]
Coordinates of the circumscribed circle: U[10; -1.75443298677]
Coordinates of the inscribed circle: I[5.5; 2.97767618499]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.7977426943° = 156°47'51″ = 0.40549612948 rad
∠ B' = β' = 123.1532887929° = 123°9'10″ = 0.99221692759 rad
∠ C' = γ' = 80.05496851286° = 80°2'59″ = 1.74444620829 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 = 8 ; ; b = 17 ; ; c = 20 ; ;

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

p = a+b+c = 8+17+20 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-8)(22.5-17)(22.5-20) } ; ; T = sqrt{ 4485.94 } = 66.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 66.98 }{ 8 } = 16.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 66.98 }{ 17 } = 7.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 66.98 }{ 20 } = 6.7 ; ;

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{ 8**2-17**2-20**2 }{ 2 * 17 * 20 } ) = 23° 12'9" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-8**2-20**2 }{ 2 * 8 * 20 } ) = 56° 50'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-8**2-17**2 }{ 2 * 17 * 8 } ) = 99° 57'1" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 66.98 }{ 22.5 } = 2.98 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 23° 12'9" } = 10.15 ; ;




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