9 18 20 triangle

Acute scalene triangle.

Sides: a = 9   b = 18   c = 20

Area: T = 80.9990354364
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 26.7440245925° = 26°44'25″ = 0.46767053342 rad
Angle ∠ B = β = 64.14439833022° = 64°8'38″ = 1.1219523704 rad
Angle ∠ C = γ = 89.11657707729° = 89°6'57″ = 1.55553636154 rad

Height: ha = 17.99878565253
Height: hb = 8.99989282627
Height: hc = 8.09990354364

Median: ma = 18.48664815473
Median: mb = 12.62993309403
Median: mc = 10.12442283657

Inradius: r = 3.4466398058
Circumradius: R = 10.00111909611

Vertex coordinates: A[20; 0] B[0; 0] C[3.925; 8.09990354364]
Centroid: CG[7.975; 2.76996784788]
Coordinates of the circumscribed circle: U[10; 0.15443393667]
Coordinates of the inscribed circle: I[5.5; 3.4466398058]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.2659754075° = 153°15'35″ = 0.46767053342 rad
∠ B' = β' = 115.8566016698° = 115°51'22″ = 1.1219523704 rad
∠ C' = γ' = 90.88442292271° = 90°53'3″ = 1.55553636154 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 = 9 ; ; b = 18 ; ; c = 20 ; ;

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

p = a+b+c = 9+18+20 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-9)(23.5-18)(23.5-20) } ; ; T = sqrt{ 6559.44 } = 80.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 80.99 }{ 9 } = 18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 80.99 }{ 18 } = 9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 80.99 }{ 20 } = 8.1 ; ;

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{ 9**2-18**2-20**2 }{ 2 * 18 * 20 } ) = 26° 44'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-9**2-20**2 }{ 2 * 9 * 20 } ) = 64° 8'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-9**2-18**2 }{ 2 * 18 * 9 } ) = 89° 6'57" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 80.99 }{ 23.5 } = 3.45 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 26° 44'25" } = 10 ; ;




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