9 20 22 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 20   c = 22

Area: T = 89.99768749457
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 24.14768479965° = 24°8'49″ = 0.42114420015 rad
Angle ∠ B = β = 65.37656816478° = 65°22'32″ = 1.14110208955 rad
Angle ∠ C = γ = 90.47774703557° = 90°28'39″ = 1.57991297566 rad

Height: ha = 19.99993055435
Height: hb = 98.9996874946
Height: hc = 8.1821534086

Median: ma = 20.53765527779
Median: mb = 13.50992560861
Median: mc = 10.93216055545

Inradius: r = 3.52992892136
Circumradius: R = 111.0003819643

Vertex coordinates: A[22; 0] B[0; 0] C[3.75; 8.1821534086]
Centroid: CG[8.58333333333; 2.72771780287]
Coordinates of the circumscribed circle: U[11; -0.09216698497]
Coordinates of the inscribed circle: I[5.5; 3.52992892136]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.8533152003° = 155°51'11″ = 0.42114420015 rad
∠ B' = β' = 114.6244318352° = 114°37'28″ = 1.14110208955 rad
∠ C' = γ' = 89.52325296443° = 89°31'21″ = 1.57991297566 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 = 20 ; ; c = 22 ; ;

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

p = a+b+c = 9+20+22 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-9)(25.5-20)(25.5-22) } ; ; T = sqrt{ 8099.44 } = 90 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 90 }{ 9 } = 20 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 90 }{ 20 } = 9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 90 }{ 22 } = 8.18 ; ;

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**2-20**2-22**2 }{ 2 * 20 * 22 } ) = 24° 8'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-9**2-22**2 }{ 2 * 9 * 22 } ) = 65° 22'32" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-9**2-20**2 }{ 2 * 20 * 9 } ) = 90° 28'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 90 }{ 25.5 } = 3.53 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 24° 8'49" } = 11 ; ;




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