12 22 30 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 22   c = 30

Area: T = 113.137708499
Perimeter: p = 64
Semiperimeter: s = 32

Angle ∠ A = α = 20.05499757242° = 20°3' = 0.35499380913 rad
Angle ∠ B = β = 38.9422441269° = 38°56'33″ = 0.68796738189 rad
Angle ∠ C = γ = 121.0087583007° = 121°27″ = 2.11219807434 rad

Height: ha = 18.85661808316
Height: hb = 10.28551895445
Height: hc = 7.54224723327

Median: ma = 25.61224969497
Median: mb = 20.02549843945
Median: mc = 9.43439811321

Inradius: r = 3.53655339059
Circumradius: R = 17.50108928344

Vertex coordinates: A[30; 0] B[0; 0] C[9.33333333333; 7.54224723327]
Centroid: CG[13.11111111111; 2.51441574442]
Coordinates of the circumscribed circle: U[15; -9.01656114601]
Coordinates of the inscribed circle: I[10; 3.53655339059]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.9550024276° = 159°57' = 0.35499380913 rad
∠ B' = β' = 141.0587558731° = 141°3'27″ = 0.68796738189 rad
∠ C' = γ' = 58.99224169931° = 58°59'33″ = 2.11219807434 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 = 22 ; ; c = 30 ; ;

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

p = a+b+c = 12+22+30 = 64 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 64 }{ 2 } = 32 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32 * (32-12)(32-22)(32-30) } ; ; T = sqrt{ 12800 } = 113.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 113.14 }{ 12 } = 18.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 113.14 }{ 22 } = 10.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 113.14 }{ 30 } = 7.54 ; ;

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**2-22**2-30**2 }{ 2 * 22 * 30 } ) = 20° 3' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-12**2-30**2 }{ 2 * 12 * 30 } ) = 38° 56'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-12**2-22**2 }{ 2 * 22 * 12 } ) = 121° 27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 113.14 }{ 32 } = 3.54 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 20° 3' } = 17.5 ; ;




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