22 27 30 triangle

Acute scalene triangle.

Sides: a = 22   b = 27   c = 30

Area: T = 286.5066435355
Perimeter: p = 79
Semiperimeter: s = 39.5

Angle ∠ A = α = 45.02656525491° = 45°1'32″ = 0.78658458848 rad
Angle ∠ B = β = 60.25502892152° = 60°15'1″ = 1.05215659221 rad
Angle ∠ C = γ = 74.72440582357° = 74°43'27″ = 1.30441808467 rad

Height: ha = 26.04660395778
Height: hb = 21.22326989152
Height: hc = 19.11004290237

Median: ma = 26.33443881645
Median: mb = 22.57876438098
Median: mc = 19.53220249846

Inradius: r = 7.25333274773
Circumradius: R = 15.54993889499

Vertex coordinates: A[30; 0] B[0; 0] C[10.91766666667; 19.11004290237]
Centroid: CG[13.63988888889; 6.36768096746]
Coordinates of the circumscribed circle: U[15; 4.09767666173]
Coordinates of the inscribed circle: I[12.5; 7.25333274773]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.9744347451° = 134°58'28″ = 0.78658458848 rad
∠ B' = β' = 119.7549710785° = 119°44'59″ = 1.05215659221 rad
∠ C' = γ' = 105.2765941764° = 105°16'33″ = 1.30441808467 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 = 22 ; ; b = 27 ; ; c = 30 ; ;

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

p = a+b+c = 22+27+30 = 79 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 79 }{ 2 } = 39.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.5 * (39.5-22)(39.5-27)(39.5-30) } ; ; T = sqrt{ 82085.94 } = 286.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 286.51 }{ 22 } = 26.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 286.51 }{ 27 } = 21.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 286.51 }{ 30 } = 19.1 ; ;

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{ 22**2-27**2-30**2 }{ 2 * 27 * 30 } ) = 45° 1'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-22**2-30**2 }{ 2 * 22 * 30 } ) = 60° 15'1" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-22**2-27**2 }{ 2 * 27 * 22 } ) = 74° 43'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 286.51 }{ 39.5 } = 7.25 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 22 }{ 2 * sin 45° 1'32" } = 15.55 ; ;




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