22 24 30 triangle

Acute scalene triangle.

Sides: a = 22   b = 24   c = 30

Area: T = 260.9522102885
Perimeter: p = 76
Semiperimeter: s = 38

Angle ∠ A = α = 46.45877809718° = 46°27'28″ = 0.81108412411 rad
Angle ∠ B = β = 52.25769610468° = 52°15'25″ = 0.91220560274 rad
Angle ∠ C = γ = 81.28552579814° = 81°17'7″ = 1.41986953851 rad

Height: ha = 23.72329184441
Height: hb = 21.74660085737
Height: hc = 17.3976806859

Median: ma = 24.83994846967
Median: mb = 23.40993998214
Median: mc = 17.46442491966

Inradius: r = 6.86771606022
Circumradius: R = 15.17551986523

Vertex coordinates: A[30; 0] B[0; 0] C[13.46766666667; 17.3976806859]
Centroid: CG[14.48988888889; 5.79989356197]
Coordinates of the circumscribed circle: U[15; 2.29992725231]
Coordinates of the inscribed circle: I[14; 6.86771606022]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.5422219028° = 133°32'32″ = 0.81108412411 rad
∠ B' = β' = 127.7433038953° = 127°44'35″ = 0.91220560274 rad
∠ C' = γ' = 98.71547420186° = 98°42'53″ = 1.41986953851 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 = 24 ; ; c = 30 ; ;

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

p = a+b+c = 22+24+30 = 76 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76 }{ 2 } = 38 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38 * (38-22)(38-24)(38-30) } ; ; T = sqrt{ 68096 } = 260.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 260.95 }{ 22 } = 23.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 260.95 }{ 24 } = 21.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 260.95 }{ 30 } = 17.4 ; ;

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**2-24**2-30**2 }{ 2 * 24 * 30 } ) = 46° 27'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-22**2-30**2 }{ 2 * 22 * 30 } ) = 52° 15'25" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-22**2-24**2 }{ 2 * 24 * 22 } ) = 81° 17'7" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 260.95 }{ 38 } = 6.87 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 22 }{ 2 * sin 46° 27'28" } = 15.18 ; ;




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