22 23 24 triangle

Acute scalene triangle.

Sides: a = 22   b = 23   c = 24

Area: T = 228.1966050579
Perimeter: p = 69
Semiperimeter: s = 34.5

Angle ∠ A = α = 55.77111336722° = 55°46'16″ = 0.97333899101 rad
Angle ∠ B = β = 59.81218683052° = 59°48'43″ = 1.04439140337 rad
Angle ∠ C = γ = 64.41769980226° = 64°25'1″ = 1.12442887097 rad

Height: ha = 20.74550955072
Height: hb = 19.8433134833
Height: hc = 19.01663375483

Median: ma = 20.77325780778
Median: mb = 19.94436706752
Median: mc = 19.03994327647

Inradius: r = 6.61443782777
Circumradius: R = 13.30443494499

Vertex coordinates: A[24; 0] B[0; 0] C[11.06325; 19.01663375483]
Centroid: CG[11.68875; 6.33987791828]
Coordinates of the circumscribed circle: U[12; 5.74550599897]
Coordinates of the inscribed circle: I[11.5; 6.61443782777]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.2298866328° = 124°13'44″ = 0.97333899101 rad
∠ B' = β' = 120.1888131695° = 120°11'17″ = 1.04439140337 rad
∠ C' = γ' = 115.5833001977° = 115°34'59″ = 1.12442887097 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 = 23 ; ; c = 24 ; ;

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

p = a+b+c = 22+23+24 = 69 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 69 }{ 2 } = 34.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.5 * (34.5-22)(34.5-23)(34.5-24) } ; ; T = sqrt{ 52073.44 } = 228.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 228.2 }{ 22 } = 20.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 228.2 }{ 23 } = 19.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 228.2 }{ 24 } = 19.02 ; ;

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**2-23**2-24**2 }{ 2 * 23 * 24 } ) = 55° 46'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-22**2-24**2 }{ 2 * 22 * 24 } ) = 59° 48'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-22**2-23**2 }{ 2 * 23 * 22 } ) = 64° 25'1" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 228.2 }{ 34.5 } = 6.61 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 22 }{ 2 * sin 55° 46'16" } = 13.3 ; ;




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