13 22 23 triangle

Acute scalene triangle.

Sides: a = 13   b = 22   c = 23

Area: T = 139.5999426933
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 33.48989663699° = 33°29'20″ = 0.58444927262 rad
Angle ∠ B = β = 69.03111994008° = 69°1'52″ = 1.20548217161 rad
Angle ∠ C = γ = 77.48798342294° = 77°28'47″ = 1.35222782112 rad

Height: ha = 21.47768349128
Height: hb = 12.69108569939
Height: hc = 12.13990806029

Median: ma = 21.54664614264
Median: mb = 15.10996688705
Median: mc = 13.93773598648

Inradius: r = 4.81437733425
Circumradius: R = 11.78801343181

Vertex coordinates: A[23; 0] B[0; 0] C[4.6522173913; 12.13990806029]
Centroid: CG[9.21773913043; 4.0466360201]
Coordinates of the circumscribed circle: U[11.5; 2.55437354116]
Coordinates of the inscribed circle: I[7; 4.81437733425]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.511103363° = 146°30'40″ = 0.58444927262 rad
∠ B' = β' = 110.9698800599° = 110°58'8″ = 1.20548217161 rad
∠ C' = γ' = 102.5220165771° = 102°31'13″ = 1.35222782112 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 = 13 ; ; b = 22 ; ; c = 23 ; ;

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

p = a+b+c = 13+22+23 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-13)(29-22)(29-23) } ; ; T = sqrt{ 19488 } = 139.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 139.6 }{ 13 } = 21.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 139.6 }{ 22 } = 12.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 139.6 }{ 23 } = 12.14 ; ;

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{ 13**2-22**2-23**2 }{ 2 * 22 * 23 } ) = 33° 29'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-13**2-23**2 }{ 2 * 13 * 23 } ) = 69° 1'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-13**2-22**2 }{ 2 * 22 * 13 } ) = 77° 28'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 139.6 }{ 29 } = 4.81 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 33° 29'20" } = 11.78 ; ;




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