13 22 24 triangle

Acute scalene triangle.

Sides: a = 13   b = 22   c = 24

Area: T = 141.6988403308
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 32.46217449672° = 32°27'42″ = 0.56765643306 rad
Angle ∠ B = β = 65.27546350727° = 65°16'29″ = 1.13992573001 rad
Angle ∠ C = γ = 82.264361996° = 82°15'49″ = 1.43657710229 rad

Height: ha = 21.87997543551
Height: hb = 12.8821673028
Height: hc = 11.80882002757

Median: ma = 22.08550628254
Median: mb = 15.85987515272
Median: mc = 13.50992560861

Inradius: r = 4.80333357054
Circumradius: R = 12.11102282026

Vertex coordinates: A[24; 0] B[0; 0] C[5.43875; 11.80882002757]
Centroid: CG[9.81325; 3.93660667586]
Coordinates of the circumscribed circle: U[12; 1.63302230273]
Coordinates of the inscribed circle: I[7.5; 4.80333357054]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.5388255033° = 147°32'18″ = 0.56765643306 rad
∠ B' = β' = 114.7255364927° = 114°43'31″ = 1.13992573001 rad
∠ C' = γ' = 97.736638004° = 97°44'11″ = 1.43657710229 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 = 24 ; ;

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

p = a+b+c = 13+22+24 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-13)(29.5-22)(29.5-24) } ; ; T = sqrt{ 20078.44 } = 141.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 141.7 }{ 13 } = 21.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 141.7 }{ 22 } = 12.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 141.7 }{ 24 } = 11.81 ; ;

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**2-24**2 }{ 2 * 22 * 24 } ) = 32° 27'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-13**2-24**2 }{ 2 * 13 * 24 } ) = 65° 16'29" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-13**2-22**2 }{ 2 * 22 * 13 } ) = 82° 15'49" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 141.7 }{ 29.5 } = 4.8 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 32° 27'42" } = 12.11 ; ;




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