23 24 29 triangle

Acute scalene triangle.

Sides: a = 23   b = 24   c = 29

Area: T = 267.9932537209
Perimeter: p = 76
Semiperimeter: s = 38

Angle ∠ A = α = 50.36222856093° = 50°21'44″ = 0.87989877027 rad
Angle ∠ B = β = 53.47329380047° = 53°28'23″ = 0.93332788289 rad
Angle ∠ C = γ = 76.1654776386° = 76°9'53″ = 1.3299326122 rad

Height: ha = 23.30436988878
Height: hb = 22.33327114341
Height: hc = 18.48222439455

Median: ma = 24.00552077683
Median: mb = 23.25994066992
Median: mc = 18.5

Inradius: r = 7.05224351897
Circumradius: R = 14.93332516557

Vertex coordinates: A[29; 0] B[0; 0] C[13.69896551724; 18.48222439455]
Centroid: CG[14.23298850575; 6.16107479818]
Coordinates of the circumscribed circle: U[14.5; 3.57109949611]
Coordinates of the inscribed circle: I[14; 7.05224351897]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.6387714391° = 129°38'16″ = 0.87989877027 rad
∠ B' = β' = 126.5277061995° = 126°31'37″ = 0.93332788289 rad
∠ C' = γ' = 103.8355223614° = 103°50'7″ = 1.3299326122 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 = 23 ; ; b = 24 ; ; c = 29 ; ;

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

p = a+b+c = 23+24+29 = 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 * (38-23)(38-24)(38-29) } ; ; T = sqrt{ 71820 } = 267.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 267.99 }{ 23 } = 23.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 267.99 }{ 24 } = 22.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 267.99 }{ 29 } = 18.48 ; ;

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{ 23**2-24**2-29**2 }{ 2 * 24 * 29 } ) = 50° 21'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-23**2-29**2 }{ 2 * 23 * 29 } ) = 53° 28'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-23**2-24**2 }{ 2 * 24 * 23 } ) = 76° 9'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 267.99 }{ 38 } = 7.05 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 50° 21'44" } = 14.93 ; ;




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