8 23 24 triangle

Acute scalene triangle.

Sides: a = 8   b = 23   c = 24

Area: T = 91.90217818108
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 19.45495958011° = 19°26'59″ = 0.33994594849 rad
Angle ∠ B = β = 73.19881624066° = 73°11'53″ = 1.27875489404 rad
Angle ∠ C = γ = 87.35222417923° = 87°21'8″ = 1.52545842283 rad

Height: ha = 22.97554454527
Height: hb = 7.99114592879
Height: hc = 7.65884818176

Median: ma = 23.16224696438
Median: mb = 13.7022189606
Median: mc = 12.34990890352

Inradius: r = 3.34218829749
Circumradius: R = 12.01328247597

Vertex coordinates: A[24; 0] B[0; 0] C[2.31325; 7.65884818176]
Centroid: CG[8.77108333333; 2.55328272725]
Coordinates of the circumscribed circle: U[12; 0.55549402742]
Coordinates of the inscribed circle: I[4.5; 3.34218829749]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.5550404199° = 160°33'1″ = 0.33994594849 rad
∠ B' = β' = 106.8021837593° = 106°48'7″ = 1.27875489404 rad
∠ C' = γ' = 92.64877582077° = 92°38'52″ = 1.52545842283 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 = 8 ; ; b = 23 ; ; c = 24 ; ;

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

p = a+b+c = 8+23+24 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-8)(27.5-23)(27.5-24) } ; ; T = sqrt{ 8445.94 } = 91.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 91.9 }{ 8 } = 22.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 91.9 }{ 23 } = 7.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 91.9 }{ 24 } = 7.66 ; ;

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{ 8**2-23**2-24**2 }{ 2 * 23 * 24 } ) = 19° 26'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-8**2-24**2 }{ 2 * 8 * 24 } ) = 73° 11'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-8**2-23**2 }{ 2 * 23 * 8 } ) = 87° 21'8" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 91.9 }{ 27.5 } = 3.34 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 19° 26'59" } = 12.01 ; ;




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