4 23 24 triangle

Obtuse scalene triangle.

Sides: a = 4   b = 23   c = 24

Area: T = 45.34224470006
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 9.4565656367° = 9°27'20″ = 0.16550323365 rad
Angle ∠ B = β = 70.84549899264° = 70°50'42″ = 1.23664783328 rad
Angle ∠ C = γ = 99.69993537065° = 99°41'58″ = 1.74400819843 rad

Height: ha = 22.67112235003
Height: hb = 3.94328214783
Height: hc = 3.779853725

Median: ma = 23.42200768573
Median: mb = 12.79664838921
Median: mc = 11.33657840488

Inradius: r = 1.77881351765
Circumradius: R = 12.17440231619

Vertex coordinates: A[24; 0] B[0; 0] C[1.31325; 3.779853725]
Centroid: CG[8.43875; 1.26595124167]
Coordinates of the circumscribed circle: U[12; -2.05110582501]
Coordinates of the inscribed circle: I[2.5; 1.77881351765]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170.5444343633° = 170°32'40″ = 0.16550323365 rad
∠ B' = β' = 109.1555010074° = 109°9'18″ = 1.23664783328 rad
∠ C' = γ' = 80.30106462935° = 80°18'2″ = 1.74400819843 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 = 4 ; ; b = 23 ; ; c = 24 ; ;

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

p = a+b+c = 4+23+24 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-4)(25.5-23)(25.5-24) } ; ; T = sqrt{ 2055.94 } = 45.34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 45.34 }{ 4 } = 22.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 45.34 }{ 23 } = 3.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 45.34 }{ 24 } = 3.78 ; ;

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{ 4**2-23**2-24**2 }{ 2 * 23 * 24 } ) = 9° 27'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-4**2-24**2 }{ 2 * 4 * 24 } ) = 70° 50'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-4**2-23**2 }{ 2 * 23 * 4 } ) = 99° 41'58" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 45.34 }{ 25.5 } = 1.78 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4 }{ 2 * sin 9° 27'20" } = 12.17 ; ;




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