10 23 24 triangle

Acute scalene triangle.

Sides: a = 10   b = 23   c = 24

Area: T = 114.2344134566
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 24.45495467105° = 24°26'58″ = 0.42767250907 rad
Angle ∠ B = β = 72.16766172902° = 72°10' = 1.26595450817 rad
Angle ∠ C = γ = 83.38438359994° = 83°23'2″ = 1.45553224811 rad

Height: ha = 22.84768269132
Height: hb = 9.93334030057
Height: hc = 9.52195112138

Median: ma = 22.96773681557
Median: mb = 14.34439882878
Median: mc = 13.05875648572

Inradius: r = 4.00882152479
Circumradius: R = 12.08804521805

Vertex coordinates: A[24; 0] B[0; 0] C[3.06325; 9.52195112138]
Centroid: CG[9.02108333333; 3.17331704046]
Coordinates of the circumscribed circle: U[12; 1.3921878186]
Coordinates of the inscribed circle: I[5.5; 4.00882152479]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.555045329° = 155°33'2″ = 0.42767250907 rad
∠ B' = β' = 107.833338271° = 107°50' = 1.26595450817 rad
∠ C' = γ' = 96.61661640006° = 96°36'58″ = 1.45553224811 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 = 10 ; ; b = 23 ; ; c = 24 ; ;

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

p = a+b+c = 10+23+24 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-10)(28.5-23)(28.5-24) } ; ; T = sqrt{ 13049.44 } = 114.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 114.23 }{ 10 } = 22.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 114.23 }{ 23 } = 9.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 114.23 }{ 24 } = 9.52 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 23**2+24**2-10**2 }{ 2 * 23 * 24 } ) = 24° 26'58" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10**2+24**2-23**2 }{ 2 * 10 * 24 } ) = 72° 10' ; ; gamma = 180° - alpha - beta = 180° - 24° 26'58" - 72° 10' = 83° 23'2" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 114.23 }{ 28.5 } = 4.01 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10 }{ 2 * sin 24° 26'58" } = 12.08 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 24**2 - 10**2 } }{ 2 } = 22.967 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 10**2 - 23**2 } }{ 2 } = 14.344 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 10**2 - 24**2 } }{ 2 } = 13.058 ; ;
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