23 24 28 triangle

Acute scalene triangle.

Sides: a = 23   b = 24   c = 28

Area: T = 264.0765628372
Perimeter: p = 75
Semiperimeter: s = 37.5

Angle ∠ A = α = 51.8087642172° = 51°48'28″ = 0.90442139336 rad
Angle ∠ B = β = 55.09658714869° = 55°5'45″ = 0.96216043617 rad
Angle ∠ C = γ = 73.09664863411° = 73°5'47″ = 1.27657743583 rad

Height: ha = 22.96330981193
Height: hb = 22.00663023643
Height: hc = 18.86325448837

Median: ma = 23.40440594769
Median: mb = 22.63884628453
Median: mc = 18.88112075885

Inradius: r = 7.04220167566
Circumradius: R = 14.63221719419

Vertex coordinates: A[28; 0] B[0; 0] C[13.16107142857; 18.86325448837]
Centroid: CG[13.72202380952; 6.28875149612]
Coordinates of the circumscribed circle: U[14; 4.25444630375]
Coordinates of the inscribed circle: I[13.5; 7.04220167566]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.1922357828° = 128°11'32″ = 0.90442139336 rad
∠ B' = β' = 124.9044128513° = 124°54'15″ = 0.96216043617 rad
∠ C' = γ' = 106.9043513659° = 106°54'13″ = 1.27657743583 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 = 28 ; ;

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

p = a+b+c = 23+24+28 = 75 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 75 }{ 2 } = 37.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.5 * (37.5-23)(37.5-24)(37.5-28) } ; ; T = sqrt{ 69735.94 } = 264.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 264.08 }{ 23 } = 22.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 264.08 }{ 24 } = 22.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 264.08 }{ 28 } = 18.86 ; ;

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{ 24**2+28**2-23**2 }{ 2 * 24 * 28 } ) = 51° 48'28" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 23**2+28**2-24**2 }{ 2 * 23 * 28 } ) = 55° 5'45" ; ; gamma = 180° - alpha - beta = 180° - 51° 48'28" - 55° 5'45" = 73° 5'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 264.08 }{ 37.5 } = 7.04 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 23 }{ 2 * sin 51° 48'28" } = 14.63 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 28**2 - 23**2 } }{ 2 } = 23.404 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 23**2 - 24**2 } }{ 2 } = 22.638 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 23**2 - 28**2 } }{ 2 } = 18.881 ; ;
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