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: h_a = 22.96330981193
Height: h_b = 22.00663023643
Height: h_c = 18.86325448837

Median: m_a = 23.40440594769
Median: m_b = 22.63884628453
Median: m_c = 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

Calculate another triangle

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

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

6. Inradius

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

7. Circumradius

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

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