24 27 28 triangle

Acute scalene triangle.

Sides: a = 24   b = 27   c = 28

Area: T = 296.6666374063
Perimeter: p = 79
Semiperimeter: s = 39.5

Angle ∠ A = α = 51.70551131611° = 51°42'18″ = 0.90224244648 rad
Angle ∠ B = β = 61.99985409159° = 61°59'55″ = 1.08220786704 rad
Angle ∠ C = γ = 66.29663459229° = 66°17'47″ = 1.15770895184 rad

Height: ha = 24.72221978386
Height: hb = 21.97552869677
Height: hc = 21.19904552903

Median: ma = 24.74987373415
Median: mb = 22.31103115173
Median: mc = 21.36658606192

Inradius: r = 7.51105411155
Circumradius: R = 15.29899027209

Vertex coordinates: A[28; 0] B[0; 0] C[11.26878571429; 21.19904552903]
Centroid: CG[13.08992857143; 7.06334850968]
Coordinates of the circumscribed circle: U[14; 6.14766352759]
Coordinates of the inscribed circle: I[12.5; 7.51105411155]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.2954886839° = 128°17'42″ = 0.90224244648 rad
∠ B' = β' = 118.0011459084° = 118°5″ = 1.08220786704 rad
∠ C' = γ' = 113.7043654077° = 113°42'13″ = 1.15770895184 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 = 24 ; ; b = 27 ; ; c = 28 ; ;

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

p = a+b+c = 24+27+28 = 79 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 79 }{ 2 } = 39.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.5 * (39.5-24)(39.5-27)(39.5-28) } ; ; T = sqrt{ 88010.94 } = 296.67 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 296.67 }{ 24 } = 24.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 296.67 }{ 27 } = 21.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 296.67 }{ 28 } = 21.19 ; ;

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{ 24**2-27**2-28**2 }{ 2 * 27 * 28 } ) = 51° 42'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-24**2-28**2 }{ 2 * 24 * 28 } ) = 61° 59'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-24**2-27**2 }{ 2 * 27 * 24 } ) = 66° 17'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 296.67 }{ 39.5 } = 7.51 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 24 }{ 2 * sin 51° 42'18" } = 15.29 ; ;




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