24 25 27 triangle

Acute scalene triangle.

Sides: a = 24   b = 25   c = 27

Area: T = 275.8198781086
Perimeter: p = 76
Semiperimeter: s = 38

Angle ∠ A = α = 54.81095367507° = 54°48'34″ = 0.95766068778 rad
Angle ∠ B = β = 58.35325320481° = 58°21'9″ = 1.01884438111 rad
Angle ∠ C = γ = 66.83879312011° = 66°50'17″ = 1.16765419647 rad

Height: ha = 22.98548984239
Height: hb = 22.06655024869
Height: hc = 20.43110208212

Median: ma = 23.08767927612
Median: mb = 22.27766694099
Median: mc = 20.45111613362

Inradius: r = 7.2588388976
Circumradius: R = 14.68435541222

Vertex coordinates: A[27; 0] B[0; 0] C[12.59325925926; 20.43110208212]
Centroid: CG[13.19875308642; 6.81103402737]
Coordinates of the circumscribed circle: U[13.5; 5.77655312881]
Coordinates of the inscribed circle: I[13; 7.2588388976]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.1990463249° = 125°11'26″ = 0.95766068778 rad
∠ B' = β' = 121.6477467952° = 121°38'51″ = 1.01884438111 rad
∠ C' = γ' = 113.1622068799° = 113°9'43″ = 1.16765419647 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 = 25 ; ; c = 27 ; ;

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

p = a+b+c = 24+25+27 = 76 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76 }{ 2 } = 38 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38 * (38-24)(38-25)(38-27) } ; ; T = sqrt{ 76076 } = 275.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 275.82 }{ 24 } = 22.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 275.82 }{ 25 } = 22.07 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 275.82 }{ 27 } = 20.43 ; ;

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{ 25**2+27**2-24**2 }{ 2 * 25 * 27 } ) = 54° 48'34" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 24**2+27**2-25**2 }{ 2 * 24 * 27 } ) = 58° 21'9" ; ; gamma = 180° - alpha - beta = 180° - 54° 48'34" - 58° 21'9" = 66° 50'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 275.82 }{ 38 } = 7.26 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 24 }{ 2 * sin 54° 48'34" } = 14.68 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 25**2+2 * 27**2 - 24**2 } }{ 2 } = 23.087 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 24**2 - 25**2 } }{ 2 } = 22.277 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 25**2+2 * 24**2 - 27**2 } }{ 2 } = 20.451 ; ;
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