5 21 24 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 21   c = 24

Area: T = 44.721135955
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 10.22221793906° = 10°13'20″ = 0.17884106871 rad
Angle ∠ B = β = 48.19896851042° = 48°11'23″ = 0.84110686706 rad
Angle ∠ C = γ = 121.5888135505° = 121°35'17″ = 2.12221132959 rad

Height: ha = 17.889854382
Height: hb = 4.25991771
Height: hc = 3.72767799625

Median: ma = 22.4110934831
Median: mb = 13.79331142241
Median: mc = 9.43439811321

Inradius: r = 1.7898854382
Circumradius: R = 14.08772282582

Vertex coordinates: A[24; 0] B[0; 0] C[3.33333333333; 3.72767799625]
Centroid: CG[9.11111111111; 1.24222599875]
Coordinates of the circumscribed circle: U[12; -7.37990243257]
Coordinates of the inscribed circle: I[4; 1.7898854382]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 169.7787820609° = 169°46'40″ = 0.17884106871 rad
∠ B' = β' = 131.8110314896° = 131°48'37″ = 0.84110686706 rad
∠ C' = γ' = 58.41218644948° = 58°24'43″ = 2.12221132959 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 = 5 ; ; b = 21 ; ; c = 24 ; ;

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

p = a+b+c = 5+21+24 = 50 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 50 }{ 2 } = 25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25 * (25-5)(25-21)(25-24) } ; ; T = sqrt{ 2000 } = 44.72 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 44.72 }{ 5 } = 17.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 44.72 }{ 21 } = 4.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 44.72 }{ 24 } = 3.73 ; ;

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{ 21**2+24**2-5**2 }{ 2 * 21 * 24 } ) = 10° 13'20" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5**2+24**2-21**2 }{ 2 * 5 * 24 } ) = 48° 11'23" ; ; gamma = 180° - alpha - beta = 180° - 10° 13'20" - 48° 11'23" = 121° 35'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 44.72 }{ 25 } = 1.79 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 10° 13'20" } = 14.09 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 24**2 - 5**2 } }{ 2 } = 22.411 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 5**2 - 21**2 } }{ 2 } = 13.793 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 5**2 - 24**2 } }{ 2 } = 9.434 ; ;
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