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: h_a = 17.889854382
Height: h_b = 4.25991771
Height: h_c = 3.72767799625

Median: m_a = 22.4110934831
Median: m_b = 13.79331142241
Median: m_c = 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

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.

$a = 5 ; ; b = 21 ; ; c = 24 ; ;$

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

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

6. Inradius

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

7. Circumradius

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

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