9 20 24 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 20   c = 24

Area: T = 86.81097776751
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 21.20550908133° = 21°12'18″ = 0.37700986529 rad
Angle ∠ B = β = 53.4943968335° = 53°29'38″ = 0.93436458774 rad
Angle ∠ C = γ = 105.3010940852° = 105°18'3″ = 1.83878481233 rad

Height: ha = 19.29110617056
Height: hb = 8.68109777675
Height: hc = 7.23441481396

Median: ma = 21.62875287539
Median: mb = 15.11662164578
Median: mc = 9.82334413522

Inradius: r = 3.2765840667
Circumradius: R = 12.44109948847

Vertex coordinates: A[24; 0] B[0; 0] C[5.35441666667; 7.23441481396]
Centroid: CG[9.78547222222; 2.41113827132]
Coordinates of the circumscribed circle: U[12; -3.28330403168]
Coordinates of the inscribed circle: I[6.5; 3.2765840667]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.7954909187° = 158°47'42″ = 0.37700986529 rad
∠ B' = β' = 126.5066031665° = 126°30'22″ = 0.93436458774 rad
∠ C' = γ' = 74.69990591483° = 74°41'57″ = 1.83878481233 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 = 9 ; ; b = 20 ; ; c = 24 ; ;

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

p = a+b+c = 9+20+24 = 53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53 }{ 2 } = 26.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.5 * (26.5-9)(26.5-20)(26.5-24) } ; ; T = sqrt{ 7535.94 } = 86.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 86.81 }{ 9 } = 19.29 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 86.81 }{ 20 } = 8.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 86.81 }{ 24 } = 7.23 ; ;

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{ 9**2-20**2-24**2 }{ 2 * 20 * 24 } ) = 21° 12'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-9**2-24**2 }{ 2 * 9 * 24 } ) = 53° 29'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-9**2-20**2 }{ 2 * 20 * 9 } ) = 105° 18'3" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 86.81 }{ 26.5 } = 3.28 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 21° 12'18" } = 12.44 ; ;




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