8 21 24 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 21   c = 24

Area: T = 82.10332124828
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 19.01444375634° = 19°52″ = 0.33218645409 rad
Angle ∠ B = β = 58.78664577123° = 58°47'11″ = 1.02660172427 rad
Angle ∠ C = γ = 102.1999104724° = 102°11'57″ = 1.784371087 rad

Height: ha = 20.52658031207
Height: hb = 7.81993535698
Height: hc = 6.84219343736

Median: ma = 22.19223410212
Median: mb = 14.4832748358
Median: mc = 10.4166333328

Inradius: r = 3.09882344333
Circumradius: R = 12.27772297151

Vertex coordinates: A[24; 0] B[0; 0] C[4.14658333333; 6.84219343736]
Centroid: CG[9.38219444444; 2.28106447912]
Coordinates of the circumscribed circle: U[12; -2.59442955648]
Coordinates of the inscribed circle: I[5.5; 3.09882344333]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.9865562437° = 160°59'8″ = 0.33218645409 rad
∠ B' = β' = 121.2143542288° = 121°12'49″ = 1.02660172427 rad
∠ C' = γ' = 77.80108952757° = 77°48'3″ = 1.784371087 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 = 8 ; ; b = 21 ; ; c = 24 ; ;

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

p = a+b+c = 8+21+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 * (26.5-8)(26.5-21)(26.5-24) } ; ; T = sqrt{ 6740.94 } = 82.1 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 82.1 }{ 8 } = 20.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 82.1 }{ 21 } = 7.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 82.1 }{ 24 } = 6.84 ; ;

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{ 8**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 19° 52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-8**2-24**2 }{ 2 * 8 * 24 } ) = 58° 47'11" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-8**2-21**2 }{ 2 * 21 * 8 } ) = 102° 11'57" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 82.1 }{ 26.5 } = 3.1 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 19° 52" } = 12.28 ; ;




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