21 24 27 triangle

Acute scalene triangle.

Sides: a = 21   b = 24   c = 27

Area: T = 241.495534157
Perimeter: p = 72
Semiperimeter: s = 36

Angle ∠ A = α = 48.19896851042° = 48°11'23″ = 0.84110686706 rad
Angle ∠ B = β = 58.41218644948° = 58°24'43″ = 1.01994793577 rad
Angle ∠ C = γ = 73.3988450401° = 73°23'54″ = 1.28110446254 rad

Height: ha = 232.99955634
Height: hb = 20.12546117975
Height: hc = 17.889854382

Median: ma = 23.28662620444
Median: mb = 21
Median: mc = 18.06223918682

Inradius: r = 6.70882039325
Circumradius: R = 14.08772282582

Vertex coordinates: A[27; 0] B[0; 0] C[11; 17.889854382]
Centroid: CG[12.66766666667; 5.963284794]
Coordinates of the circumscribed circle: U[13.5; 4.02549223595]
Coordinates of the inscribed circle: I[12; 6.70882039325]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.8110314896° = 131°48'37″ = 0.84110686706 rad
∠ B' = β' = 121.5888135505° = 121°35'17″ = 1.01994793577 rad
∠ C' = γ' = 106.6021549599° = 106°36'6″ = 1.28110446254 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 = 21 ; ; b = 24 ; ; c = 27 ; ;

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

p = a+b+c = 21+24+27 = 72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72 }{ 2 } = 36 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36 * (36-21)(36-24)(36-27) } ; ; T = sqrt{ 58320 } = 241.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 241.5 }{ 21 } = 23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 241.5 }{ 24 } = 20.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 241.5 }{ 27 } = 17.89 ; ;

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{ 21**2-24**2-27**2 }{ 2 * 24 * 27 } ) = 48° 11'23" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 58° 24'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-21**2-24**2 }{ 2 * 24 * 21 } ) = 73° 23'54" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 241.5 }{ 36 } = 6.71 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 48° 11'23" } = 14.09 ; ;




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