20 21 25 triangle

Acute scalene triangle.

Sides: a = 20   b = 21   c = 25

Area: T = 202.9388414303
Perimeter: p = 66
Semiperimeter: s = 33

Angle ∠ A = α = 50.63329726582° = 50°37'59″ = 0.8843712083 rad
Angle ∠ B = β = 54.26876240296° = 54°16'3″ = 0.94771487166 rad
Angle ∠ C = γ = 75.09994033122° = 75°5'58″ = 1.31107318541 rad

Height: ha = 20.29438414303
Height: hb = 19.32774680289
Height: hc = 16.23550731443

Median: ma = 20.80986520467
Median: mb = 20.05661711201
Median: mc = 16.2565768207

Inradius: r = 6.15496489183
Circumradius: R = 12.93549586623

Vertex coordinates: A[25; 0] B[0; 0] C[11.68; 16.23550731443]
Centroid: CG[12.22766666667; 5.41216910481]
Coordinates of the circumscribed circle: U[12.5; 3.32661322274]
Coordinates of the inscribed circle: I[12; 6.15496489183]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.3677027342° = 129°22'1″ = 0.8843712083 rad
∠ B' = β' = 125.732237597° = 125°43'57″ = 0.94771487166 rad
∠ C' = γ' = 104.9010596688° = 104°54'2″ = 1.31107318541 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 = 20 ; ; b = 21 ; ; c = 25 ; ;

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

p = a+b+c = 20+21+25 = 66 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66 }{ 2 } = 33 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33 * (33-20)(33-21)(33-25) } ; ; T = sqrt{ 41184 } = 202.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 202.94 }{ 20 } = 20.29 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 202.94 }{ 21 } = 19.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 202.94 }{ 25 } = 16.24 ; ;

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{ 20**2-21**2-25**2 }{ 2 * 21 * 25 } ) = 50° 37'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-20**2-25**2 }{ 2 * 20 * 25 } ) = 54° 16'3" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-20**2-21**2 }{ 2 * 21 * 20 } ) = 75° 5'58" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 202.94 }{ 33 } = 6.15 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 50° 37'59" } = 12.93 ; ;




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