8 21 25 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 21   c = 25

Area: T = 78.46601809837
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 17.3911302046° = 17°23'29″ = 0.30435354819 rad
Angle ∠ B = β = 51.68438655263° = 51°41'2″ = 0.90220536236 rad
Angle ∠ C = γ = 110.9254832428° = 110°55'29″ = 1.93660035481 rad

Height: ha = 19.61550452459
Height: hb = 7.47223981889
Height: hc = 6.27768144787

Median: ma = 22.73876340018
Median: mb = 15.3055227865
Median: mc = 9.81107084352

Inradius: r = 2.9065932629
Circumradius: R = 13.38325844758

Vertex coordinates: A[25; 0] B[0; 0] C[4.96; 6.27768144787]
Centroid: CG[9.98766666667; 2.09222714929]
Coordinates of the circumscribed circle: U[12.5; -4.77994944556]
Coordinates of the inscribed circle: I[6; 2.9065932629]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.6098697954° = 162°36'31″ = 0.30435354819 rad
∠ B' = β' = 128.3166134474° = 128°18'58″ = 0.90220536236 rad
∠ C' = γ' = 69.07551675724° = 69°4'31″ = 1.93660035481 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 = 25 ; ;

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

p = a+b+c = 8+21+25 = 54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54 }{ 2 } = 27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27 * (27-8)(27-21)(27-25) } ; ; T = sqrt{ 6156 } = 78.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 78.46 }{ 8 } = 19.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 78.46 }{ 21 } = 7.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 78.46 }{ 25 } = 6.28 ; ;

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**2-25**2 }{ 2 * 21 * 25 } ) = 17° 23'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-8**2-25**2 }{ 2 * 8 * 25 } ) = 51° 41'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-8**2-21**2 }{ 2 * 21 * 8 } ) = 110° 55'29" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 78.46 }{ 27 } = 2.91 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 17° 23'29" } = 13.38 ; ;




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