9 21 25 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 21   c = 25

Area: T = 90.92440754696
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 20.26659019313° = 20°15'57″ = 0.35437067146 rad
Angle ∠ B = β = 53.92218000282° = 53°55'18″ = 0.94111129491 rad
Angle ∠ C = γ = 105.8122298041° = 105°48'44″ = 1.84767729899 rad

Height: ha = 20.20553501044
Height: hb = 8.6599435759
Height: hc = 7.27439260376

Median: ma = 22.64439837484
Median: mb = 15.58804364509
Median: mc = 10.23547447452

Inradius: r = 3.30663300171
Circumradius: R = 12.9921608591

Vertex coordinates: A[25; 0] B[0; 0] C[5.3; 7.27439260376]
Centroid: CG[10.1; 2.42546420125]
Coordinates of the circumscribed circle: U[12.5; -3.54400414944]
Coordinates of the inscribed circle: I[6.5; 3.30663300171]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.7344098069° = 159°44'3″ = 0.35437067146 rad
∠ B' = β' = 126.0788199972° = 126°4'42″ = 0.94111129491 rad
∠ C' = γ' = 74.18877019595° = 74°11'16″ = 1.84767729899 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 = 21 ; ; c = 25 ; ;

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

p = a+b+c = 9+21+25 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-9)(27.5-21)(27.5-25) } ; ; T = sqrt{ 8267.19 } = 90.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 90.92 }{ 9 } = 20.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 90.92 }{ 21 } = 8.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 90.92 }{ 25 } = 7.27 ; ;

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**2-21**2-25**2 }{ 2 * 21 * 25 } ) = 20° 15'57" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-9**2-25**2 }{ 2 * 9 * 25 } ) = 53° 55'18" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 25**2-9**2-21**2 }{ 2 * 21 * 9 } ) = 105° 48'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 90.92 }{ 27.5 } = 3.31 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 20° 15'57" } = 12.99 ; ;




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