12 17 21 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 17   c = 21

Area: T = 101.9880390272
Perimeter: p = 50
Semiperimeter: s = 25

Angle ∠ A = α = 34.84222349066° = 34°50'32″ = 0.60881117179 rad
Angle ∠ B = β = 54.03442464357° = 54°2'3″ = 0.94330755091 rad
Angle ∠ C = γ = 91.12435186577° = 91°7'25″ = 1.59904054266 rad

Height: ha = 16.9976731712
Height: hb = 11.99876929732
Height: hc = 9.71224181211

Median: ma = 18.13883571472
Median: mb = 14.84108220797
Median: mc = 10.3087764064

Inradius: r = 4.07992156109
Circumradius: R = 10.50220190366

Vertex coordinates: A[21; 0] B[0; 0] C[7.04876190476; 9.71224181211]
Centroid: CG[9.34992063492; 3.2377472707]
Coordinates of the circumscribed circle: U[10.5; -0.20659219419]
Coordinates of the inscribed circle: I[8; 4.07992156109]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.1587765093° = 145°9'28″ = 0.60881117179 rad
∠ B' = β' = 125.9665753564° = 125°57'57″ = 0.94330755091 rad
∠ C' = γ' = 88.87664813423° = 88°52'35″ = 1.59904054266 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 = 12 ; ; b = 17 ; ; c = 21 ; ;

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

p = a+b+c = 12+17+21 = 50 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 50 }{ 2 } = 25 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25 * (25-12)(25-17)(25-21) } ; ; T = sqrt{ 10400 } = 101.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 101.98 }{ 12 } = 17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 101.98 }{ 17 } = 12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 101.98 }{ 21 } = 9.71 ; ;

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{ 12**2-17**2-21**2 }{ 2 * 17 * 21 } ) = 34° 50'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-12**2-21**2 }{ 2 * 12 * 21 } ) = 54° 2'3" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-12**2-17**2 }{ 2 * 17 * 12 } ) = 91° 7'25" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 101.98 }{ 25 } = 4.08 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 34° 50'32" } = 10.5 ; ;




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