12 16 17 triangle

Acute scalene triangle.

Sides: a = 12   b = 16   c = 17

Area: T = 91.90217818108
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 42.51222944223° = 42°30'44″ = 0.74219795102 rad
Angle ∠ B = β = 64.29895431711° = 64°17'22″ = 1.12220642029 rad
Angle ∠ C = γ = 73.19881624066° = 73°11'53″ = 1.27875489404 rad

Height: ha = 15.31769636351
Height: hb = 11.48877227264
Height: hc = 10.81219743307

Median: ma = 15.37985564992
Median: mb = 12.34990890352
Median: mc = 11.30326545555

Inradius: r = 4.0854523636
Circumradius: R = 8.87990443876

Vertex coordinates: A[17; 0] B[0; 0] C[5.20658823529; 10.81219743307]
Centroid: CG[7.40219607843; 3.60439914436]
Coordinates of the circumscribed circle: U[8.5; 2.56765987683]
Coordinates of the inscribed circle: I[6.5; 4.0854523636]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.4887705578° = 137°29'16″ = 0.74219795102 rad
∠ B' = β' = 115.7110456829° = 115°42'38″ = 1.12220642029 rad
∠ C' = γ' = 106.8021837593° = 106°48'7″ = 1.27875489404 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 = 16 ; ; c = 17 ; ;

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

p = a+b+c = 12+16+17 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-12)(22.5-16)(22.5-17) } ; ; T = sqrt{ 8445.94 } = 91.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 91.9 }{ 12 } = 15.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 91.9 }{ 16 } = 11.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 91.9 }{ 17 } = 10.81 ; ;

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**2-16**2-17**2 }{ 2 * 16 * 17 } ) = 42° 30'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-12**2-17**2 }{ 2 * 12 * 17 } ) = 64° 17'22" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 17**2-12**2-16**2 }{ 2 * 16 * 12 } ) = 73° 11'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 91.9 }{ 22.5 } = 4.08 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 42° 30'44" } = 8.88 ; ;




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