12 13 16 triangle

Acute scalene triangle.

Sides: a = 12   b = 13   c = 16

Area: T = 76.68772707821
Perimeter: p = 41
Semiperimeter: s = 20.5

Angle ∠ A = α = 47.50985041692° = 47°30'31″ = 0.82991798204 rad
Angle ∠ B = β = 53.01881143809° = 53°1'5″ = 0.92553406591 rad
Angle ∠ C = γ = 79.47333814499° = 79°28'24″ = 1.3877072174 rad

Height: ha = 12.7811211797
Height: hb = 11.79880416588
Height: hc = 9.58659088478

Median: ma = 13.28553302556
Median: mb = 12.56598566871
Median: mc = 9.61876920308

Inradius: r = 3.74108424772
Circumradius: R = 8.13769436366

Vertex coordinates: A[16; 0] B[0; 0] C[7.219875; 9.58659088478]
Centroid: CG[7.74395833333; 3.19553029493]
Coordinates of the circumscribed circle: U[8; 1.48765570105]
Coordinates of the inscribed circle: I[7.5; 3.74108424772]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.4911495831° = 132°29'29″ = 0.82991798204 rad
∠ B' = β' = 126.9821885619° = 126°58'55″ = 0.92553406591 rad
∠ C' = γ' = 100.527661855° = 100°31'36″ = 1.3877072174 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 = 13 ; ; c = 16 ; ;

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

p = a+b+c = 12+13+16 = 41 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 41 }{ 2 } = 20.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.5 * (20.5-12)(20.5-13)(20.5-16) } ; ; T = sqrt{ 5880.94 } = 76.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 76.69 }{ 12 } = 12.78 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 76.69 }{ 13 } = 11.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 76.69 }{ 16 } = 9.59 ; ;

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**2-13**2-16**2 }{ 2 * 13 * 16 } ) = 47° 30'31" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-12**2-16**2 }{ 2 * 12 * 16 } ) = 53° 1'5" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16**2-12**2-13**2 }{ 2 * 13 * 12 } ) = 79° 28'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 76.69 }{ 20.5 } = 3.74 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 47° 30'31" } = 8.14 ; ;




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