13 15 16 triangle

Acute scalene triangle.

Sides: a = 13   b = 15   c = 16

Area: T = 91.19221049214
Perimeter: p = 44
Semiperimeter: s = 22

Angle ∠ A = α = 49.45883981265° = 49°27'30″ = 0.86332118901 rad
Angle ∠ B = β = 61.26443462551° = 61°15'52″ = 1.06992645562 rad
Angle ∠ C = γ = 69.27772556184° = 69°16'38″ = 1.20991162073 rad

Height: ha = 14.03295546033
Height: hb = 12.15989473229
Height: hc = 11.39990131152

Median: ma = 14.08801278403
Median: mb = 12.5
Median: mc = 11.53325625947

Inradius: r = 4.14550956782
Circumradius: R = 8.55333720345

Vertex coordinates: A[16; 0] B[0; 0] C[6.25; 11.39990131152]
Centroid: CG[7.41766666667; 3.87996710384]
Coordinates of the circumscribed circle: U[8; 3.02765777968]
Coordinates of the inscribed circle: I[7; 4.14550956782]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.5421601874° = 130°32'30″ = 0.86332118901 rad
∠ B' = β' = 118.7365653745° = 118°44'8″ = 1.06992645562 rad
∠ C' = γ' = 110.7232744382° = 110°43'22″ = 1.20991162073 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 = 13 ; ; b = 15 ; ; c = 16 ; ;

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

p = a+b+c = 13+15+16 = 44 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 44 }{ 2 } = 22 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22 * (22-13)(22-15)(22-16) } ; ; T = sqrt{ 8316 } = 91.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 91.19 }{ 13 } = 14.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 91.19 }{ 15 } = 12.16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 91.19 }{ 16 } = 11.4 ; ;

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{ 13**2-15**2-16**2 }{ 2 * 15 * 16 } ) = 49° 27'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-13**2-16**2 }{ 2 * 13 * 16 } ) = 61° 15'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16**2-13**2-15**2 }{ 2 * 15 * 13 } ) = 69° 16'38" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 91.19 }{ 22 } = 4.15 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 49° 27'30" } = 8.55 ; ;




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