9 13 16 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 13   c = 16

Area: T = 58.48107660689
Perimeter: p = 38
Semiperimeter: s = 19

Angle ∠ A = α = 34.21660511313° = 34°12'58″ = 0.59771827493 rad
Angle ∠ B = β = 54.31546652873° = 54°18'53″ = 0.94879697414 rad
Angle ∠ C = γ = 91.46992835814° = 91°28'9″ = 1.59664401629 rad

Height: ha = 12.99657257931
Height: hb = 8.99770409337
Height: hc = 7.31100957586

Median: ma = 13.86554246239
Median: mb = 11.23661025271
Median: mc = 7.81102496759

Inradius: r = 3.07879350563
Circumradius: R = 8.00326311463

Vertex coordinates: A[16; 0] B[0; 0] C[5.25; 7.31100957586]
Centroid: CG[7.08333333333; 2.43766985862]
Coordinates of the circumscribed circle: U[8; -0.20551956704]
Coordinates of the inscribed circle: I[6; 3.07879350563]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.7843948869° = 145°47'2″ = 0.59771827493 rad
∠ B' = β' = 125.6855334713° = 125°41'7″ = 0.94879697414 rad
∠ C' = γ' = 88.53107164186° = 88°31'51″ = 1.59664401629 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 = 13 ; ; c = 16 ; ;

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

p = a+b+c = 9+13+16 = 38 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38 }{ 2 } = 19 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19 * (19-9)(19-13)(19-16) } ; ; T = sqrt{ 3420 } = 58.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 58.48 }{ 9 } = 13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 58.48 }{ 13 } = 9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 58.48 }{ 16 } = 7.31 ; ;

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**2-13**2-16**2 }{ 2 * 13 * 16 } ) = 34° 12'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-9**2-16**2 }{ 2 * 9 * 16 } ) = 54° 18'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16**2-9**2-13**2 }{ 2 * 13 * 9 } ) = 91° 28'9" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 58.48 }{ 19 } = 3.08 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 34° 12'58" } = 8 ; ;




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