9 15 16 triangle

Acute scalene triangle.

Sides: a = 9   b = 15   c = 16

Area: T = 66.33224958071
Perimeter: p = 40
Semiperimeter: s = 20

Angle ∠ A = α = 33.55773097619° = 33°33'26″ = 0.58656855435 rad
Angle ∠ B = β = 67.11546195238° = 67°6'53″ = 1.17113710869 rad
Angle ∠ C = γ = 79.32880707142° = 79°19'41″ = 1.38545360232 rad

Height: ha = 14.74105546238
Height: hb = 8.84443327743
Height: hc = 8.29215619759

Median: ma = 14.84108220797
Median: mb = 10.59548100502
Median: mc = 9.43439811321

Inradius: r = 3.31766247904
Circumradius: R = 8.14108063036

Vertex coordinates: A[16; 0] B[0; 0] C[3.5; 8.29215619759]
Centroid: CG[6.5; 2.7643853992]
Coordinates of the circumscribed circle: U[8; 1.50875567229]
Coordinates of the inscribed circle: I[5; 3.31766247904]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.4432690238° = 146°26'34″ = 0.58656855435 rad
∠ B' = β' = 112.8855380476° = 112°53'7″ = 1.17113710869 rad
∠ C' = γ' = 100.6721929286° = 100°40'19″ = 1.38545360232 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 = 15 ; ; c = 16 ; ;

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

p = a+b+c = 9+15+16 = 40 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40 }{ 2 } = 20 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20 * (20-9)(20-15)(20-16) } ; ; T = sqrt{ 4400 } = 66.33 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 66.33 }{ 9 } = 14.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 66.33 }{ 15 } = 8.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 66.33 }{ 16 } = 8.29 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 15**2+16**2-9**2 }{ 2 * 15 * 16 } ) = 33° 33'26" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9**2+16**2-15**2 }{ 2 * 9 * 16 } ) = 67° 6'53" ; ; gamma = 180° - alpha - beta = 180° - 33° 33'26" - 67° 6'53" = 79° 19'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 66.33 }{ 20 } = 3.32 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 33° 33'26" } = 8.14 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15**2+2 * 16**2 - 9**2 } }{ 2 } = 14.841 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 16**2+2 * 9**2 - 15**2 } }{ 2 } = 10.595 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15**2+2 * 9**2 - 16**2 } }{ 2 } = 9.434 ; ;
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