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: h_a = 14.74105546238
Height: h_b = 8.84443327743
Height: h_c = 8.29215619759

Median: m_a = 14.84108220797
Median: m_b = 10.59548100502
Median: m_c = 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

Calculate another triangle

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

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

6. Inradius

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

7. Circumradius

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

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