12 15 16 triangle

Acute scalene triangle.

Sides: a = 12   b = 15   c = 16

Area: T = 85.45113750621
Perimeter: p = 43
Semiperimeter: s = 21.5

Angle ∠ A = α = 45.40656099767° = 45°24'20″ = 0.7922477393 rad
Angle ∠ B = β = 62.88881400862° = 62°53'17″ = 1.0987605105 rad
Angle ∠ C = γ = 71.70662499371° = 71°42'22″ = 1.25215101557 rad

Height: ha = 14.24218958437
Height: hb = 11.39435166749
Height: hc = 10.68114218828

Median: ma = 14.33003496461
Median: mb = 11.99895788083
Median: mc = 10.97772492001

Inradius: r = 3.9744482561
Circumradius: R = 8.42658445166

Vertex coordinates: A[16; 0] B[0; 0] C[5.469875; 10.68114218828]
Centroid: CG[7.156625; 3.56604739609]
Coordinates of the circumscribed circle: U[8; 2.64547789733]
Coordinates of the inscribed circle: I[6.5; 3.9744482561]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.5944390023° = 134°35'40″ = 0.7922477393 rad
∠ B' = β' = 117.1121859914° = 117°6'43″ = 1.0987605105 rad
∠ C' = γ' = 108.2943750063° = 108°17'37″ = 1.25215101557 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 = 15 ; ; c = 16 ; ;

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

p = a+b+c = 12+15+16 = 43 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43 }{ 2 } = 21.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.5 * (21.5-12)(21.5-15)(21.5-16) } ; ; T = sqrt{ 7301.94 } = 85.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 85.45 }{ 12 } = 14.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 85.45 }{ 15 } = 11.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 85.45 }{ 16 } = 10.68 ; ;

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**2-15**2-16**2 }{ 2 * 15 * 16 } ) = 45° 24'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-12**2-16**2 }{ 2 * 12 * 16 } ) = 62° 53'17" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 16**2-12**2-15**2 }{ 2 * 15 * 12 } ) = 71° 42'22" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 85.45 }{ 21.5 } = 3.97 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 45° 24'20" } = 8.43 ; ;




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