10 11 12 triangle

Acute scalene triangle.

Sides: a = 10   b = 11   c = 12

Area: T = 51.52112334868
Perimeter: p = 33
Semiperimeter: s = 16.5

Angle ∠ A = α = 51.31878125465° = 51°19'4″ = 0.89656647939 rad
Angle ∠ B = β = 59.17695025682° = 59°10'10″ = 1.03327026366 rad
Angle ∠ C = γ = 69.51326848853° = 69°30'46″ = 1.21332252231 rad

Height: ha = 10.30442466974
Height: hb = 9.36774969976
Height: hc = 8.58768722478

Median: ma = 10.36882206767
Median: mb = 9.57986220303
Median: mc = 8.63113382508

Inradius: r = 3.12224989992
Circumradius: R = 6.40551261522

Vertex coordinates: A[12; 0] B[0; 0] C[5.125; 8.58768722478]
Centroid: CG[5.70883333333; 2.86222907493]
Coordinates of the circumscribed circle: U[6; 2.24217941533]
Coordinates of the inscribed circle: I[5.5; 3.12224989992]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.6822187453° = 128°40'56″ = 0.89656647939 rad
∠ B' = β' = 120.8330497432° = 120°49'50″ = 1.03327026366 rad
∠ C' = γ' = 110.4877315115° = 110°29'14″ = 1.21332252231 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 = 10 ; ; b = 11 ; ; c = 12 ; ;

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

p = a+b+c = 10+11+12 = 33 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33 }{ 2 } = 16.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.5 * (16.5-10)(16.5-11)(16.5-12) } ; ; T = sqrt{ 2654.44 } = 51.52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 51.52 }{ 10 } = 10.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 51.52 }{ 11 } = 9.37 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 51.52 }{ 12 } = 8.59 ; ;

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{ 10**2-11**2-12**2 }{ 2 * 11 * 12 } ) = 51° 19'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11**2-10**2-12**2 }{ 2 * 10 * 12 } ) = 59° 10'10" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12**2-10**2-11**2 }{ 2 * 11 * 10 } ) = 69° 30'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 51.52 }{ 16.5 } = 3.12 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 51° 19'4" } = 6.41 ; ;




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