12 15 22 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 15   c = 22

Area: T = 85.28444505171
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 31.12328957773° = 31°7'22″ = 0.54331970041 rad
Angle ∠ B = β = 40.2487773648° = 40°14'52″ = 0.70224561668 rad
Angle ∠ C = γ = 108.6299330575° = 108°37'46″ = 1.89659394828 rad

Height: ha = 14.21440750862
Height: hb = 11.37112600689
Height: hc = 7.75331318652

Median: ma = 17.84765682976
Median: mb = 16.0554594358
Median: mc = 7.96986887253

Inradius: r = 3.48109979803
Circumradius: R = 11.60882122122

Vertex coordinates: A[22; 0] B[0; 0] C[9.15990909091; 7.75331318652]
Centroid: CG[10.38663636364; 2.58443772884]
Coordinates of the circumscribed circle: U[11; -3.70881789011]
Coordinates of the inscribed circle: I[9.5; 3.48109979803]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.8777104223° = 148°52'38″ = 0.54331970041 rad
∠ B' = β' = 139.7522226352° = 139°45'8″ = 0.70224561668 rad
∠ C' = γ' = 71.37106694253° = 71°22'14″ = 1.89659394828 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 = 22 ; ;

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

p = a+b+c = 12+15+22 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-12)(24.5-15)(24.5-22) } ; ; T = sqrt{ 7273.44 } = 85.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 85.28 }{ 12 } = 14.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 85.28 }{ 15 } = 11.37 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 85.28 }{ 22 } = 7.75 ; ;

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**2-22**2 }{ 2 * 15 * 22 } ) = 31° 7'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-12**2-22**2 }{ 2 * 12 * 22 } ) = 40° 14'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 22**2-12**2-15**2 }{ 2 * 15 * 12 } ) = 108° 37'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 85.28 }{ 24.5 } = 3.48 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 31° 7'22" } = 11.61 ; ;




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