12 22 27 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 22   c = 27

Area: T = 129.5622484925
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 25.8644052825° = 25°51'51″ = 0.45114128797 rad
Angle ∠ B = β = 53.10879943014° = 53°6'29″ = 0.92769093597 rad
Angle ∠ C = γ = 101.0287952874° = 101°1'41″ = 1.76332704142 rad

Height: ha = 21.59437474875
Height: hb = 11.77884077205
Height: hc = 9.59772211056

Median: ma = 23.8855141825
Median: mb = 17.76223196683
Median: mc = 11.47882402832

Inradius: r = 4.24879503254
Circumradius: R = 13.75439813398

Vertex coordinates: A[27; 0] B[0; 0] C[7.20437037037; 9.59772211056]
Centroid: CG[11.40112345679; 3.19990737019]
Coordinates of the circumscribed circle: U[13.5; -2.63109699154]
Coordinates of the inscribed circle: I[8.5; 4.24879503254]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.1365947175° = 154°8'9″ = 0.45114128797 rad
∠ B' = β' = 126.8922005699° = 126°53'31″ = 0.92769093597 rad
∠ C' = γ' = 78.97220471265° = 78°58'19″ = 1.76332704142 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 = 22 ; ; c = 27 ; ;

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

p = a+b+c = 12+22+27 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-12)(30.5-22)(30.5-27) } ; ; T = sqrt{ 16786.44 } = 129.56 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 129.56 }{ 12 } = 21.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 129.56 }{ 22 } = 11.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 129.56 }{ 27 } = 9.6 ; ;

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**2-22**2-27**2 }{ 2 * 22 * 27 } ) = 25° 51'51" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-12**2-27**2 }{ 2 * 12 * 27 } ) = 53° 6'29" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-12**2-22**2 }{ 2 * 22 * 12 } ) = 101° 1'41" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 129.56 }{ 30.5 } = 4.25 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 25° 51'51" } = 13.75 ; ;




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