20 21 27 triangle

Acute scalene triangle.

Sides: a = 20   b = 21   c = 27

Area: T = 208.1254962463
Perimeter: p = 68
Semiperimeter: s = 34

Angle ∠ A = α = 47.23334876223° = 47°14'1″ = 0.82443798762 rad
Angle ∠ B = β = 50.42987680384° = 50°25'44″ = 0.888014804 rad
Angle ∠ C = γ = 82.33877443392° = 82°20'16″ = 1.43770647374 rad

Height: ha = 20.81224962462
Height: hb = 19.82114249964
Height: hc = 15.41766638861

Median: ma = 22.02327155455
Median: mb = 21.31331414859
Median: mc = 15.43553490404

Inradius: r = 6.12113224254
Circumradius: R = 13.62216240784

Vertex coordinates: A[27; 0] B[0; 0] C[12.74107407407; 15.41766638861]
Centroid: CG[13.24769135802; 5.1398887962]
Coordinates of the circumscribed circle: U[13.5; 1.81662165438]
Coordinates of the inscribed circle: I[13; 6.12113224254]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.7676512378° = 132°45'59″ = 0.82443798762 rad
∠ B' = β' = 129.5711231962° = 129°34'16″ = 0.888014804 rad
∠ C' = γ' = 97.66222556608° = 97°39'44″ = 1.43770647374 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 = 20 ; ; b = 21 ; ; c = 27 ; ;

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

p = a+b+c = 20+21+27 = 68 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 68 }{ 2 } = 34 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34 * (34-20)(34-21)(34-27) } ; ; T = sqrt{ 43316 } = 208.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 208.12 }{ 20 } = 20.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 208.12 }{ 21 } = 19.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 208.12 }{ 27 } = 15.42 ; ;

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{ 20**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 47° 14'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-20**2-27**2 }{ 2 * 20 * 27 } ) = 50° 25'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-20**2-21**2 }{ 2 * 21 * 20 } ) = 82° 20'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 208.12 }{ 34 } = 6.12 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 47° 14'1" } = 13.62 ; ;




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