21 22 30 triangle

Acute scalene triangle.

Sides: a = 21   b = 22   c = 30

Area: T = 230.9155433655
Perimeter: p = 73
Semiperimeter: s = 36.5

Angle ∠ A = α = 44.40664477069° = 44°24'23″ = 0.77550387216 rad
Angle ∠ B = β = 47.14439519766° = 47°8'38″ = 0.82328171844 rad
Angle ∠ C = γ = 88.45496003164° = 88°26'59″ = 1.54437367476 rad

Height: ha = 21.99219460624
Height: hb = 20.99223121504
Height: hc = 15.39443622437

Median: ma = 24.11994941904
Median: mb = 23.44114163395
Median: mc = 15.41110350074

Inradius: r = 6.32664502371
Circumradius: R = 15.00554933322

Vertex coordinates: A[30; 0] B[0; 0] C[14.28333333333; 15.39443622437]
Centroid: CG[14.76111111111; 5.13114540812]
Coordinates of the circumscribed circle: U[15; 0.4065992785]
Coordinates of the inscribed circle: I[14.5; 6.32664502371]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.5943552293° = 135°35'37″ = 0.77550387216 rad
∠ B' = β' = 132.8566048023° = 132°51'22″ = 0.82328171844 rad
∠ C' = γ' = 91.55503996836° = 91°33'1″ = 1.54437367476 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 = 21 ; ; b = 22 ; ; c = 30 ; ;

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

p = a+b+c = 21+22+30 = 73 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 73 }{ 2 } = 36.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36.5 * (36.5-21)(36.5-22)(36.5-30) } ; ; T = sqrt{ 53321.94 } = 230.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 230.92 }{ 21 } = 21.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 230.92 }{ 22 } = 20.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 230.92 }{ 30 } = 15.39 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 22**2+30**2-21**2 }{ 2 * 22 * 30 } ) = 44° 24'23" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 21**2+30**2-22**2 }{ 2 * 21 * 30 } ) = 47° 8'38" ; ; gamma = 180° - alpha - beta = 180° - 44° 24'23" - 47° 8'38" = 88° 26'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 230.92 }{ 36.5 } = 6.33 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 21 }{ 2 * sin 44° 24'23" } = 15.01 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 22**2+2 * 30**2 - 21**2 } }{ 2 } = 24.119 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 30**2+2 * 21**2 - 22**2 } }{ 2 } = 23.441 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 22**2+2 * 21**2 - 30**2 } }{ 2 } = 15.411 ; ;
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