20 24 28 triangle

Acute scalene triangle.

Sides: a = 20   b = 24   c = 28

Area: T = 235.1511015307
Perimeter: p = 72
Semiperimeter: s = 36

Angle ∠ A = α = 44.41553085972° = 44°24'55″ = 0.77551933733 rad
Angle ∠ B = β = 57.12216504356° = 57°7'18″ = 0.99769608743 rad
Angle ∠ C = γ = 78.46330409672° = 78°27'47″ = 1.3699438406 rad

Height: ha = 23.51551015307
Height: hb = 19.59659179423
Height: hc = 16.79765010934

Median: ma = 24.08331891576
Median: mb = 21.16660104885
Median: mc = 17.08880074906

Inradius: r = 6.53219726474
Circumradius: R = 14.28986901662

Vertex coordinates: A[28; 0] B[0; 0] C[10.85771428571; 16.79765010934]
Centroid: CG[12.95223809524; 5.59988336978]
Coordinates of the circumscribed circle: U[14; 2.85877380332]
Coordinates of the inscribed circle: I[12; 6.53219726474]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.5854691403° = 135°35'5″ = 0.77551933733 rad
∠ B' = β' = 122.8788349564° = 122°52'42″ = 0.99769608743 rad
∠ C' = γ' = 101.5376959033° = 101°32'13″ = 1.3699438406 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 = 24 ; ; c = 28 ; ;

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

p = a+b+c = 20+24+28 = 72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72 }{ 2 } = 36 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36 * (36-20)(36-24)(36-28) } ; ; T = sqrt{ 55296 } = 235.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 235.15 }{ 20 } = 23.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 235.15 }{ 24 } = 19.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 235.15 }{ 28 } = 16.8 ; ;

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**2-24**2-28**2 }{ 2 * 24 * 28 } ) = 44° 24'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-20**2-28**2 }{ 2 * 20 * 28 } ) = 57° 7'18" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-20**2-24**2 }{ 2 * 24 * 20 } ) = 78° 27'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 235.15 }{ 36 } = 6.53 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 44° 24'55" } = 14.29 ; ;




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