10 15 20 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 15   c = 20

Area: T = 72.61884377414
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 28.95550243719° = 28°57'18″ = 0.50553605103 rad
Angle ∠ B = β = 46.56774634422° = 46°34'3″ = 0.81327555614 rad
Angle ∠ C = γ = 104.4787512186° = 104°28'39″ = 1.82334765819 rad

Height: ha = 14.52436875483
Height: hb = 9.68224583655
Height: hc = 7.26218437741

Median: ma = 16.95658249578
Median: mb = 13.91994109071
Median: mc = 7.90656941504

Inradius: r = 3.22774861218
Circumradius: R = 10.32879555899

Vertex coordinates: A[20; 0] B[0; 0] C[6.875; 7.26218437741]
Centroid: CG[8.95883333333; 2.42106145914]
Coordinates of the circumscribed circle: U[10; -2.58219888975]
Coordinates of the inscribed circle: I[7.5; 3.22774861218]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.0454975628° = 151°2'42″ = 0.50553605103 rad
∠ B' = β' = 133.4332536558° = 133°25'57″ = 0.81327555614 rad
∠ C' = γ' = 75.52224878141° = 75°31'21″ = 1.82334765819 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 = 10 ; ; b = 15 ; ; c = 20 ; ;

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

p = a+b+c = 10+15+20 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-10)(22.5-15)(22.5-20) } ; ; T = sqrt{ 5273.44 } = 72.62 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 72.62 }{ 10 } = 14.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 72.62 }{ 15 } = 9.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 72.62 }{ 20 } = 7.26 ; ;

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{ 10**2-15**2-20**2 }{ 2 * 15 * 20 } ) = 28° 57'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-10**2-20**2 }{ 2 * 10 * 20 } ) = 46° 34'3" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-10**2-15**2 }{ 2 * 15 * 10 } ) = 104° 28'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 72.62 }{ 22.5 } = 3.23 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 28° 57'18" } = 10.33 ; ;




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