13 16 20 triangle

Acute scalene triangle.

Sides: a = 13   b = 16   c = 20

Area: T = 103.812202965
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 40.4533084335° = 40°27'11″ = 0.70660395142 rad
Angle ∠ B = β = 52.99222476015° = 52°59'32″ = 0.92548891987 rad
Angle ∠ C = γ = 86.55546680635° = 86°33'17″ = 1.51106639407 rad

Height: ha = 15.97110814846
Height: hb = 12.97765037062
Height: hc = 10.3811202965

Median: ma = 16.90441415044
Median: mb = 14.84992424049
Median: mc = 10.60766017178

Inradius: r = 4.23772257
Circumradius: R = 10.01881067985

Vertex coordinates: A[20; 0] B[0; 0] C[7.825; 10.3811202965]
Centroid: CG[9.275; 3.46604009883]
Coordinates of the circumscribed circle: U[10; 0.60220496874]
Coordinates of the inscribed circle: I[8.5; 4.23772257]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.5476915665° = 139°32'49″ = 0.70660395142 rad
∠ B' = β' = 127.0087752398° = 127°28″ = 0.92548891987 rad
∠ C' = γ' = 93.44553319365° = 93°26'43″ = 1.51106639407 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 = 13 ; ; b = 16 ; ; c = 20 ; ;

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

p = a+b+c = 13+16+20 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-13)(24.5-16)(24.5-20) } ; ; T = sqrt{ 10776.94 } = 103.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 103.81 }{ 13 } = 15.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 103.81 }{ 16 } = 12.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 103.81 }{ 20 } = 10.38 ; ;

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{ 13**2-16**2-20**2 }{ 2 * 16 * 20 } ) = 40° 27'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-13**2-20**2 }{ 2 * 13 * 20 } ) = 52° 59'32" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-13**2-16**2 }{ 2 * 16 * 13 } ) = 86° 33'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 103.81 }{ 24.5 } = 4.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 40° 27'11" } = 10.02 ; ;




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