4 10 13 triangle

Obtuse scalene triangle.

Sides: a = 4   b = 10   c = 13

Area: T = 14.98112382666
Perimeter: p = 27
Semiperimeter: s = 13.5

Angle ∠ A = α = 13.3255367656° = 13°19'31″ = 0.23325715396 rad
Angle ∠ B = β = 35.18438154883° = 35°11'2″ = 0.61440734237 rad
Angle ∠ C = γ = 131.4910816856° = 131°29'27″ = 2.29549476903 rad

Height: ha = 7.49106191333
Height: hb = 2.99662476533
Height: hc = 2.30548058872

Median: ma = 11.42436596588
Median: mb = 8.21658383626
Median: mc = 3.96986269666

Inradius: r = 1.11097213531
Circumradius: R = 8.67875203549

Vertex coordinates: A[13; 0] B[0; 0] C[3.26992307692; 2.30548058872]
Centroid: CG[5.42330769231; 0.76882686291]
Coordinates of the circumscribed circle: U[6.5; -5.74988572351]
Coordinates of the inscribed circle: I[3.5; 1.11097213531]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 166.6754632344° = 166°40'29″ = 0.23325715396 rad
∠ B' = β' = 144.8166184512° = 144°48'58″ = 0.61440734237 rad
∠ C' = γ' = 48.50991831443° = 48°30'33″ = 2.29549476903 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 = 4 ; ; b = 10 ; ; c = 13 ; ;

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

p = a+b+c = 4+10+13 = 27 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27 }{ 2 } = 13.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.5 * (13.5-4)(13.5-10)(13.5-13) } ; ; T = sqrt{ 224.44 } = 14.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14.98 }{ 4 } = 7.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14.98 }{ 10 } = 3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14.98 }{ 13 } = 2.3 ; ;

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{ 4**2-10**2-13**2 }{ 2 * 10 * 13 } ) = 13° 19'31" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10**2-4**2-13**2 }{ 2 * 4 * 13 } ) = 35° 11'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-4**2-10**2 }{ 2 * 10 * 4 } ) = 131° 29'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14.98 }{ 13.5 } = 1.11 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4 }{ 2 * sin 13° 19'31" } = 8.68 ; ;




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