Triangle calculator SSS - result

Please enter the triangle sides:


Acute scalene triangle.

Sides: a = 13   b = 23   c = 24

Area: T = 146.3565731012
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 32.02439959746° = 32°1'26″ = 0.55989241694 rad
Angle ∠ B = β = 69.74877532578° = 69°44'52″ = 1.21773279402 rad
Angle ∠ C = γ = 78.22882507676° = 78°13'42″ = 1.3655340544 rad

Height: ha = 22.51662663095
Height: hb = 12.72765853054
Height: hc = 12.19663109177

Median: ma = 22.58987139962
Median: mb = 15.5
Median: mc = 14.31878210633

Inradius: r = 4.87985243671
Circumradius: R = 12.25878049223

Vertex coordinates: A[24; 0] B[0; 0] C[4.5; 12.19663109177]
Centroid: CG[9.5; 4.06554369726]
Coordinates of the circumscribed circle: U[12; 2.50107561882]
Coordinates of the inscribed circle: I[7; 4.87985243671]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.9766004025° = 147°58'34″ = 0.55989241694 rad
∠ B' = β' = 110.2522246742° = 110°15'8″ = 1.21773279402 rad
∠ C' = γ' = 101.7721749232° = 101°46'18″ = 1.3655340544 rad

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How did we calculate this triangle?

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

p = a+b+c = 13+23+24 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-13)(30-23)(30-24) } ; ; T = sqrt{ 21420 } = 146.36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 146.36 }{ 13 } = 22.52 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 146.36 }{ 23 } = 12.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 146.36 }{ 24 } = 12.2 ; ;

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{ 23**2+24**2-13**2 }{ 2 * 23 * 24 } ) = 32° 1'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13**2+24**2-23**2 }{ 2 * 13 * 24 } ) = 69° 44'52" ; ; gamma = 180° - alpha - beta = 180° - 32° 1'26" - 69° 44'52" = 78° 13'42" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 146.36 }{ 30 } = 4.88 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13 }{ 2 * sin 32° 1'26" } = 12.26 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 24**2 - 13**2 } }{ 2 } = 22.589 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 13**2 - 23**2 } }{ 2 } = 15.5 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 13**2 - 24**2 } }{ 2 } = 14.318 ; ;
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