Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, b and c.

Acute scalene triangle.

Sides: a = 76   b = 85   c = 111

Area: T = 3225.523321337
Perimeter: p = 272
Semiperimeter: s = 136

Angle ∠ A = α = 43.13662588713° = 43°8'11″ = 0.75328697443 rad
Angle ∠ B = β = 49.88107024385° = 49°52'51″ = 0.87105824908 rad
Angle ∠ C = γ = 86.98330386902° = 86°58'59″ = 1.51881404185 rad

Height: ha = 84.88221898256
Height: hb = 75.8954663844
Height: hc = 58.11875353761

Median: ma = 91.26333551871
Median: mb = 85.10114100941
Median: mc = 58.48329034847

Inradius: r = 23.71770824513
Circumradius: R = 55.57770298775

Vertex coordinates: A[111; 0] B[0; 0] C[48.9732972973; 58.11875353761]
Centroid: CG[53.32443243243; 19.3732511792]
Coordinates of the circumscribed circle: U[55.5; 2.92551068357]
Coordinates of the inscribed circle: I[51; 23.71770824513]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.8643741129° = 136°51'49″ = 0.75328697443 rad
∠ B' = β' = 130.1199297561° = 130°7'9″ = 0.87105824908 rad
∠ C' = γ' = 93.01769613098° = 93°1'1″ = 1.51881404185 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and c.

a = 76 ; ; b = 85 ; ; c = 111 ; ;


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 = 76 ; ; b = 85 ; ; c = 111 ; ; : Nr. 1

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

p = a+b+c = 76+85+111 = 272 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 272 }{ 2 } = 136 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 136 * (136-76)(136-85)(136-111) } ; ; T = sqrt{ 10404000 } = 3225.52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3225.52 }{ 76 } = 84.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3225.52 }{ 85 } = 75.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3225.52 }{ 111 } = 58.12 ; ;

6. 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{ 85**2+111**2-76**2 }{ 2 * 85 * 111 } ) = 43° 8'11" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 76**2+111**2-85**2 }{ 2 * 76 * 111 } ) = 49° 52'51" ; ; gamma = 180° - alpha - beta = 180° - 43° 8'11" - 49° 52'51" = 86° 58'59" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3225.52 }{ 136 } = 23.72 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 76 }{ 2 * sin 43° 8'11" } = 55.58 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 85**2+2 * 111**2 - 76**2 } }{ 2 } = 91.263 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 111**2+2 * 76**2 - 85**2 } }{ 2 } = 85.101 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 85**2+2 * 76**2 - 111**2 } }{ 2 } = 58.483 ; ;
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