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 = 33   b = 51   c = 42

Area: T = 690.1330422457
Perimeter: p = 126
Semiperimeter: s = 63

Angle ∠ A = α = 40.11991668984° = 40°7'9″ = 0.77002115555 rad
Angle ∠ B = β = 84.78440914295° = 84°47'3″ = 1.48797615488 rad
Angle ∠ C = γ = 55.0976741672° = 55°5'48″ = 0.96216195493 rad

Height: ha = 41.82660862095
Height: hb = 27.06439381355
Height: hc = 32.86333534503

Median: ma = 43.7066406853
Median: mb = 27.86112634315
Median: mc = 37.47699879904

Inradius: r = 10.95444511501
Circumradius: R = 25.60660295634

Vertex coordinates: A[42; 0] B[0; 0] C[3; 32.86333534503]
Centroid: CG[15; 10.95444511501]
Coordinates of the circumscribed circle: U[21; 14.65215784133]
Coordinates of the inscribed circle: I[12; 10.95444511501]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.8810833102° = 139°52'51″ = 0.77002115555 rad
∠ B' = β' = 95.21659085705° = 95°12'57″ = 1.48797615488 rad
∠ C' = γ' = 124.9033258328° = 124°54'12″ = 0.96216195493 rad

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How did we calculate this triangle?

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

a = 33 ; ; b = 51 ; ; c = 42 ; ;

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

p = a+b+c = 33+51+42 = 126 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 126 }{ 2 } = 63 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 63 * (63-33)(63-51)(63-42) } ; ; T = sqrt{ 476280 } = 690.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 690.13 }{ 33 } = 41.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 690.13 }{ 51 } = 27.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 690.13 }{ 42 } = 32.86 ; ;

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{ 51**2+42**2-33**2 }{ 2 * 51 * 42 } ) = 40° 7'9" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 33**2+42**2-51**2 }{ 2 * 33 * 42 } ) = 84° 47'3" ; ;
 gamma = 180° - alpha - beta = 180° - 40° 7'9" - 84° 47'3" = 55° 5'48" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 690.13 }{ 63 } = 10.95 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 33 }{ 2 * sin 40° 7'9" } = 25.61 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 51**2+2 * 42**2 - 33**2 } }{ 2 } = 43.706 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 33**2 - 51**2 } }{ 2 } = 27.861 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 51**2+2 * 33**2 - 42**2 } }{ 2 } = 37.47 ; ;
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