Triangle calculator

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

You have entered side b, c and angle α.

Right scalene triangle.

Sides: a = 5.88221764679   b = 3.4   c = 4.8

Area: T = 8.16
Perimeter: p = 14.08221764679
Semiperimeter: s = 7.0411088234

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 35.31112134396° = 35°18'40″ = 0.61662969374 rad
Angle ∠ C = γ = 54.68987865604° = 54°41'20″ = 0.95444993894 rad

Height: ha = 2.77444832357
Height: hb = 4.8
Height: hc = 3.4

Median: ma = 2.9411088234
Median: mb = 5.09221508226
Median: mc = 4.16217304093

Inradius: r = 1.1598911766
Circumradius: R = 2.9411088234

Vertex coordinates: A[4.8; 0] B[0; 0] C[4.8; 3.4]
Centroid: CG[3.2; 1.13333333333]
Coordinates of the circumscribed circle: U[2.4; 1.7]
Coordinates of the inscribed circle: I[3.6411088234; 1.1598911766]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 144.689878656° = 144°41'20″ = 0.61662969374 rad
∠ C' = γ' = 125.311121344° = 125°18'40″ = 0.95444993894 rad

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How did we calculate this triangle?

1. Input data entered: side b, c and angle α.

b = 3.4 ; ; c = 4.8 ; ; alpha = 90° ; ;

2. Calculation of the third side a of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 3.4**2+4.8**2 - 2 * 3.4 * 4.8 * cos 90° } ; ; a = 5.88 ; ;


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 = 5.88 ; ; b = 3.4 ; ; c = 4.8 ; ;

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

p = a+b+c = 5.88+3.4+4.8 = 14.08 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 14.08 }{ 2 } = 7.04 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.04 * (7.04-5.88)(7.04-3.4)(7.04-4.8) } ; ; T = sqrt{ 66.59 } = 8.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8.16 }{ 5.88 } = 2.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8.16 }{ 3.4 } = 4.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8.16 }{ 4.8 } = 3.4 ; ;

7. 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{ 3.4**2+4.8**2-5.88**2 }{ 2 * 3.4 * 4.8 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.88**2+4.8**2-3.4**2 }{ 2 * 5.88 * 4.8 } ) = 35° 18'40" ; ; gamma = 180° - alpha - beta = 180° - 90° - 35° 18'40" = 54° 41'20" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8.16 }{ 7.04 } = 1.16 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.88 }{ 2 * sin 90° } = 2.94 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.4**2+2 * 4.8**2 - 5.88**2 } }{ 2 } = 2.941 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.8**2+2 * 5.88**2 - 3.4**2 } }{ 2 } = 5.092 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.4**2+2 * 5.88**2 - 4.8**2 } }{ 2 } = 4.162 ; ;
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