Triangle calculator

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

You have entered side a, c and angle β.

Right scalene triangle.

Sides: a = 606.2   b = 625.2109916748   c = 153

Area: T = 46374.3
Perimeter: p = 1384.410991675
Semiperimeter: s = 692.2054958374

Angle ∠ A = α = 75.8354842357° = 75°50'5″ = 1.32435676869 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 14.1655157643° = 14°9'55″ = 0.24772286399 rad

Height: ha = 153
Height: hb = 148.3487934854
Height: hc = 606.2

Median: ma = 339.5277038688
Median: mb = 312.6054958374
Median: mc = 611.0087929572

Inradius: r = 66.9955041626
Circumradius: R = 312.6054958374

Vertex coordinates: A[153; 0] B[0; 0] C[-0; 606.2]
Centroid: CG[51; 202.0676666667]
Coordinates of the circumscribed circle: U[76.5; 303.1]
Coordinates of the inscribed circle: I[66.9955041626; 66.9955041626]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 104.1655157643° = 104°9'55″ = 1.32435676869 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 165.8354842357° = 165°50'5″ = 0.24772286399 rad

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How did we calculate this triangle?

1. Input data entered: side a, c and angle β.

a = 606.2 ; ; c = 153 ; ; beta = 90° ; ;

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

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 606.2**2+153**2 - 2 * 606.2 * 153 * cos 90° } ; ; b = 625.21 ; ;
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 = 606.2 ; ; b = 625.21 ; ; c = 153 ; ;

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

p = a+b+c = 606.2+625.21+153 = 1384.41 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1384.41 }{ 2 } = 692.2 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 692.2 * (692.2-606.2)(692.2-625.21)(692.2-153) } ; ; T = sqrt{ 2150575700.49 } = 46374.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 46374.3 }{ 606.2 } = 153 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 46374.3 }{ 625.21 } = 148.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 46374.3 }{ 153 } = 606.2 ; ;

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{ 625.21**2+153**2-606.2**2 }{ 2 * 625.21 * 153 } ) = 75° 50'5" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 606.2**2+153**2-625.21**2 }{ 2 * 606.2 * 153 } ) = 90° ; ;
 gamma = 180° - alpha - beta = 180° - 75° 50'5" - 90° = 14° 9'55" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 46374.3 }{ 692.2 } = 67 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 606.2 }{ 2 * sin 75° 50'5" } = 312.6 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 625.21**2+2 * 153**2 - 606.2**2 } }{ 2 } = 339.527 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 153**2+2 * 606.2**2 - 625.21**2 } }{ 2 } = 312.605 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 625.21**2+2 * 606.2**2 - 153**2 } }{ 2 } = 611.008 ; ;
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