Triangle calculator

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

You have entered side a, b and angle γ.

Right scalene triangle.

Sides: a = 400   b = 900   c = 984.886578018

Area: T = 180000
Perimeter: p = 2284.886578018
Semiperimeter: s = 1142.443289009

Angle ∠ A = α = 23.96224889746° = 23°57'45″ = 0.41882243296 rad
Angle ∠ B = β = 66.03875110254° = 66°2'15″ = 1.15325719972 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 900
Height: hb = 400
Height: hc = 365.5254619448

Median: ma = 921.9544445729
Median: mb = 602.087972894
Median: mc = 492.443289009

Inradius: r = 157.557710991
Circumradius: R = 492.443289009

Vertex coordinates: A[984.886578018; 0] B[0; 0] C[162.4555386421; 365.5254619448]
Centroid: CG[382.4477055534; 121.8421539816]
Coordinates of the circumscribed circle: U[492.443289009; 0]
Coordinates of the inscribed circle: I[242.443289009; 157.557710991]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.0387511025° = 156°2'15″ = 0.41882243296 rad
∠ B' = β' = 113.9622488975° = 113°57'45″ = 1.15325719972 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and angle γ.

a = 400 ; ; b = 900 ; ; gamma = 90° ; ;

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

c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 400**2+900**2 - 2 * 400 * 900 * cos 90° } ; ; c = 984.89 ; ;
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 = 400 ; ; b = 900 ; ; c = 984.89 ; ;

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

p = a+b+c = 400+900+984.89 = 2284.89 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2284.89 }{ 2 } = 1142.44 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1142.44 * (1142.44-400)(1142.44-900)(1142.44-984.89) } ; ; T = sqrt{ 32400000000 } = 180000 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 180000 }{ 400 } = 900 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 180000 }{ 900 } = 400 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 180000 }{ 984.89 } = 365.52 ; ;

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{ 900**2+984.89**2-400**2 }{ 2 * 900 * 984.89 } ) = 23° 57'45" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 400**2+984.89**2-900**2 }{ 2 * 400 * 984.89 } ) = 66° 2'15" ; ; gamma = 180° - alpha - beta = 180° - 23° 57'45" - 66° 2'15" = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 180000 }{ 1142.44 } = 157.56 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 400 }{ 2 * sin 23° 57'45" } = 492.44 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 900**2+2 * 984.89**2 - 400**2 } }{ 2 } = 921.954 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 984.89**2+2 * 400**2 - 900**2 } }{ 2 } = 602.08 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 900**2+2 * 400**2 - 984.89**2 } }{ 2 } = 492.443 ; ;
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