Triangle calculator

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

You have entered side a, b and angle γ.

Acute isosceles triangle.

Sides: a = 300   b = 300   c = 360.001085748

Area: T = 43200.04550174
Perimeter: p = 960.001085748
Semiperimeter: s = 4800.00042874

Angle ∠ A = α = 53.13° = 53°7'48″ = 0.92772934316 rad
Angle ∠ B = β = 53.13° = 53°7'48″ = 0.92772934316 rad
Angle ∠ C = γ = 73.74° = 73°44'24″ = 1.28770057904 rad

Height: ha = 2888.000300116
Height: hb = 2888.000300116
Height: hc = 2409.999678445

Median: ma = 295.4666256437
Median: mb = 295.4666256437
Median: mc = 2409.999678445

Inradius: r = 900.0000133976
Circumradius: R = 187.5500251215

Vertex coordinates: A[360.001085748; 0] B[0; 0] C[1800.00042874; 2409.999678445]
Centroid: CG[1800.00042874; 809.9998928149]
Coordinates of the circumscribed circle: U[1800.00042874; 52.4999427229]
Coordinates of the inscribed circle: I[1800.00042874; 900.0000133976]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.87° = 126°52'12″ = 0.92772934316 rad
∠ B' = β' = 126.87° = 126°52'12″ = 0.92772934316 rad
∠ C' = γ' = 106.26° = 106°15'36″ = 1.28770057904 rad

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How did we calculate this triangle?

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

a = 300 ; ; b = 300 ; ; gamma = 73.74° ; ;

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{ 300**2+300**2 - 2 * 300 * 300 * cos(73° 44'24") } ; ; c = 360 ; ;


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 = 300 ; ; b = 300 ; ; c = 360 ; ;

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

p = a+b+c = 300+300+360 = 960 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 960 }{ 2 } = 480 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 480 * (480-300)(480-300)(480-360) } ; ; T = sqrt{ 1866243889.51 } = 43200.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 43200.05 }{ 300 } = 288 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 43200.05 }{ 300 } = 288 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 43200.05 }{ 360 } = 240 ; ;

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{ 300**2+360**2-300**2 }{ 2 * 300 * 360 } ) = 53° 7'48" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 300**2+360**2-300**2 }{ 2 * 300 * 360 } ) = 53° 7'48" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 300**2+300**2-360**2 }{ 2 * 300 * 300 } ) = 73° 44'24" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 43200.05 }{ 480 } = 90 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 300 }{ 2 * sin 53° 7'48" } = 187.5 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 300**2+2 * 360**2 - 300**2 } }{ 2 } = 295.466 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 360**2+2 * 300**2 - 300**2 } }{ 2 } = 295.466 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 300**2+2 * 300**2 - 360**2 } }{ 2 } = 240 ; ;
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