Right triangle calculator (a,c)

Please enter two properties of the right triangle

Use symbols: a, b, c, A, B, h, T, p, r, R


You have entered cathetus a and hypotenuse c.

Right scalene triangle.

Sides: a = 1000   b = 2291.288784748   c = 2500

Area: T = 1145643.924374
Perimeter: p = 5791.288784748
Semiperimeter: s = 2895.644392374

Angle ∠ A = α = 23.57881784782° = 23°34'41″ = 0.41215168461 rad
Angle ∠ B = β = 66.42218215218° = 66°25'19″ = 1.15992794807 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 2291.288784748
Height: hb = 1000
Height: hc = 916.5155138991

Median: ma = 2345.208787991
Median: mb = 1520.691063257
Median: mc = 1250

Inradius: r = 395.6443923739
Circumradius: R = 1250

Vertex coordinates: A[2500; 0] B[0; 0] C[400; 916.5155138991]
Centroid: CG[966.6676666667; 305.505504633]
Coordinates of the circumscribed circle: U[1250; -0]
Coordinates of the inscribed circle: I[604.3566076261; 395.6443923739]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.4221821522° = 156°25'19″ = 0.41215168461 rad
∠ B' = β' = 113.5788178478° = 113°34'41″ = 1.15992794807 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus a and hypotenuse c

a = 1000 ; ; c = 2500 ; ;

2. From cathetus a and hypotenuse c we calculate cathetus b - Pythagorean theorem:

c**2 = a**2+b**2 ; ; b = sqrt{ c**2 - a**2 } = sqrt{ 2500**2 - 1000**2 } = 2291.288 ; ;
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 = 1000 ; ; b = 2291.29 ; ; c = 2500 ; ;

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

p = a+b+c = 1000+2291.29+2500 = 5791.29 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 5791.29 }{ 2 } = 2895.64 ; ;

5. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 1000 * 2291.29 }{ 2 } = 1145643.92 ; ;

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

h _a = b = 2291.29 ; ; h _b = a = 1000 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1145643.92 }{ 2500 } = 916.52 ; ;

7. Calculation of the inner angles of the triangle - basic use of sine function

 sin alpha = fraction{ a }{ c } ; ; alpha = arcsin( fraction{ a }{ c } ) = arcsin( fraction{ 1000 }{ 2500 } ) = 23° 34'41" ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 2291.29 }{ 2500 } ) = 66° 25'19" ; ; gamma = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1145643.92 }{ 2895.64 } = 395.64 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1000 }{ 2 * sin 23° 34'41" } = 1250 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2291.29**2+2 * 2500**2 - 1000**2 } }{ 2 } = 2345.208 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2500**2+2 * 1000**2 - 2291.29**2 } }{ 2 } = 1520.691 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2291.29**2+2 * 1000**2 - 2500**2 } }{ 2 } = 1250 ; ;
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Trigonometry right triangle solver. You can calculate angles, sides (adjacent, opposite, hypotenuse) and area of any right-angled triangle and use it in real world to find height. A right-angled triangle is entirely determined by two independent properties. Step-by-step explanations are provided for each calculation.

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