Right triangle calculator (c,v)

Please enter two properties of the right triangle

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


You have entered hypotenuse c and height h.

Right scalene triangle.

Sides: a = 33.63302049227   b = 12.84655952318   c = 36

Area: T = 216
Perimeter: p = 82.47658001545
Semiperimeter: s = 41.23879000772

Angle ∠ A = α = 69.09548425521° = 69°5'41″ = 1.20659324987 rad
Angle ∠ B = β = 20.90551574479° = 20°54'19″ = 0.36548638281 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 12.84655952318
Height: hb = 33.63302049227
Height: hc = 12

Median: ma = 21.16602690825
Median: mb = 34.23880345866
Median: mc = 18

Inradius: r = 5.23879000772
Circumradius: R = 18

Vertex coordinates: A[36; 0] B[0; 0] C[31.4166407865; 12]
Centroid: CG[22.4722135955; 4]
Coordinates of the circumscribed circle: U[18; 0]
Coordinates of the inscribed circle: I[28.39223048454; 5.23879000772]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 110.9055157448° = 110°54'19″ = 1.20659324987 rad
∠ B' = β' = 159.0954842552° = 159°5'41″ = 0.36548638281 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: hypotenuse c height h

c = 36 ; ; hc = 12 ; ;

2. From hypotenuse c and we calculate a,b - Pythagorean theorem, Euclid's theorem:

c = c_1+c_2 ; ; h**2 = c_1 * c_2 ; ; ; ; h**2 = c_1 * (c-c_1) ; ; h**2 = c_1 * c-c_1 **2 ; ; ; ; c_1**2 -c_1 * c + h**2 = 0 ; ; ; ; c_1**2 -36 * c_1 + 144 = 0 ; ; ; ; c_1 = 31.416 ; ; c_2 = 4.584 ; ; ; ; a = sqrt{ c_1**2+h**2 } = sqrt{ 31.416**2+12**2 } = 33.63 ; ; b = sqrt{ c_2**2+h**2 } = sqrt{ 4.584**2+12**2 } = 12.846 ; ;


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 = 33.63 ; ; b = 12.85 ; ; c = 36 ; ;

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

p = a+b+c = 33.63+12.85+36 = 82.48 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 82.48 }{ 2 } = 41.24 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 41.24 * (41.24-33.63)(41.24-12.85)(41.24-36) } ; ; T = sqrt{ 46656 } = 216 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 216 }{ 33.63 } = 12.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 216 }{ 12.85 } = 33.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 216 }{ 36 } = 12 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 33.63**2-12.85**2-36**2 }{ 2 * 12.85 * 36 } ) = 69° 5'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.85**2-33.63**2-36**2 }{ 2 * 33.63 * 36 } ) = 20° 54'19" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 36**2-33.63**2-12.85**2 }{ 2 * 12.85 * 33.63 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 216 }{ 41.24 } = 5.24 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 33.63 }{ 2 * sin 69° 5'41" } = 18 ; ;
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