Right triangle calculator (c) - result

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 area T.

Right scalene Pythagorean triangle.

Sides: a = 4   b = 3   c = 5

Area: T = 6
Perimeter: p = 12
Semiperimeter: s = 6

Angle ∠ A = α = 53.13301023542° = 53°7'48″ = 0.9277295218 rad
Angle ∠ B = β = 36.87698976458° = 36°52'12″ = 0.64435011088 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 3
Height: hb = 4
Height: hc = 2.4

Median: ma = 3.60655512755
Median: mb = 4.27220018727
Median: mc = 2.5

Inradius: r = 1
Circumradius: R = 2.5

Vertex coordinates: A[5; 0] B[0; 0] C[3.2; 2.4]
Centroid: CG[2.73333333333; 0.8]
Coordinates of the circumscribed circle: U[2.5; -0]
Coordinates of the inscribed circle: I[3; 1]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.8769897646° = 126°52'12″ = 0.9277295218 rad
∠ B' = β' = 143.1330102354° = 143°7'48″ = 0.64435011088 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: hypotenuse c area T

c = 5 ; ; S = 6 ; ;

2. From area T and hypotenuse c we calculate h:

S = fraction{ c * h }{ 2 } ; ; h = 2 * S / c = 2 * 6 / c = 2.4 ; ;

3. 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 -5 * c_1 + 5.76 = 0 ; ; ; ; c_1 = 3.2 ; ; c_2 = 1.8 ; ; ; ; a = sqrt{ c_1**2+h**2 } = sqrt{ 3.2**2+2.4**2 } = 4 ; ; b = sqrt{ c_2**2+h**2 } = sqrt{ 1.8**2+2.4**2 } = 3 ; ;


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 = 4 ; ; b = 3 ; ; c = 5 ; ;

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

p = a+b+c = 4+3+5 = 12 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12 }{ 2 } = 6 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6 * (6-4)(6-3)(6-5) } ; ; T = sqrt{ 36 } = 6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6 }{ 4 } = 3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6 }{ 3 } = 4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6 }{ 5 } = 2.4 ; ;

8. 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{ 4**2-3**2-5**2 }{ 2 * 3 * 5 } ) = 53° 7'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3**2-4**2-5**2 }{ 2 * 4 * 5 } ) = 36° 52'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5**2-4**2-3**2 }{ 2 * 3 * 4 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6 }{ 6 } = 1 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4 }{ 2 * sin 53° 7'48" } = 2.5 ; ;
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