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 Pythagorean triangle.

Sides: a = 84   b = 63   c = 105

Area: T = 2646
Perimeter: p = 252
Semiperimeter: s = 126

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 = 63
Height: hb = 84
Height: hc = 50.4

Median: ma = 75.71765767847
Median: mb = 89.71220393258
Median: mc = 52.5

Inradius: r = 21
Circumradius: R = 52.5

Vertex coordinates: A[105; 0] B[0; 0] C[67.2; 50.4]
Centroid: CG[57.4; 16.8]
Coordinates of the circumscribed circle: U[52.5; -0]
Coordinates of the inscribed circle: I[63; 21]

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: cathetus a and hypotenuse c

a = 84 ; ; c = 105 ; ;

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{ 105**2 - 84**2 } = 63 ; ;


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 = 84 ; ; b = 63 ; ; c = 105 ; ;

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

p = a+b+c = 84+63+105 = 252 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 252 }{ 2 } = 126 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 126 * (126-84)(126-63)(126-105) } ; ; T = sqrt{ 7001316 } = 2646 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2646 }{ 84 } = 63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2646 }{ 63 } = 84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2646 }{ 105 } = 50.4 ; ;

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{ 84**2-63**2-105**2 }{ 2 * 63 * 105 } ) = 53° 7'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 63**2-84**2-105**2 }{ 2 * 84 * 105 } ) = 36° 52'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 105**2-84**2-63**2 }{ 2 * 63 * 84 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2646 }{ 126 } = 21 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 84 }{ 2 * sin 53° 7'48" } = 52.5 ; ;
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