Triangle calculator

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

You have entered side a, c and angle β.

Right scalene triangle.

Sides: a = 32   b = 116.4821758228   c = 112

Area: T = 1792
Perimeter: p = 260.4821758228
Semiperimeter: s = 130.2410879114

Angle ∠ A = α = 15.94553959009° = 15°56'43″ = 0.2788299659 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 74.05546040991° = 74°3'17″ = 1.29224966678 rad

Height: ha = 112
Height: hb = 30.76987663245
Height: hc = 32

Median: ma = 113.137708499
Median: mb = 58.24108791142
Median: mc = 64.49880619864

Inradius: r = 13.75991208858
Circumradius: R = 58.24108791142

Vertex coordinates: A[112; 0] B[0; 0] C[-0; 32]
Centroid: CG[37.33333333333; 10.66766666667]
Coordinates of the circumscribed circle: U[56; 16]
Coordinates of the inscribed circle: I[13.75991208858; 13.75991208858]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164.0554604099° = 164°3'17″ = 0.2788299659 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 105.9455395901° = 105°56'43″ = 1.29224966678 rad

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How did we calculate this triangle?

1. Input data entered: side a, c and angle β.

a = 32 ; ; c = 112 ; ; beta = 90° ; ;

2. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 32**2+112**2 - 2 * 32 * 112 * cos(90° ) } ; ; b = 116.48 ; ;


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 = 32 ; ; b = 116.48 ; ; c = 112 ; ;

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

p = a+b+c = 32+116.48+112 = 260.48 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 260.48 }{ 2 } = 130.24 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 130.24 * (130.24-32)(130.24-116.48)(130.24-112) } ; ; T = sqrt{ 3211264 } = 1792 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1792 }{ 32 } = 112 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1792 }{ 116.48 } = 30.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1792 }{ 112 } = 32 ; ;

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{ 32**2-116.48**2-112**2 }{ 2 * 116.48 * 112 } ) = 15° 56'43" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 116.48**2-32**2-112**2 }{ 2 * 32 * 112 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 112**2-32**2-116.48**2 }{ 2 * 116.48 * 32 } ) = 74° 3'17" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1792 }{ 130.24 } = 13.76 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 32 }{ 2 * sin 15° 56'43" } = 58.24 ; ;




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