Triangle calculator

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

You have entered side a, b and angle γ.

Acute scalene triangle.

Sides: a = 39   b = 114   c = 111.8788216042

Area: T = 2166.025465402
Perimeter: p = 264.8788216042
Semiperimeter: s = 132.4399108021

Angle ∠ A = α = 19.85660941825° = 19°51'22″ = 0.34765542201 rad
Angle ∠ B = β = 83.14439058175° = 83°8'38″ = 1.45111349095 rad
Angle ∠ C = γ = 77° = 1.3443903524 rad

Height: ha = 111.0788187386
Height: hb = 388.0004325266
Height: hc = 38.72111153457

Median: ma = 111.2488000487
Median: mb = 61.39992476533
Median: mc = 64.26597556314

Inradius: r = 16.35548719588
Circumradius: R = 57.41105363483

Vertex coordinates: A[111.8788216042; 0] B[0; 0] C[4.6565666052; 38.72111153457]
Centroid: CG[38.84546273647; 12.90770384486]
Coordinates of the circumscribed circle: U[55.93991080211; 12.9154560682]
Coordinates of the inscribed circle: I[18.43991080211; 16.35548719588]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.1443905818° = 160°8'38″ = 0.34765542201 rad
∠ B' = β' = 96.85660941825° = 96°51'22″ = 1.45111349095 rad
∠ C' = γ' = 103° = 1.3443903524 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and angle γ.

a = 39 ; ; b = 114 ; ; gamma = 77° ; ;

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

c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 39**2+114**2 - 2 * 39 * 114 * cos(77° ) } ; ; c = 111.88 ; ;


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 = 39 ; ; b = 114 ; ; c = 111.88 ; ;

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

p = a+b+c = 39+114+111.88 = 264.88 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 264.88 }{ 2 } = 132.44 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 132.44 * (132.44-39)(132.44-114)(132.44-111.88) } ; ; T = sqrt{ 4691662.8 } = 2166.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2166.02 }{ 39 } = 111.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2166.02 }{ 114 } = 38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2166.02 }{ 111.88 } = 38.72 ; ;

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{ 39**2-114**2-111.88**2 }{ 2 * 114 * 111.88 } ) = 19° 51'22" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 114**2-39**2-111.88**2 }{ 2 * 39 * 111.88 } ) = 83° 8'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 111.88**2-39**2-114**2 }{ 2 * 114 * 39 } ) = 77° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2166.02 }{ 132.44 } = 16.35 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39 }{ 2 * sin 19° 51'22" } = 57.41 ; ;




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