Triangle calculator

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

You have entered side a, b and angle β.

Triangle has two solutions: a=54; b=49; c=7.03329551737 and a=54; b=49; c=73.22766859779.

#1 Obtuse scalene triangle.

Sides: a = 54   b = 49   c = 7.03329551737

Area: T = 127.0611070116
Perimeter: p = 110.0332955174
Semiperimeter: s = 55.01664775868

Angle ∠ A = α = 132.4898812327° = 132°29'20″ = 2.31223659972 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 5.51111876725° = 5°30'40″ = 0.09661883706 rad

Height: ha = 4.70659655599
Height: hb = 5.18661661272
Height: hc = 36.13330527434

Median: ma = 22.27662480961
Median: mb = 29.70765856207
Median: mc = 51.44105908343

Inradius: r = 2.31095093632
Circumradius: R = 36.61546754717

Vertex coordinates: A[7.03329551737; 0] B[0; 0] C[40.13298205758; 36.13330527434]
Centroid: CG[15.72109252498; 12.04443509145]
Coordinates of the circumscribed circle: U[3.51664775868; 36.44554228303]
Coordinates of the inscribed circle: I[6.01664775868; 2.31095093632]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 47.51111876725° = 47°30'40″ = 2.31223659972 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 174.4898812327° = 174°29'20″ = 0.09661883706 rad




How did we calculate this triangle?

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 = 54 ; ; b = 49 ; ; c = 7.03 ; ;

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

p = a+b+c = 54+49+7.03 = 110.03 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 110.03 }{ 2 } = 55.02 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 55.02 * (55.02-54)(55.02-49)(55.02-7.03) } ; ; T = sqrt{ 16144.52 } = 127.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 127.06 }{ 54 } = 4.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 127.06 }{ 49 } = 5.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 127.06 }{ 7.03 } = 36.13 ; ;

5. 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{ 54**2-49**2-7.03**2 }{ 2 * 49 * 7.03 } ) = 132° 29'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 49**2-54**2-7.03**2 }{ 2 * 54 * 7.03 } ) = 42° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.03**2-54**2-49**2 }{ 2 * 49 * 54 } ) = 5° 30'40" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 127.06 }{ 55.02 } = 2.31 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 54 }{ 2 * sin 132° 29'20" } = 36.61 ; ;





#2 Obtuse scalene triangle.

Sides: a = 54   b = 49   c = 73.22766859779

Area: T = 1322.952185333
Perimeter: p = 176.2276685978
Semiperimeter: s = 88.11333429889

Angle ∠ A = α = 47.51111876725° = 47°30'40″ = 0.82992266564 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 90.48988123275° = 90°29'20″ = 1.57993277113 rad

Height: ha = 48.998821679
Height: hb = 53.99880348298
Height: hc = 36.13330527434

Median: ma = 56.14877850823
Median: mb = 59.48880136637
Median: mc = 36.30437617221

Inradius: r = 15.01442056635
Circumradius: R = 36.61546754717

Vertex coordinates: A[73.22766859779; 0] B[0; 0] C[40.13298205758; 36.13330527434]
Centroid: CG[37.78655021846; 12.04443509145]
Coordinates of the circumscribed circle: U[36.61333429889; -0.31223700869]
Coordinates of the inscribed circle: I[39.11333429889; 15.01442056635]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.4898812327° = 132°29'20″ = 0.82992266564 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 89.51111876725° = 89°30'40″ = 1.57993277113 rad

Calculate another triangle

How did we calculate this triangle?

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 = 54 ; ; b = 49 ; ; c = 73.23 ; ;

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

p = a+b+c = 54+49+73.23 = 176.23 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 176.23 }{ 2 } = 88.11 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 88.11 * (88.11-54)(88.11-49)(88.11-73.23) } ; ; T = sqrt{ 1750201.61 } = 1322.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1322.95 }{ 54 } = 49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1322.95 }{ 49 } = 54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1322.95 }{ 73.23 } = 36.13 ; ;

5. 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{ 54**2-49**2-73.23**2 }{ 2 * 49 * 73.23 } ) = 47° 30'40" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 49**2-54**2-73.23**2 }{ 2 * 54 * 73.23 } ) = 42° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 73.23**2-54**2-49**2 }{ 2 * 49 * 54 } ) = 90° 29'20" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1322.95 }{ 88.11 } = 15.01 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 54 }{ 2 * sin 47° 30'40" } = 36.61 ; ;




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