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=800; b=250; c=560.0187787443 and a=800; b=250; c=1031.217724515.

#1 Obtuse scalene triangle.

Sides: a = 800   b = 250   c = 560.0187787443

Area: T = 23415.11994896
Perimeter: p = 1610.018778744
Semiperimeter: s = 805.0098893722

Angle ∠ A = α = 160.4588406089° = 160°27'30″ = 2.80105274988 rad
Angle ∠ B = β = 6° = 0.10547197551 rad
Angle ∠ C = γ = 13.54215939107° = 13°32'30″ = 0.23663453997 rad

Height: ha = 58.5387798724
Height: hb = 187.3210955917
Height: hc = 83.62327706141

Median: ma = 167.5111077623
Median: mb = 679.1066001392
Median: mc = 522.3465689593

Inradius: r = 29.08767836023
Circumradius: R = 1195.847652919

Vertex coordinates: A[560.0187787443; 0] B[0; 0] C[795.6187516295; 83.62327706141]
Centroid: CG[451.8788434579; 27.87442568714]
Coordinates of the circumscribed circle: U[280.0098893722; 1162.602222811]
Coordinates of the inscribed circle: I[555.0098893722; 29.08767836023]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 19.54215939107° = 19°32'30″ = 2.80105274988 rad
∠ B' = β' = 174° = 0.10547197551 rad
∠ C' = γ' = 166.4588406089° = 166°27'30″ = 0.23663453997 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 = 800 ; ; b = 250 ; ; c = 560.02 ; ;

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

p = a+b+c = 800+250+560.02 = 1610.02 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1610.02 }{ 2 } = 805.01 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 805.01 * (805.01-800)(805.01-250)(805.01-560.02) } ; ; T = sqrt{ 548267820.71 } = 23415.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 23415.12 }{ 800 } = 58.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 23415.12 }{ 250 } = 187.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 23415.12 }{ 560.02 } = 83.62 ; ;

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{ 800**2-250**2-560.02**2 }{ 2 * 250 * 560.02 } ) = 160° 27'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 250**2-800**2-560.02**2 }{ 2 * 800 * 560.02 } ) = 6° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 560.02**2-800**2-250**2 }{ 2 * 250 * 800 } ) = 13° 32'30" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 23415.12 }{ 805.01 } = 29.09 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 800 }{ 2 * sin 160° 27'30" } = 1195.85 ; ;





#2 Obtuse scalene triangle.

Sides: a = 800   b = 250   c = 1031.217724515

Area: T = 43116.62215721
Perimeter: p = 2081.217724515
Semiperimeter: s = 1040.609862257

Angle ∠ A = α = 19.54215939107° = 19°32'30″ = 0.34110651548 rad
Angle ∠ B = β = 6° = 0.10547197551 rad
Angle ∠ C = γ = 154.4588406089° = 154°27'30″ = 2.69658077436 rad

Height: ha = 107.792155393
Height: hb = 344.9332972577
Height: hc = 83.62327706141

Median: ma = 634.7876974774
Median: mb = 914.3743831287
Median: mc = 292.2298931368

Inradius: r = 41.43440421911
Circumradius: R = 1195.847652919

Vertex coordinates: A[1031.217724515; 0] B[0; 0] C[795.6187516295; 83.62327706141]
Centroid: CG[608.945492048; 27.87442568714]
Coordinates of the circumscribed circle: U[515.6098622573; -1078.97994575]
Coordinates of the inscribed circle: I[790.6098622573; 41.43440421911]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.4588406089° = 160°27'30″ = 0.34110651548 rad
∠ B' = β' = 174° = 0.10547197551 rad
∠ C' = γ' = 25.54215939107° = 25°32'30″ = 2.69658077436 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 = 800 ; ; b = 250 ; ; c = 1031.22 ; ;

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

p = a+b+c = 800+250+1031.22 = 2081.22 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2081.22 }{ 2 } = 1040.61 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1040.61 * (1040.61-800)(1040.61-250)(1040.61-1031.22) } ; ; T = sqrt{ 1859043055.79 } = 43116.62 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 43116.62 }{ 800 } = 107.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 43116.62 }{ 250 } = 344.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 43116.62 }{ 1031.22 } = 83.62 ; ;

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{ 800**2-250**2-1031.22**2 }{ 2 * 250 * 1031.22 } ) = 19° 32'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 250**2-800**2-1031.22**2 }{ 2 * 800 * 1031.22 } ) = 6° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1031.22**2-800**2-250**2 }{ 2 * 250 * 800 } ) = 154° 27'30" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 43116.62 }{ 1040.61 } = 41.43 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 800 }{ 2 * sin 19° 32'30" } = 1195.85 ; ;




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