Triangle calculator

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

You have entered side a, b and area T.

Triangle has two solutions: a=5.2; b=5.2; c=10.28329442806 and a=5.2; b=5.2; c=1.55659745889.

#1 Obtuse isosceles triangle.

Sides: a = 5.2   b = 5.2   c = 10.28329442806

Area: T = 4
Perimeter: p = 20.68329442806
Semiperimeter: s = 10.34114721403

Angle ∠ A = α = 8.6044496611° = 8°36'16″ = 0.15501767963 rad
Angle ∠ B = β = 8.6044496611° = 8°36'16″ = 0.15501767963 rad
Angle ∠ C = γ = 162.7911006778° = 162°47'28″ = 2.84112390609 rad

Height: ha = 1.53884615385
Height: hb = 1.53884615385
Height: hc = 0.77879872945

Median: ma = 7.72220121432
Median: mb = 7.72220121432
Median: mc = 0.77879872945

Inradius: r = 0.38767921265
Circumradius: R = 17.37881758343

Vertex coordinates: A[10.28329442806; 0] B[0; 0] C[5.14114721403; 0.77879872945]
Centroid: CG[5.14114721403; 0.25993290982]
Coordinates of the circumscribed circle: U[5.14114721403; -16.66001885398]
Coordinates of the inscribed circle: I[5.14114721403; 0.38767921265]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 171.3965503389° = 171°23'44″ = 0.15501767963 rad
∠ B' = β' = 171.3965503389° = 171°23'44″ = 0.15501767963 rad
∠ C' = γ' = 17.20989932219° = 17°12'32″ = 2.84112390609 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 = 5.2 ; ; b = 5.2 ; ; c = 10.28 ; ;

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

p = a+b+c = 5.2+5.2+10.28 = 20.68 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.68 }{ 2 } = 10.34 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.34 * (10.34-5.2)(10.34-5.2)(10.34-10.28) } ; ; T = sqrt{ 16 } = 4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4 }{ 5.2 } = 1.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4 }{ 5.2 } = 1.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4 }{ 10.28 } = 0.78 ; ;

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{ 5.2**2-5.2**2-10.28**2 }{ 2 * 5.2 * 10.28 } ) = 8° 36'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.2**2-5.2**2-10.28**2 }{ 2 * 5.2 * 10.28 } ) = 8° 36'16" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.28**2-5.2**2-5.2**2 }{ 2 * 5.2 * 5.2 } ) = 162° 47'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4 }{ 10.34 } = 0.39 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.2 }{ 2 * sin 8° 36'16" } = 17.38 ; ;





#2 Acute isosceles triangle.

Sides: a = 5.2   b = 5.2   c = 1.55659745889

Area: T = 4
Perimeter: p = 11.95659745889
Semiperimeter: s = 5.97879872945

Angle ∠ A = α = 81.3965503389° = 81°23'44″ = 1.42106195305 rad
Angle ∠ B = β = 81.3965503389° = 81°23'44″ = 1.42106195305 rad
Angle ∠ C = γ = 17.20989932219° = 17°12'32″ = 0.33003535927 rad

Height: ha = 1.53884615385
Height: hb = 1.53884615385
Height: hc = 5.14114721403

Median: ma = 2.82332124363
Median: mb = 2.82332124363
Median: mc = 5.14114721403

Inradius: r = 0.66991215292
Circumradius: R = 2.63295970553

Vertex coordinates: A[1.55659745889; 0] B[0; 0] C[0.77879872945; 5.14114721403]
Centroid: CG[0.77879872945; 1.71438240468]
Coordinates of the circumscribed circle: U[0.77879872945; 2.5121875085]
Coordinates of the inscribed circle: I[0.77879872945; 0.66991215292]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 98.6044496611° = 98°36'16″ = 1.42106195305 rad
∠ B' = β' = 98.6044496611° = 98°36'16″ = 1.42106195305 rad
∠ C' = γ' = 162.7911006778° = 162°47'28″ = 0.33003535927 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 = 5.2 ; ; b = 5.2 ; ; c = 1.56 ; ;

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

p = a+b+c = 5.2+5.2+1.56 = 11.96 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.96 }{ 2 } = 5.98 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.98 * (5.98-5.2)(5.98-5.2)(5.98-1.56) } ; ; T = sqrt{ 16 } = 4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4 }{ 5.2 } = 1.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4 }{ 5.2 } = 1.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4 }{ 1.56 } = 5.14 ; ;

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{ 5.2**2-5.2**2-1.56**2 }{ 2 * 5.2 * 1.56 } ) = 81° 23'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.2**2-5.2**2-1.56**2 }{ 2 * 5.2 * 1.56 } ) = 81° 23'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.56**2-5.2**2-5.2**2 }{ 2 * 5.2 * 5.2 } ) = 17° 12'32" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4 }{ 5.98 } = 0.67 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.2 }{ 2 * sin 81° 23'44" } = 2.63 ; ;




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