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?

1. Input data entered: side a, b and area T.

a = 5.2 ; ; b = 5.2 ; ; T = 4 ; ;

2. From area T and side a we calculate height ha - The area of the triangle is the product of the length of the base and the height divided by two:

T = fraction{ a h_a }{ 2 } ; ; ; ; h_a = fraction{ 2 T }{ a } = fraction{ 2 * 4 }{ 5.2 } = 1.538 ; ;
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 ; ;

3. 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 ; ;

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 5.2**2+10.28**2-5.2**2 }{ 2 * 5.2 * 10.28 } ) = 8° 36'16" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.2**2+10.28**2-5.2**2 }{ 2 * 5.2 * 10.28 } ) = 8° 36'16" ; ;
 gamma = 180° - alpha - beta = 180° - 8° 36'16" - 8° 36'16" = 162° 47'28" ; ;

8. Inradius

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

9. Circumradius

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

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.2**2+2 * 10.28**2 - 5.2**2 } }{ 2 } = 7.722 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.28**2+2 * 5.2**2 - 5.2**2 } }{ 2 } = 7.722 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.2**2+2 * 5.2**2 - 10.28**2 } }{ 2 } = 0.778 ; ;



#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?

1. Input data entered: side a, b and area T.

a = 5.2 ; ; b = 5.2 ; ; T = 4 ; ; : Nr. 1

2. From area T and side a we calculate height ha - The area of the triangle is the product of the length of the base and the height divided by two:

T = fraction{ a h_a }{ 2 } ; ; ; ; h_a = fraction{ 2 T }{ a } = fraction{ 2 * 4 }{ 5.2 } = 1.538 ; ; : Nr. 1
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 ; ;

3. 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 ; ;

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 5.2**2+1.56**2-5.2**2 }{ 2 * 5.2 * 1.56 } ) = 81° 23'44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.2**2+1.56**2-5.2**2 }{ 2 * 5.2 * 1.56 } ) = 81° 23'44" ; ;
 gamma = 180° - alpha - beta = 180° - 81° 23'44" - 81° 23'44" = 17° 12'32" ; ;

8. Inradius

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

9. Circumradius

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

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.2**2+2 * 1.56**2 - 5.2**2 } }{ 2 } = 2.823 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.56**2+2 * 5.2**2 - 5.2**2 } }{ 2 } = 2.823 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.2**2+2 * 5.2**2 - 1.56**2 } }{ 2 } = 5.141 ; ;
Calculate another triangle


Look also our friend's collection of math examples and problems:

See more information about triangles or more details on solving triangles.