Triangle calculator

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

You have entered side a, b and area S.

Triangle has two solutions: a=8.94; b=7.5; c=14.99217714647 and a=8.94; b=7.5; c=6.8998839638.

#1 Obtuse scalene triangle.

Sides: a = 8.94   b = 7.5   c = 14.99217714647

Area: T = 25.17
Perimeter: p = 31.43217714647
Semiperimeter: s = 15.71658857323

Angle ∠ A = α = 26.59769998929° = 26°35'49″ = 0.46442052193 rad
Angle ∠ B = β = 22.06112492706° = 22°3'40″ = 0.38550414369 rad
Angle ∠ C = γ = 131.3421750837° = 131°20'30″ = 2.29223459974 rad

Height: ha = 5.63108724832
Height: hb = 6.712
Height: hc = 3.35878420081

Median: ma = 10.9788192284
Median: mb = 11.75990775924
Median: mc = 3.4499419819

Inradius: r = 1.60215642025
Circumradius: R = 9.98440909486

Vertex coordinates: A[14.99217714647; 0] B[0; 0] C[8.28554388567; 3.35878420081]
Centroid: CG[7.75990701071; 1.11992806694]
Coordinates of the circumscribed circle: U[7.49658857323; -6.59549806033]
Coordinates of the inscribed circle: I[8.21658857323; 1.60215642025]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.4033000107° = 153°24'11″ = 0.46442052193 rad
∠ B' = β' = 157.9398750729° = 157°56'20″ = 0.38550414369 rad
∠ C' = γ' = 48.65882491635° = 48°39'30″ = 2.29223459974 rad




How did we calculate this triangle?

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

a = 8.94 ; ; b = 7.5 ; ; S = 25.17 ; ;

2. From area T and side a we calculate h_a - 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 * 25.17 }{ 8.94 } = 5.631 ; ;


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 = 8.94 ; ; b = 7.5 ; ; c = 14.99 ; ;

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

p = a+b+c = 8.94+7.5+14.99 = 31.43 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.43 }{ 2 } = 15.72 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.72 * (15.72-8.94)(15.72-7.5)(15.72-14.99) } ; ; T = sqrt{ 633.53 } = 25.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 25.17 }{ 8.94 } = 5.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 25.17 }{ 7.5 } = 6.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 25.17 }{ 14.99 } = 3.36 ; ;

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{ 7.5**2+14.99**2-8.94**2 }{ 2 * 7.5 * 14.99 } ) = 26° 35'49" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.94**2+14.99**2-7.5**2 }{ 2 * 8.94 * 14.99 } ) = 22° 3'40" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 8.94**2+7.5**2-14.99**2 }{ 2 * 8.94 * 7.5 } ) = 131° 20'30" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 25.17 }{ 15.72 } = 1.6 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.94 }{ 2 * sin 26° 35'49" } = 9.98 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.5**2+2 * 14.99**2 - 8.94**2 } }{ 2 } = 10.978 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.99**2+2 * 8.94**2 - 7.5**2 } }{ 2 } = 11.759 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.5**2+2 * 8.94**2 - 14.99**2 } }{ 2 } = 3.449 ; ;







#2 Acute scalene triangle.

Sides: a = 8.94   b = 7.5   c = 6.8998839638

Area: T = 25.17
Perimeter: p = 23.3398839638
Semiperimeter: s = 11.6699419819

Angle ∠ A = α = 76.63549934795° = 76°38'6″ = 1.33875329585 rad
Angle ∠ B = β = 54.7076757357° = 54°42'24″ = 0.9554813039 rad
Angle ∠ C = γ = 48.65882491635° = 48°39'30″ = 0.84992466562 rad

Height: ha = 5.63108724832
Height: hb = 6.712
Height: hc = 7.29768792785

Median: ma = 5.65216452627
Median: mb = 7.05495598568
Median: mc = 7.49658857323

Inradius: r = 2.15769195719
Circumradius: R = 4.59444298542

Vertex coordinates: A[6.8998839638; 0] B[0; 0] C[5.16551866176; 7.29768792785]
Centroid: CG[4.02113420852; 2.43222930928]
Coordinates of the circumscribed circle: U[3.4499419819; 3.03548457289]
Coordinates of the inscribed circle: I[4.1699419819; 2.15769195719]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103.365500652° = 103°21'54″ = 1.33875329585 rad
∠ B' = β' = 125.2933242643° = 125°17'36″ = 0.9554813039 rad
∠ C' = γ' = 131.3421750837° = 131°20'30″ = 0.84992466562 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 8.94 ; ; b = 7.5 ; ; S = 25.17 ; ; : Nr. 1

2. From area T and side a we calculate h_a - 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 * 25.17 }{ 8.94 } = 5.631 ; ; : 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 = 8.94 ; ; b = 7.5 ; ; c = 6.9 ; ;

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

p = a+b+c = 8.94+7.5+6.9 = 23.34 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.34 }{ 2 } = 11.67 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.67 * (11.67-8.94)(11.67-7.5)(11.67-6.9) } ; ; T = sqrt{ 633.53 } = 25.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 25.17 }{ 8.94 } = 5.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 25.17 }{ 7.5 } = 6.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 25.17 }{ 6.9 } = 7.3 ; ;

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{ 7.5**2+6.9**2-8.94**2 }{ 2 * 7.5 * 6.9 } ) = 76° 38'6" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.94**2+6.9**2-7.5**2 }{ 2 * 8.94 * 6.9 } ) = 54° 42'24" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 8.94**2+7.5**2-6.9**2 }{ 2 * 8.94 * 7.5 } ) = 48° 39'30" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 25.17 }{ 11.67 } = 2.16 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.94 }{ 2 * sin 76° 38'6" } = 4.59 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.5**2+2 * 6.9**2 - 8.94**2 } }{ 2 } = 5.652 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.9**2+2 * 8.94**2 - 7.5**2 } }{ 2 } = 7.05 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.5**2+2 * 8.94**2 - 6.9**2 } }{ 2 } = 7.496 ; ;
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