How To Find Angles And Sides Of A Triangle

Triangle Figure
Bending-Side-Bending (ASA)

A = angle A
B = angle B
C = angle C
a = side a
b = side b
c = side c

P = perimeter
s = semi-perimeter
K = area
r = radius of inscribed circle
R = radius of confining circle

Calculator Use

Each calculation option, shown below, has sub-bullets that list the sequence of methods used in this calculator to solve for unknown bending and side values including Sum of Angles in a Triangle, Law of Sines and Police force of Cosines.  These are Not the ONLY sequences you could use to solve these types of problems.

• See also these Trigonometry Calculators:
• Law of Cosines Computer
• Constabulary of Sines Figurer

Solving Triangle Theorems

AAA is Angle, Bending, Bending

Specifying the three angles of a triangle does not uniquely identify ane triangle. Therefore, specifying two angles of a tringle allows y'all to summate the 3rd bending only.

Given the sizes of 2 angles of a triangle you can summate the size of the third angle. The total will equal 180° or π radians.

C = 180° - A - B (in degrees)

C = π - A - B (in radians)

AAS is Angle, Angle, Side

Given the size of two angles and 1 side opposite ane of the given angles, you lot tin summate the sizes of the remaining ane bending and two sides.

use the Sum of Angles Rule to notice the other angle, then

use The Law of Sines to solve for each of the other ii sides.

ASA is Angle, Side, Angle

Given the size of 2 angles and the size of the side that is in betwixt those 2 angles y'all can calculate the sizes of the remaining 1 angle and two sides.

use the Sum of Angles Rule to observe the other bending, then

use The Police of Sines to solve for each of the other two sides.

ASS (or SSA) is Angle, Side, Side

Given the size of 2 sides (a and c where a < c) and the size of the angle A that is non in betwixt those two sides yous might exist able to calculate the sizes of the remaining ane side and ii angles, depending on the post-obit conditions.

For A ≥ 90° (A ≥ π/two):

If a ≤ c there in that location are no possible triangles

Instance:

If a > c there is 1 possible solution

• utilize The Law of Sines to solve for angle C
• use the Sum of Angles Dominion to find the other angle, B
• use The Law of Sines to solve for the last side, b
• Example:
  

For A < xc° (A < π/ii):

If a ≥ c there is one possible solution

• utilize The Constabulary of Sines to solve for angle C
• utilise the Sum of Angles Rule to discover the other angle, B
• use The Law of Sines to solve for the last side, b
• Example:
  

If a < c we have 3 potential situations.  "If sin(A) < a/c, there are ii possible triangles satisfying the given conditions. If sin(A) = a/c, there is i possible triangle. If sin(A) > a/c, at that place are no possible triangles." ^[i]

sin(A) < a/c, at that place are two possible triangles

solve for the ii possible values of the third side b = c*cos(A) ± √[ a² - c^two sin² (A) ]^[1]

for each set of solutions, utilise The Law of Cosines to solve for each of the other two angles

present 2 full solutions

Example:

sin(A) = a/c, there is i possible triangle

use The Police force of Sines to solve for an angle, C

use the Sum of Angles Dominion to find the other angle, B

use The Law of Sines to solve for the last side, b

Example:

sin(A) > a/c, there are no possible triangles

Mistake Notice: sin(A) > a/c then there are no solutions and no triangle!

Case:

SAS is Side, Angle, Side

Given the size of 2 sides (c and a) and the size of the angle B that is in betwixt those 2 sides you tin calculate the sizes of the remaining one side and 2 angles.

use The Police of Cosines to solve for the remaining side, b

determine which side, a or c, is smallest and use the Police of Sines to solve for the size of the reverse angle, A or C respectively.^[two]

use the Sum of Angles Rule to discover the last angle

SSS is Side, Side, Side

Given the sizes of the 3 sides you can calculate the sizes of all 3 angles in the triangle.

utilise The Law of Cosines to solve for the angles.  You could also use the Sum of Angles Rule to find the final angle once you know 2 of them.

Sum of Angles in a Triangle

In Degrees A + B + C = 180°

In Radians A + B + C = π

Law of Sines

If a, b and c are the lengths of the legs of a triangle opposite to the angles A, B and C respectively; so the police of sines states:

a/sin A = b/sin B = c/sin C

Solving, for example, for an angle, A = sin^-1 [ a*sin(B) / b ]

Police force of Cosines

If a, b and c are the lengths of the legs of a triangle opposite to the angles A, B and C respectively; and so the law of cosines states:

aⁱⁱ = cⁱⁱ + bⁱⁱ - 2bc cos A,   solving for cos A,   cos A = ( b² + c² - aⁱⁱ ) / 2bc

bⁱⁱ = a² + c² - 2ca cos B,   solving for cos B,   cos B = ( cⁱⁱ + a² - b^two ) / 2ca

c² = b² + a^two - 2ab cos C,   solving for cos C,   cos C = ( a² + bⁱⁱ - c² ) / 2ab

Solving, for example, for an angle, A = cos^-one [ ( b^two + c² - a² ) / 2bc ]

Other Triangle Characteristics

Triangle perimeter, P = a + b + c

Triangle semi-perimeter, s = 0.5 * (a + b + c)

Triangle area, One thousand = √[ due south*(s-a)*(s-b)*(s-c)]

Radius of inscribed circumvolve in the triangle, r = √[ (southward-a)*(s-b)*(s-c) / southward ]

Radius of circumscribed circle around triangle, R = (abc) / (4K)

References/ Further Reading

[ane] Weisstein, Eric W. "Ass Theorem." From MathWorld-- A Wolfram Web Resource. ASS Theorem.

[2] Math is Fun - Solving SAS Triangles

Zwillinger, Daniel (Editor-in-Chief). CRC Standard Mathematical Tables and Formulae, 31st Edition New York, NY: CRC Press, p. 512, 2003.

Weisstein, Eric Due west. "Triangle Backdrop." From MathWorld-- A Wolfram Web Resource. Triangle Properties.

Math is Fun at Solving Triangles.

Source: https://www.calculatorsoup.com/calculators/geometry-plane/triangle-theorems.php

Posted by: hillneho1973.blogspot.com

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