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AAS Triangle Calculator

Table of contents

What is AAS triangle congruence?How to solve an AAS triangle — Angle and sidesHow to find the area of an AAS triangleHow to use the AAS triangle calculatorMore triangle calculatorsFAQs

Use the AAS triangle calculator to calculate the area and the rest of the dimensions of an AAS triangle.

In the accompanying text of this tool, we discuss:

  • The AAS triangle congruence;
  • How to solve an AAS triangle;
  • How to calculate the height for an AAS triangle; and
  • How to obtain the AAS triangle formula for area.

Enjoy! 🔺

What is AAS triangle congruence?

AAS stands for Angle-Angle-Side. An AAS triangle is a triangle in which one side aa, the opposite α\alpha, and adjacent β\beta angles are known.

AAS triangle: One side an oposite and adjacent angles.

In geometry, two figures or objects are said to be congruent if their shape and size are the same. Triangles classification in SAS, SSS, ASA, or AAS simplifies the study of triangle congruence.

In the case of AAS triangles, two triangles are congruent if two consecutive angles and the non-included side of one triangle are equivalent to the corresponding two angles and side of the second triangle. These are AAS congruent triangles.

How to solve an AAS triangle — Angle and sides

If we have an AAS triangle and want to know the other three dimensions: bb, cc, and γ\gamma, we can resort to some good old never-out-of-style trigonometry.

Oblique triangle with all its sides and angles indicated

Let's see how to calculate them:

Third angle γ\gamma

For this one, we need to remember that the internal angles of any triangle will always add π\pi or 180°180°. With this in mind, simply subtract the two known values of the angles from 180°180° to find the missing angle:

π=α+β+γγ=παβ\small \begin{aligned} \pi &= \alpha + \beta + \gamma \\ \gamma &= \pi - \alpha - \beta \end{aligned}

Sides bb and cc

For these two, we'll be using the law of sines:

asin(α)=bsin(β)=csin(γ)\cfrac{a}{\text{sin}(\alpha)} = \cfrac{b}{\text{sin}(\beta)} = \cfrac{c}{\text{sin}(\gamma)}

We may start by finding the expression to calculate the base bb, since the side aa, angles α\alpha, and β\beta are all known:

b=asin(β)sin(α)\small b = a \cdot \cfrac{\text{sin}(\beta)}{\text{sin}(\alpha)}

Similarly, given that we have an expression for the angle γ\gamma, we can use the following equation derived from the law of sines to determine the side cc:

c=asin(γ)sin(α)\small c = a \cdot \cfrac{\text{sin}(\gamma)}{\text{sin}(\alpha)}


c=bsin(γ)sin(β)\small c = b \cdot \cfrac{\text{sin}(\gamma)}{\text{sin}(\beta)}

How to find the area of an AAS triangle

To find the AAS triangle's area, we'll be using the general triangle area formula:

A=12bh\small \text{A} = \cfrac{1}{2} \cdot \text{b} \cdot \text{h}
Triangle with base and height

In the case of our AAS triangle, the initial known dimensions are the side aa and angles α\alpha and β\beta:

AAS triangle: One side an oposite and adjacent angles.

Notice from the area equation that we'll need to know the base bb and height hh to be able to calculate the area. In the previous section, we already mentioned how to calculate bb with the law of sines:

b=asin(β)sin(α)\small b = a \cdot \cfrac{\text{sin}(\beta)}{\text{sin}(\alpha)}

We need to figure out how to obtain the height hh. By drawing the height of the triangle, our triangle will be divided into two right triangles:

  • The triangle of hypotenuse aa and height hh; and
  • The triangle with hypotenuse cc and height hh.
Oblique triangle with all its sides and angles indicated

In this case, since we have the value for the side aa, we'll use the first triangle. By employing the expression for the sine of the angle γ\gamma, we can clear the height hh:

sin(γ)=hah=asin(γ)\small \begin{aligned} \text{sin}(\gamma) &=\cfrac{h}{a} \\ h &= a \cdot \text{sin}(\gamma) \end{aligned}

Even though γ\gamma is not initially provided, we already saw that we can easily obtain its value by subtracting α\alpha and β\beta from π\piγ=παβ\gamma = \pi - \alpha - \beta.

All that's left is to replace all of these expressions in our original area formula:

A=12(asin(β)sin(α))(asin(γ))A=a22sin(β)sin(γ)sin(α)\small \begin{aligned} A &= \cfrac{1}{2} \cdot \left(a \cdot \cfrac{\text{sin}(\beta)}{\text{sin}(\alpha)} \right) \cdot \left( a \cdot \text{sin}(\gamma) \right) \\ A &= \cfrac{a^2}{2} \cdot \cfrac{\text{sin}(\beta) \cdot \text{sin}(\gamma)}{\text{sin}(\alpha)} \end{aligned}

To express the area only in terms of the initial knowns, we can use the properties of sines: sin(πθ)=+ sin(θ)\text{sin}(\pi - \theta) = +\ \text{sin}(\theta). Therefore sin(γ)=sin(παβ)=+ sin(α+β)\text{sin}(\gamma) = \text{sin}(\pi - \alpha- \beta) = +\ \text{sin}(\alpha + \beta). Then the AAS triangle area formula is:

A=a22sin(β)sin(α+β)sin(α)\small A = \cfrac{a^2}{2} \cdot \cfrac{\text{sin}(\beta) \cdot \text{sin}(\alpha + \beta)}{\text{sin}(\alpha)}

How to use the AAS triangle calculator

Use the AAS triangle calculator to determine the area, third angle, and the two missing sides of this type of triangle. Let's have a look at how to use this tool:

  1. In the first section of the calculator, enter the known values of the AAS triangle. These are the two consecutive angles β and α and the non-included side a.
  2. In the section Results, the calculator will show you the results of the AAS triangle. Here you'll get the angle γ, other sides b and c, height h, and the Area of the triangle.
  3. You are done! 😀

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How do I know if a triangle is AAS or ASA?

AAS and ASA are triangles with two known angles and one known side. The difference is in the order of the knowns. In the AAS, one side and the opposite and adjacent angles are known. In the ASA, the knowns are two angles and the side between them.

How do I calculate the height of an AAS triangle?

If the known dimensions are the consecutive angles α and β and the non-included side a, to calculate the height h of an AAS triangle:

  1. Determine the missing angle γ:
    γ = π - α - β
  2. The formula for height reads:
    h = a × sin(γ)
  3. That's it!

P.S. When substituting values, make sure that all angles are in the same unit, either degrees or radians.

What is the area of an AAS triangle of 40°, 25° and 16 cm?

The area is 76.27 cm2. To calculate the area of an AAS triangle of dimensions a = 16 cm, α = 40° and β = 25°:

  1. Use the area formula:
    A = (1/2) × a² × sin(β) × sin(α+ β) / sin(α)
  2. Substitute the known values:
    A = (1/2) × (16 cm)² × sin(25°) × sin(40° + 25°) / sin(40°)
  3. Execute the calculations to obtain area's value:
    A = 76.27 cm²
Triangle area: triangle with two angles and side (AAS).

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Triangle with three sides and three angles.

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