# ASA Triangle Calculator

Created by Gabriela Diaz
Reviewed by Łucja Zaborowska, MD, PhD candidate
Last updated: Oct 11, 2022

With Omni's ASA triangle calculator you'll be able to determine the area, remaining sides, and angle of an ASA triangle 🔺

Are you interested in finding out how to obtain the formula to calculate the area of this type of triangle? Keep on reading and discover answers to all these questions and more:

• What is an ASA triangle?;
• What is ASA triangle congruence?; and
• How to solve an ASA triangle.

## ASA triangle congruence

An ASA triangle is an oblique triangle in which two angles $\beta$ and $\gamma$ and the side $a$ in between them are known. ASA stands for angle-side-angle.

With these dimensions and by use of some trigonometry is possible to determine its area, third angle, and the other two sides.

ASA and other types of oblique triangles, such as SAS, SSS, or AAS, are utilized to study triangle congruence. If all three corresponding sides and angles are equal in measure, two triangles are said to be congruent.

In the case of ASA triangles, two triangles are congruent if any two angles and the side between them of one triangle are equivalent to the corresponding two angles and side of the second triangle. These are ASA congruent triangles.

## How to find the area of an ASA triangle - Formula for area

To find the area of an ASA triangle, we'll use the general triangle area formula and some trigonometry. First, let's start by refreshing the triangle area formula:

$\small \text{A} = \cfrac{1}{2} \cdot \text{b} \cdot \text{h}$

In the case of an ASA triangle where we know the angles $\beta$ and $\gamma$ and the side $a$, we'll need to determine its base $b$ and height $h$. But how can we do this? 🤔

Remember that we just mentioned something about trigonometry right? Well, this is where we'll start using it. Fear not, I promise it'll be simple.

First, let's determine the height $h$. If we draw the height of the triangle, our oblique triangle will be divided into two right triangles:

• One of them being the triangle of hypotenuse $a$, height $h$ and base $b - x$; and
• The other one is with hypotenuse $c$, height $h$, and base $x$.

To calculate $h$, we can use either of these triangles. For our solution, we'll be using the first one. By employing the expression to calculate the sine of the angle $\gamma$, we can clear the height $h$:

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

Now let's see how to calculate the base $b$ of our triangle. For this, we'll use the law of sines:

$\small \cfrac{a}{\text{sin}(\alpha)} = \cfrac{b}{\text{sin}(\beta)} = \cfrac{c}{\text{sin}(\gamma)}$

Since $a$ and $\beta$ are known, and we are interested in $b$, we'll use the $a$ $\alpha$ and $b$ $\beta$ ratios. From which the expression to calculate the base $b$ is:

$\small b = a \cdot \cfrac{\text{sin}(\beta)}{\text{sin}(\alpha)}$

We still need the value of the angle $\alpha$. Remembering that the sum of the interior angles of a triangle is always equal to $\pi$ or $180°$:

\small \begin{aligned} \pi &=\alpha + \beta + \gamma \\ \alpha &= \pi - \beta - \gamma \end{aligned}

Finally, we replace all of these expressions in our original area formula:

\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) \\[1.5em] A &= \cfrac{a^2}{2} \cdot \cfrac{\text{sin}(\beta) \cdot \text{sin}(\gamma)}{\text{sin}(\alpha)} \end{aligned}

According to the properties of sines, $\text{sin}(\pi - \theta) = +\ \text{sin}(\theta)$. Thus $\text{sin}(\alpha) = \text{sin}(\pi - \beta - \gamma) = +\ \text{sin}(\beta + \gamma)$, then the ASA triangle formula for area is:

$\small A = \cfrac{a^2}{2} \cdot \cfrac{\text{sin}(\beta) \cdot \text{sin}(\gamma)}{\text{sin}(\beta + \gamma)}$

## Find the sides of an ASA triangle - Example of an ASA triangle

We've already showed you how to obtain one of the sides of an ASA triangle using the law of sines, step by step. We established that the expression to calculate the length of the side $b$ is given by:

$\small b = a \cdot \cfrac{\text{sin}(\beta)}{\text{sin}(\beta + \gamma)}$

Where the length $a$ and angles $\beta$ and $\gamma$ are known.

Similarly, we can obtain an expression to determine the side $c$. Let's return to the law of sines. This time, we'll use the ratios that contain the side $c$ and angle $\gamma$:

$\small \cfrac{a}{\text{sin}(\alpha)} = \cfrac{c}{\text{sin}(\gamma)}$

From here, we can obtain the length of $c$ with:

$\small c = a \cdot \cfrac{\text{sin}(\gamma)}{\text{sin}(\beta + \gamma)}$

Let's employ these expressions and see how to solve an ASA triangle of known: $a = 5 \ \text{cm}$, $\beta = 50°$ and $\gamma = 32°$:

1. To calculate the length of the side $b$, substitute these dimensions in the respective formula from above:
\qquad \small \begin{aligned} b &= a \cdot \cfrac{\text{sin}(\beta)}{\text{sin}(\beta + \gamma)} \\ b &= 5 \ \text{cm} \cdot \cfrac{\text{sin}(\, 50°)}{\text{sin}(\, 50° + 32°)} \\ b &= 3.868 \ \text{cm} \end{aligned}
1. Likewise, we calculate the side $c$:
\qquad \small \begin{aligned} c &= a \cdot \cfrac{\text{sin}(\gamma)}{\text{sin}(\beta + \gamma)} \\ c &= 5 \ \text{cm} \cdot \cfrac{\text{sin}(\, 32°)}{\text{sin}(\, 50° + 32°)} \\ c &= 2.6756 \ \text{cm} \end{aligned}

## How to use the ASA triangle calculator

With the ASA triangle calculator, you'll be able to determine the area and the rest of the dimensions of this type of oblique triangle. You'll see that using this tool is pretty simple:

1. In the first section of the calculator, enter the known angles β and γ and side a of the triangle.
2. In the section below, the calculator will display the results of your ASA triangle. Here you'll find the angle α, other sides b and c, and the area of the triangle.
3. Yes, that's it! We made it that simple 😉

## Let's keep solving triangles!

If you found the ASA triangle calculator interesting, you might like to take a look at some of our other triangles calculators:

## FAQ

### What is the difference between ASA and AAS triangle?

The difference between ASA and AAS triangles is that ASA refers to a triangle in which two angles and the side between them are known. In the AAS triangle, one side and two angles are known, one of them being opposite and the other adjacent to the known angle.

### What is the area of an ASA triangle of 7 cm, 71° and 34°?

The area is 13.41 cm2. To obtain the area of an ASA triangle with dimensions a= 7 cm, β= 34° and γ= 71°:

1. Use the area formula:
A = (1/2) × a² × sin(β) × sin(γ)/ sin(β + γ)
2. Substitute the known values:
A = (1/2) × (7 cm)² × sin(34°) × sin(71°)/ sin(34° + 71°)
3. Perform the calculations to determine the area:
A = 13. 41 cm²

### How do I find the remaining sides of an ASA triangle?

To find the remaining sides of an ASA triangle which angles β and γ and side a are known:

1. Calculate the unknown angle α:
α = π - β - γ
2. With the law of sines, determine the side b:
b = a × sin(β) / sin(α)
3. To obtain the side c, use the law of sines once again:
c = a × sin(γ) / sin(α)
Gabriela Diaz
Enter dimensions of ASA triangle

Angle β
deg
a
in
Angle γ
deg
Results

Angle α
deg
b
in
c
in
Area
in²
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