Two sides and an angle are enough to uniquely define a triangle: learn why and how to calculate the remaining quantities in a triangle with our SAS triangle calculator.

With our SAS triangle calculator, you will learn:

  • What is a SAS triangle;
  • How to calculate the missing side and angles in a SAS triangle;
  • How to calculate the area of a SAS triangle and its perimeter;
  • The rules of congruence in SAS triangles.

You won't learn about sassy triangles, though.

What's so special about triangles?

Triangles are rather special polygons. Undeformable, ubiquitous, neverending source of mathematical theorems, hell, they constantly appear in trigonometry, too!

One of the reasons for these peculiarities is that sides and vertices of a triangle are related by strong constraints. For example, the interior angles of a triangle always sum to 180°180\degree:

α+β+γ=180°\alpha+\beta+\gamma = 180\degree

Also, the small number of vertices makes it possible to fix the distances in a triangle by assigning the values of just two sides. Once the values are set, the triangle is uniquely identified. We will talk later about another important concept in the study of triangles, congruence.

We can calculate the entire set of characteristic quantities of a triangle in many ways. Here we will focus on the calculation of a SAS triangle. We are not talking of British special forces but of a specific set of known values.

What is a SAS triangle?

If you define the values of two adjacent sides and the angle between them you've given enough material to calculate everything in a triangle.

SAS triangle stands for side-angle-side triangle. There's another possible combination of two sides and an angle, the "SSA". In the first case, the angle is included between the two sides. The latter case may allow for multiple solutions; thus, we are not going to deal with it in this tool. Be careful not to shuffle the letters around one more time!

How to solve a SAS triangle

To solve a SAS triangle, we need to use both trigonometric rules of a triangle. We are talking aboutthe:

  • Rule of the sine; and
  • Rule of cosines.

The rule of the sines states that the corresponding sides and angles of a triangle are related by the formula:

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

While the rule of the cosines states that the three sides of a triangle cooperate to define the value of each angle in this way:

c ⁣= ⁣a2+b22 ⁣ ⁣a ⁣ ⁣b ⁣ ⁣cos(γ)b ⁣= ⁣a2+c22 ⁣ ⁣a ⁣ ⁣c ⁣ ⁣cos(β)a ⁣= ⁣c2+b22 ⁣ ⁣c ⁣ ⁣b ⁣ ⁣cos(α)\begin{align*} c\! &=\! \sqrt{a^2+b^2-2\!\cdot\! a\!\cdot\! b\!\cdot\!\cos{(\gamma)}}\\ b\! &=\! \sqrt{a^2+c^2-2\!\cdot\! a\!\cdot \!c\!\cdot\!\cos{(\beta)}}\\ a\! &=\! \sqrt{c^2+b^2-2\!\cdot\! c\!\cdot\! b\!\cdot\!\cos{(\alpha)}} \end{align*}

We can show you how to solve a SAS triangle. Assume to know the sides aa and bb, and the angle γ\gamma. Here is the first SAS triangle formula:

c=a2 ⁣+ ⁣b2 ⁣ ⁣2 ⁣ ⁣a ⁣ ⁣b ⁣cos(γ)c = \sqrt{a^2\! +\! b^2 \!-\! 2\!\cdot\!a\!\cdot\!b\!\cdot\cos{(\gamma)}}

Here we used the law of cosines to find the third side of the SAS triangle. Now, with this new knowledge, we can proceed to find the other quantities, this time using the law of sines:

sin(α)=sin(γ)ac\sin{(\alpha)} = \sin{(\gamma)}\cdot \frac{a}{c}

Now, applying the inverse trigonometric function we can find the value of the angle α\alpha, and thanks to the property of the internal angles of a triangle, we can find the last angle:

β=180αγ\beta = 180-\alpha-\gamma

This is it! Now all quantities are known, and we can complete our SAS triangle calculations. For the perimeter of the SAS triangle, we have:

P=a+b+cP =a+b+c

Defining the semi-perimeter as p=P/2p =P/2, we can use the Heron's formula to calculate the area:

A=p ⁣(p ⁣ ⁣a) ⁣(p ⁣ ⁣b) ⁣(p ⁣ ⁣c)A = \sqrt{p\!\left(p\!-\!a\right)\!\left(p\!-\!b\right)\!\left(p\!-\!c\right)}

There is another way to calculate the area of a SAS triangle:

A=12 ⁣ ⁣a ⁣ ⁣b ⁣ ⁣sin(γ)A = \frac{1}{2}\!\cdot\!a\!\cdot\!b\!\cdot\!\sin{(\gamma)}

Which, rather conveniently, uses the original quantities to find the value.

Congruence in SAS triangles

A SAS triangle is uniquely defined. You've just seen it: with the values you gave us, we could find all the other quantities.

This implies that SAS triangles are congruent: the fact that the angle is comprised between the two sides constrains the construction of the shape. With such a set of values, the only thing we can do is connect the ends of the sides and close the triangle: there is no opportunity to change the shape (because of the angle) or scale (because of the given length of two sides.

How to use our SAS triangle calculator

Our SAS triangle calculator can solve any congruent SAS triangle. Insert a combination of two adjacent sides and the angle between them.

For example, let's take a triangle with the following parameters:

  • a=4 cma = 4\ \text{cm};
  • b=3 cmb = 3\ \text{cm}; and
  • γ=42°\gamma = 42\degree.

Input these values in the SAS triangle calculator. First thing, we will find the value of the third side:

c=a2 ⁣+ ⁣b2 ⁣ ⁣2 ⁣ ⁣a ⁣ ⁣b ⁣cos(γ)=42 ⁣+ ⁣32 ⁣ ⁣2 ⁣ ⁣4 ⁣ ⁣3 ⁣cos(42°)=2.68 cm\begin{align*} c &= \sqrt{a^2\! +\! b^2 \!-\! 2\!\cdot\!a\!\cdot\!b\!\cdot\cos{(\gamma)}}\\ &= \sqrt{4^2\! +\!3^2 \!-\! 2\!\cdot\!4\!\cdot\!3\!\cdot\cos{(42\degree)}} \\ &= 2.68\ \text{cm} \end{align*}

We now calculate the remaining angles. Let's take β\beta:

β=arcsin(sin(γ)bc)=arcsin(sin(42)42.68)=48.59°\begin{align*} \beta& = \arcsin{\left(\sin{(\gamma)}\cdot \frac{b}{c}\right)}\\ \\ &= \arcsin{\left(\sin{(42)}\cdot \frac{4}{2.68}\right)}\\ \\ &= 48.59\degree \end{align*}

And with the simple relation between the angles of a triangle, we find α\alpha:

α=180°β=γ=180°42°48.59°=89.41°\begin{align*} \alpha &= 180\degree-\beta=\gamma\\ &=180\degree-42\degree-48.59\degree\\ &= 89.41\degree \end{align*}

What if you don't have a SAS triangle?

Don't worry about knowing different parameters than side-angle-side. We covered almost all the possible combinations of parameters and types of triangles. You will find what you need among our triangle calculators:

  • Triangle area;
  • AAA triangle;
  • Midsegment of a triangle ;
  • Acute triangle;
  • Circumcenter of a triangle;
  • Triangle congruence;
  • Obtuse triangle;
  • Oblique triangle;
  • Base of a triangle;
  • AAS triangle;
  • SSS triangle; and
  • ASA triangle.

FAQ

What is a SAS triangle?

A SAS triangle is a way to define a triangle by assigning the values to two adjacent sides and to the angle between them. Thanks to the properties of triangles, such a combination uniquely defines a triangle;; thus, we can calculate all the remaining quantities: sides, angles, area, and perimeter.

What is the SAS triangle formula for the area?

We can calculate the area of a triangle if the given quantities are two adjacent sides and the angle between them with a particular SAS triangle formula for the area:
A = 1/2 × z × b × sin(γ)
This is nothing but the more known expression A = b × h/2!

How do I calculate a SAS triangle with sides 4 cm and 5 cm, and angle between them 30°?

Follow the next steps to calculate entirely the SAS triangle defined by s₁ = 4 cm, s₂ = 5 cm, and a₃ = 30°.

  1. Find the value of the missing side:
    c = sqrt(4² + 5² - 2 × 4 × 5 × cos(30°)) = 2.52 cm
  2. Use the law of sines to find one of the other angles:
    ß = arcsin(sin(γ) × (b/c))
  3. Complete the triangle finding the last angle: α = 180° - ß - γ

Are SAS triangles congruent?

SAS triangles are congruent. The combination of two adjacent sides and the angle between them defines at the same time both the shape and the scale of the triangle.

Other similar combinations don't guarantee congruence. The combination SSA, for example, allows the creation of both acute and obtuse angles with the same set of initial values.

Davide Borchia
sas_triangle_image
a
cm
b
cm
γ
deg
output
c
cm
α
deg
β
deg
Area
cm²
Perimeter
cm
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