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# SAS Triangle Calculator

a:

b:

Y:

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Give a try to our free SAS triangle calculator to find each and every entity of a triangle. Just provide a couple of adjacent sides of the triangle and their mutual angle and let the calculator do the rest for you.

## What Is A SAS Triangle?

In Trigonometry, SAS corresponds to a Side-Angle-Side triangle.

### How to Solve SAS Triangle?

What if you think about how to find the angle of a triangle given 2 sides and 1 angle. Let us tell you!

Suppose we have to solve triangle SAS given as under:

In this triangle, we are given:

$$a = 1$$

$$b = 4$$

$$γ = 30^{\text{o}}$$

#### Third Side c:

By using law of cosines:

$$c=\sqrt{\left(1\right)^{2}+\left(4\right)^{2}−2 \text{1*4 } cos\left(30^{\text{o}}\right)}$$

$$c=3.17$$ For calculations, tap law of cosine calculator.

For other sides, you may use:

$$a=\sqrt{b^2+c^2−2 \text{ b c } cos(α)}$$

$$b=\sqrt{a^2+c^2−2 \text{ a c } cos(ꞵ)}$$

#### Perimeter:

$$p=a+b+c$$

$$p=1+4+3.17$$

$$p=8.17$$

#### Semiperimeter:

$$s=\dfrac{p}{2}$$

$$s=\dfrac{8.17}{2}$$

$$s=4.085$$

#### Area:

By using the Heron’s formula:

$$A=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$$

$$A=\sqrt{4.085\left(4.085-1\right)\left(4.085-4\right)\left(4.085-3.17\right)}$$

$$A=\sqrt{1}$$

$$A=1$$

#### Height of Triangle:

$$h_{a}=\dfrac{2A}{a}$$

$$h_{a}=\dfrac{2*1}{1}$$

$$h_{a}=2$$

$$h_{b}=\dfrac{2A}{b}$$

$$h_{b}=\dfrac{2*1}{4}$$

$$h_{b}=\dfrac{1}{2}$$

$$h_{b}=0.5$$

$$h_{c}=\dfrac{2A}{c}$$

$$h_{c}=\dfrac{2*1}{3.17}$$

$$h_{c}=0.630$$

#### Inner Angles:

By using law of sines here:

$$\dfrac{b}{sinꞵ}=\dfrac{c}{sin𝛾}$$

Rearranging for missing angle:

$$sinꞵ=\dfrac{b}{c}*sin𝛾$$

$$sinꞵ=\dfrac{4}{3.17}*sin\left(30^{\text{o}}\right)$$

$$sinꞵ=1.261*0.5$$

$$sinꞵ=0.6305$$

$$ꞵ=sin^{-1}\left(0.6305\right)$$

$$ꞵ=140°$$

Now using the supplementary angle measurement

$$𝛼+𝛽+𝛾=180^{\text{o}}$$

$$𝛼+140°56′7″+30°=180^{\text{o}}$$

$$𝛼=180^{\text{o}}-140°-30°$$

$$𝛼=10^{\text{o}}$$

$$r=\dfrac{A}{s}$$

$$r=\dfrac{1}{4.085}$$

$$r=0.244$$

$$R=\dfrac{a*b*c}{4r*s}$$

$$R=\dfrac{1*4*3.17}{4*0.244*4.085}$$

$$R=3.17$$

#### Medians:

$$m_{a}=\sqrt{\dfrac{2b^2+2c^2-a^2}{2}}$$

$$m_{a}=\sqrt{\dfrac{2*4^2+23.17^2-1^2}{2}}$$

$$m_{a}=3.576$$

$$m_{b}=\sqrt{\dfrac{2c^2+2a^2-b^2}{2}}$$

$$m_{b}=\sqrt{\dfrac{23.17^2+21^2-4^2}{2}}$$

$$m_{b}=1.239$$

$$m_{c}=\sqrt{\dfrac{2a^2+2b^2-c^2}{2}}$$

$$m_{c}=\sqrt{\dfrac{21^2+24^2-3.17^2}{2}}$$

$$m_{c}=2.446$$

### How Does SAS Triangle Calculator Work?

If you wish to use our SAS calculator, read on and understand the following guide!

Input:

• Enter two side and their associated angle measure
• Hit the calculate button

Output:

• The side angle side calculator complete the solution of SAS triangle

## References:

From the source of Wikipedia: Triangle, Types of triangle, Basic facts, Existence of a triangle, Points, lines, and circles associated with a triangle, Computing the sides and angles

From the source of Tutors.com: Triangle Congruence Postulates: SAS, ASA, SSS, AAS, HL

From the source of Lumen Learning: A Bit of Geometry, Similar Triangles