To find the area of a non-right triangle, let’s first review the standard area formula of a right triangle. A right triangle is made up of three sides: the base, the height, and the hypotenuse. To get the area of a triangle you must multiply the two adjacent side lengths of the 90° angle, which are the base and the height of the triangle, and divide this quantity by half. This is the formula for the area of a right triangle:
However, this formula doesn't work as effectively for acute and obtuse angles. So here's how to find the area of a non-right triangle.
Formulas for the Area of a Non-Right Triangle
When you use trigonometry, there's another group of formulas that can be used to find the area of a triangle with no right angles. Using these formulas, you can find the area of a non-right triangle even when there's a missing side length. There are different ways to find the areas of an obtuse triangle vs. an acute triangle.
Finding the Area of an Acute Triangle
When you need to find the area of an acute triangle, you must use the law of sines in place of a missing side length. Depending on which are the known sides or known angles, one of the following formulas can be used to find the area of an acute non-right triangle:
See the below acute triangle ΔABC. You can see that we do not know the length of all the sides of the triangle, but we do know that the acute angle measures 56° and that the two adjacent side lengths are 18 and 12. We would use the sine function on a calculator to plug this into the formula
When using a trigonometric formula for finding the area of an acute non-right triangle, a capital "C" is used to represent the known angle that is across from the opposite side length represented by lowercase "c".
Finding the Area of an Obtuse Triangle
The functions of sine, cosine, and tangents, can only be used to find the area of a triangle with an acute angle. So you must use a different method to find the area of a triangle with an obtuse angle. In order to find the area of the below triangle, you must draw a straight line from point C and point A, creating a right triangle where the two lines intersect.
The angle of CAE is supplementary to angle CAB, meaning that the two angles add up to 180 degrees. We can now assume that ∠CAE = 180 -∠A and from the area of ΔCAE, we can tell that sin∠CAE =sin (180-∠A). We can now substitute this formula to get
Formula for Success
Whether you need to find the area of an oblique triangle, obtuse triangle, or if you have two missing angles, if we know two sides and the included angle, we can find the area of a non-right triangle. If you’re stuck on this problem, feel free to bookmark this page as a guide.
More Math Homework Help:
- How To Find the Base of a Triangle in 4 Different Ways
- What Is the Converse of the Pythagorean Theorem?
In order to use these rules, we require a technique for labelling the sides and angles of the non-right angled triangle. This may mean that a relabelling of the features given in the actual question is needed. See the non-right angled triangle given here. Angle A is opposite side a, angle B is opposite side B and angle C is opposite side c. We determine the best choice by which formula you remember in the case of the cosine rule and what information is given in the question but you must always have the UPPER CASE angle OPPOSITE the LOWER CASE side.
The Cosine Rule
These formulae represent the cosine rule. Note that it is not necessary to memorise all of them – one will suffice, since a relabelling of the angles and sides will give you the others. Students tend to memorise the bottom one as it is the one that looks most like Pythagoras.
We use the cosine rule to find a missing side when all sides and an angle are involved in the question. It may also be used to find a missing angle if all the sides of a non-right angled triangle are known. See Examples 1 and 2.
The Cosine Rule
$a^2=b^2+c^2-2bc\cos(A)$
$b^2=a^2+c^2-2ac\cos(B)$
$c^2=a^2+b^2-2ab\cos(C)$
The Sine Rule
This formula represents the sine rule. The sine rule can be used to find a missing angle or a missing side when two corresponding pairs of angles and sides are involved in the question. This is different to the cosine rule since two angles are involved. This is a good indicator to use the sine rule in a question rather than the cosine rule. See Example 3.
Note that when using the sine rule, it is sometimes possible to get two answers for a given angle\side length, both of which are valid. See Example 4.
The Sine Rule
$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$
or
$\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$
The Area of a Non-Right Angled Triangle
These formulae represent the area of a non-right angled triangle. Again, it is not necessary to memorise them all – one will suffice (see Example 2 for relabelling). It is the analogue of a half base times height for non-right angled triangles.
Note that to maintain accuracy, store values on your calculator and leave rounding until the end of the question. You can round when jotting down working but you should retain accuracy throughout calculations. See Examples 5 and 6.
The Area of a Non-Right Angled Triangle
$\frac{1}{2}ab\sin(C)$
$\frac{1}{2}bc\sin(A)$
$\frac{1}{2}ac\sin(B)$
Questions by Topic
Examples of Non Right Angled Triangles
Find the length of the side marked x in the following
triangle:
The triangle PQR has sides $PQ=6.5$cm, $QR=9.7$cm and $PR = c$cm. Angle $QPR$ is $122^\circ$. Find the value of $c$.
Find the angle marked $x$ in the following triangle to 3 decimal
places:
In triangle $XYZ$, length $XY=6.14$m, length
$YZ=3.8$m and the angle at $X$ is $27^\circ$. Sketch the two possibilities for this triangle and find the two possible values of the angle at $Y$ to 2 decimal places.
Find the area of this triangle.
Find the area of the triangle with sides 22km, 36km and 47km to 1 decimal place.
Videos of Non Right Angled Triangles
What Next?
- Go back to PURE MATHS
- See QUESTIONS BY TOPIC
- Go to PAST and PRACTICE PAPERS