Determinant To Find Area Of A Triangle – Solved Examples-Class 12 Math Determinants Notes Study Material free pdf download

We know what a determinant is, let us know how to use Determinant to find Area of a Triangle

Suppose we are given three points in the Cartesian plane as

\(\begin{array}{l} (x_1 , y_1) , (x_2,y_2) \space and \space (x_3 , y_3) \end{array} \)

. The area of the triangle obtained by joining these points is given by,

\(\begin{array}{l} \alpha = \frac 12 [ x_1 (y_2 – y_3) + x_2 (y_3 – y_1) + x_3 (y_1 – y_2)] \end{array} \)

Where  

\(\begin{array}{l} \alpha \end{array} \)

denotes the area of the triangle and

\(\begin{array}{l} (x_1 , y_1) , (x_2,y_2) \space and \space (x_3 , y_3) \end{array} \)

,  represent the vertices of the triangle.

Best JEE & NEET Physics, Chemistry, Biology, and Math Study Material – Anand Classes by Neeraj Anand (Published by Anand Technical Publishers)

If you are preparing for JEE Mains, JEE Advanced, or NEET, having the right study material can make all the difference. Anand Classes, authored by Neeraj Anand and published by Anand Technical Publishers, provides one of the most comprehensive, structured, and exam-oriented study materials for Physics, Chemistry, Biology, and Mathematics Subjects according to latest trends of JEE & NEET Entrance Exams.

Why Choose Anand Classes Study Material for JEE & NEET Preparation?

The study materials by Neeraj Anand are designed to simplify complex concepts, provide in-depth explanations, and offer ample practice questions to help students achieve top scores in competitive exams.

Comprehensive Coverage – Detailed explanations of Physics, Chemistry, Biology, and Mathematics concepts.
JEE & NEET Focused – Designed specifically for competitive exam success.
Solved Examples & Practice Questions – Strengthen your understanding with concept-based problems.
Short Tricks & Formulas – Easy-to-remember techniques for quick problem-solving.
Concept Clarity: Easy-to-understand theory with step-by-step explanations.
Topic-Wise Breakdown: Well-structured chapters following the latest syllabus of JEE Mains, JEE Advanced, and NEET.
Solved Examples & Practice Questions: Includes previous year questions (PYQs), important formulas, and shortcut techniques.
NCERT-Based & Advanced Level: Covers both board exam preparation and entrance exam syllabus for JEE & NEET aspirants.
Time-Saving Tricks: Quick formulas, memory techniques, and problem-solving strategies.

📚 Subjects Covered in Anand Classes Study Material

🔵 Physics (For JEE & NEET)

  • Kinematics & Laws of Motion
  • Work, Power & Energy
  • Gravitation & Fluid Mechanics
  • Thermodynamics & Heat Transfer
  • Electrostatics & Magnetism
  • Optics & Modern Physics
  • Semiconductor Electronics & Communication

🟠 Chemistry (For JEE & NEET)

Physical Chemistry

  • Mole Concept & Stoichiometry
  • Thermodynamics & Chemical Equilibrium
  • Electrochemistry & Chemical Kinetics

Inorganic Chemistry

  • Periodic Table & Chemical Bonding
  • Coordination Compounds & Metallurgy
  • P-Block, D-Block & F-Block Elements

Organic Chemistry

  • Hydrocarbons & Functional Groups
  • Reaction Mechanisms & Named Reactions
  • Biomolecules & Polymers

🟢 Biology (For NEET)

  • Diversity in the Living World
  • Cell Structure & Function
  • Genetics & Evolution
  • Human Physiology & Reproduction
  • Ecology & Environment

🔴 Mathematics (For JEE Mains & Advanced)

Permutation, Combination & Complex Numbers

Algebra & Probability

Trigonometry & Coordinate Geometry

Calculus (Differential & Integral)

Vectors & 3D Geometry

📥 Download PDF & Purchase the Study Material

The Anand Classes Study Material is available in PDF format and hardcopy. You can access high-quality notes, question banks, and practice tests to boost your JEE & NEET preparation.

🔹 Download Now: Get the latest edition PDF for easy access on mobile, tablet, or laptop.
🔹 Buy Hardcopy: Order the printed book for detailed study and offline preparation.

📌 Published by: Anand Technical Publishers
📌 Author: Neeraj Anand


Determinant - Area of a triangle

The formula for finding area could be represented in the form of determinants as given below.

\(\begin{array}{l} \alpha =\frac 12 \left|
\begin{matrix}
x_1 & y_1 & 1\cr
x_2 & y_2 & 1 \cr
x_3 & y_3 & 1 \cr
\end{matrix}
\right|
\end{array} \)

As we know the value of a determinant can either be negative or a positive value but since we are talking about area and it can never be taken as a negative value, therefore we take the absolute value of the determinant so obtained.

If the area of the triangle is already given then we make use of both the positive and negative values of the determinant.

Also, if three points are collinear we would be left with a straight line instead of a triangle and as the area enclosed by a straight line is zero hence the value of the determinant will also be zero.

Keeping the above-mentioned points in mind,  let us try to expand the determinant which denoted the area by using determinant expansion techniques using minors and cofactors.

Therefore, 

\(\begin{array}{l} \alpha = \frac 12 [ x_1 (y_2 – y_3) + x_2 (y_3 – y_1) + x_3 (y_1 – y_2)] \end{array} \)

Hence we see that how determinants are applied to make calculations easy. Now let us try our hands at this application of determinants to find out the area of triangles.

Example To find Area of Triangle using Determinant

Example: Find out the area of the triangle whose vertices are given by A(0,0) , B (3,1) and C (2,4).

Solution: Using determinants we can easily find out the area of the triangle obtained by joining these points using the formula

\(\begin{array}{l} \alpha = \frac 12\left|
\begin{matrix}
x_1 & y_1 & 1\cr
x_2 & y_2 & 1 \cr
x_3 & y_3 & 1 \cr
\end{matrix}
\right|
\end{array} \)

.

Substituting the respective values in the determinant we have

\(\begin{array}{l} \alpha = \frac 12\left|
\begin{matrix}
0 & 0 & 1\cr
3 & 1 & 1 \cr
2 & 4 & 1 \cr
\end{matrix}
\right|
\end{array} \)

Expanding the above determinant by using expansion techniques of determinant we get,

\(\begin{array}{l} \alpha = \frac 12 [ 0 (1-4) – 0(3-2) + 1 (12-2)] \end{array} \)

\(\begin{array}{l} \Rightarrow \alpha = 5 units \end{array} \)

Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations. In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS". He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.