Tricks to Calculate Number of Triangles in Any Figure

Duration: 11 min

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AI Summary

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This video is a tutorial on solving a common aptitude test problem: counting the number of triangles in a given figure. The instructor, Yash Jain Sir, begins by presenting a sequence of four diagrams, each showing a large triangle subdivided into smaller triangles by horizontal lines. He explains that the number of triangles in each figure follows a pattern based on the number of horizontal divisions. For the first figure with one horizontal line, he counts the triangles as 1 (small) + 2 (medium) = 3. For the second with two horizontal lines, he counts 1 + 2 + 3 = 6. He then generalizes this pattern, stating that for a figure with 'n' horizontal lines, the total number of triangles is the sum of the first 'n+1' natural numbers, which is given by the formula n(n+1)/2. He demonstrates this with a final example of a triangle divided into 4 horizontal levels, where the total is 1+2+3+4 = 10. The video uses on-screen diagrams and handwritten annotations to illustrate the counting process and the mathematical formula.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card reading "SPEED MATHS". It then transitions to a screen with a daily quiz announcement and a sequence of four diagrams. The diagrams show a large triangle divided into smaller triangles by horizontal lines. The first has one horizontal line, the second has two, the third has three, and the fourth has four. The instructor, Yash Jain Sir, appears in a small window and begins to explain the problem, which is to count the number of triangles in each figure. He states that the quiz is about questions asked in IT companies and encourages viewers to follow their Instagram.

  2. 2:00 – 5:00 02:00-05:00

    The instructor begins to solve the first problem. He focuses on the first diagram, which has one horizontal line dividing the large triangle. He counts the small triangles at the top (1) and the larger triangles formed by combining the top and bottom sections (2), arriving at a total of 3. He then moves to the second diagram, which has two horizontal lines. He counts the triangles: 1 small one at the top, 2 medium ones in the middle, and 3 large ones at the bottom, summing to 6. He writes the equation 1+2+3=6 on the screen. He then explains that the pattern is the sum of the first 'n' natural numbers, where 'n' is the number of horizontal lines plus one.

  3. 5:00 – 10:00 05:00-10:00

    The instructor generalizes the pattern. He explains that for a figure with 'n' horizontal lines, the total number of triangles is the sum of the first (n+1) natural numbers. He writes the formula n(n+1)/2 on the screen. He then applies this to the third diagram, which has three horizontal lines. He counts the triangles as 1+2+3+4=10. He then moves to the fourth diagram, which has four horizontal lines, and states that the total number of triangles is 1+2+3+4+5=15. He uses red lines to highlight the different levels of triangles in the diagrams to aid the counting process.

  4. 10:00 – 10:32 10:00-10:32

    The video concludes with a final summary of the formula. The instructor reiterates that the total number of triangles in a figure with 'n' horizontal lines is given by the formula n(n+1)/2. He shows a final example of a triangle with 4 horizontal lines, where the total is 1+2+3+4+5=15. The video ends with a "THANKS FOR WATCHING" screen over a blue, abstract background with mathematical equations.

The video presents a clear, step-by-step method for solving a classic aptitude problem. It starts with a visual pattern of triangles, guides the viewer through the counting process for the first few cases, and then reveals the underlying mathematical formula. The progression from a specific example to a general rule is logical and effective. The use of on-screen diagrams and handwritten annotations makes the explanation easy to follow, demonstrating that a complex counting problem can be solved efficiently using a simple arithmetic series formula.