Logical Reasoningतार्किक क्षमता · आकृतियों की गिनती
Counting Figuresआकृतियों की गिनतीquestions and answersप्रश्न और उत्तर
Counting questions draw a figure from straight lines and ask how many triangles it contains. The total is always bigger than it looks, because triangles made of several smaller pieces count too.
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1subtopicउप-टॉपिक
SSC CGL ~1
SSC CHSL ~1
Practice questions are in English for now.अभ्यास के प्रश्न अभी अंग्रेज़ी में हैं।
How counting figures worksआकृतियों की गिनती को समझिए
Count by size. First count the smallest triangles, the single pieces that no line passes through. Then look for triangles made of two pieces, then larger ones, ending with the biggest. Writing a tally for each size stops you from counting any triangle twice.
Lines that fan out from one vertex have a shortcut. If n lines leave the top vertex and all reach the base (the two sides count as lines), any two of them together with the base make a triangle, so there are n(n − 1)/2 triangles. Each extra line drawn across, parallel to the base, adds another full set.
Symmetric figures, such as a square with its diagonals and the lines joining midpoints of opposite sides, can be counted in one quarter and multiplied, as long as you separately count the larger triangles that cross from one quarter into another.
Each trick says when to use it, why it works and when it fails. Formula tricks are checked by code over their whole range.हर ट्रिक बताती है कि उसे कब इस्तेमाल करें, वह क्यों काम करती है और कब नहीं। सूत्र वाली ट्रिक्स को कोड से उनकी पूरी रेंज पर जाँचा गया है।
Formulaसूत्रRarely taughtकम सिखाई जाने वाली✓ Checked on 3 cases✓ 3 मामलों पर जाँची गई
Remember three square figures: 8, 16 and 44
Use it when:कब इस्तेमाल करें: A square is drawn with its diagonals, with or without the midlines and the inner square joining the midpoints.
Square + both diagonals: 8 triangles.
Add both lines joining midpoints of opposite sides: 16.
Also join the four midpoints (an inner square): 44.
Exampleउदाहरण
A square with both diagonals and both midlines. How many triangles?
→ 16: 8 small, 4 of two pieces, 4 halves of the square.
Why it works, and when it failsयह क्यों काम करती है, और कब नहीं
Why:क्यों: Counted by size, the full figure has 16 smallest triangles, 16 made of two pieces, 8 of four pieces and 4 halves of the square: 16 + 16 + 8 + 4 = 44. The smaller figures are the same count with fewer lines.
Limits:सीमाएँ: Only for these exact figures. A single extra line changes the total, so count by size whenever the figure differs.
Formulaसूत्र✓ Checked on 28 cases✓ 28 मामलों पर जाँची गई
Lines from one vertex: n(n − 1)/2 per level
Use it when:कब इस्तेमाल करें: A triangle has several lines from its top vertex to the base, possibly with lines drawn across parallel to the base.
Count the lines leaving the top vertex, including both sides: n.
Each level has n(n − 1)/2 triangles.
With m lines across, there are m + 1 levels (the base counts): total n(n − 1)/2 × (m + 1).
Exampleउदाहरण
A triangle with 2 lines from the top to the base and 1 line across.
→ n = 4: 6 per level × 2 levels = 12.
Why it works, and when it failsयह क्यों काम करती है, और कब नहीं
Why:क्यों: Every triangle in such a figure has the top vertex as one corner, two of the n lines as sides, and one horizontal level (a cross line or the base) as its third side. Choosing 2 lines from n and 1 level from m + 1 gives the count.
Limits:सीमाएँ: Every cross line must run from one side to the other and cut all the lines from the vertex. Lines from other vertices break the formula; count by size instead.
Solved examplesहल किए गए उदाहरण
Answers and full solutions are shown. Hide them to test yourself first.उत्तर और पूरे हल दिखाए गए हैं। पहले खुद को परखने के लिए इन्हें छिपाइए।
Example 1उदाहरण 1 · Counting triangles in a figure · Easyआसान
How many triangles are there in the figure?
4
6
9
10
Answer:उत्तर:10
The figure is a triangle with 3 lines from the top vertex to the base.
By size (in units of the smallest triangle's area): 4 of size 1, 3 of size 2, 2 of size 3, 1 of size 4.
Total: 10 triangles.
Example 2उदाहरण 2 · Counting triangles in a figure · Mediumमध्यम
How many triangles are there in the figure?
5
18
45
60
Answer:उत्तर:60
The figure is a triangle with 4 lines from the top vertex to the base and 3 lines across it, parallel to the base.
By size (in units of the smallest triangle's area): 5 of size 1, 4 of size 2, 3 of size 3, 7 of size 4, 1 of size 5, 4 of size 8, 5 of size 9, 3 of size 12, 7 of size 16, 4 of size 18, 1 of size 20, 3 of size 27, 4 of size 32, 2 of size 36, 1 of size 45, 3 of size 48, 2 of size 64, 1 of size 80.
Total: 60 triangles.
Example 3उदाहरण 3 · Counting triangles in a figure · Hardकठिन
How many triangles are there in the figure?
8
16
18
20
Answer:उत्तर:18
The figure is a rectangle made of two squares side by side, each with both diagonals.
By size (in units of the smallest triangle's area): 8 of size 1, 8 of size 2, 2 of size 4.
Total: 18 triangles.
Example 4उदाहरण 4 · Counting triangles in a figure · Mediumमध्यम
How many triangles are there in the figure?
4
14
15
17
Answer:उत्तर:15
The figure is a triangle with 2 lines from the bottom-left vertex and 1 line from the bottom-right vertex to the opposite sides.
By size (in units of the smallest triangle's area): 1 of size 1, 1 of size 1.2, 1 of size 1.8, 2 of size 3, 1 of size 3.2, 3 of size 4, 1 of size 4.8, 2 of size 6, 2 of size 8, 1 of size 12.
Total: 15 triangles.
Example 5उदाहरण 5 · Counting triangles in a figure · Mediumमध्यम
How many triangles are there in the figure?
5
15
16
30
Answer:उत्तर:30
The figure is a triangle with 4 lines from the top vertex to the base and 1 line across it, parallel to the base.
By size (in units of the smallest triangle's area): 5 of size 1, 4 of size 2, 3 of size 3, 7 of size 4, 1 of size 5, 4 of size 8, 3 of size 12, 2 of size 16, 1 of size 20.
Total: 30 triangles.
Example 6उदाहरण 6 · Counting triangles in a figure · Mediumमध्यम
How many triangles are there in the figure?
5
23
24
26
Answer:उत्तर:24
The figure is a triangle with 3 lines from the bottom-left vertex and 1 line from the bottom-right vertex to the opposite sides.
By size (in units of the smallest triangle's area): 1 of size 1, 1 of size 1.43, 1 of size 1.9, 1 of size 2.67, 2 of size 3.33, 1 of size 4, 1 of size 4.57, 4 of size 5, 1 of size 6, 1 of size 6.43, 1 of size 6.67, 1 of size 8.57, 5 of size 10, 2 of size 15, 1 of size 20.
Total: 24 triangles.
Common mistakesआम गलतियाँ
Counting only the smallest triangles and missing those made by joining pieces.
Forgetting the outer, largest triangles in the figure.
Using n(n − 1)/2 with n = the lines inside the triangle; the two sides must be included.
Frequently asked questionsअक्सर पूछे जाने वाले प्रश्न
How many triangles does a square with both diagonals have?
Eight: four small triangles meeting at the centre, and four larger ones, each half of the square, cut off by one diagonal.
How many triangles are in a square with diagonals and both midlines?
Sixteen: eight small ones, four made of two small ones (one on each side, meeting at the centre), and four halves of the square cut off by a diagonal. Adding the square that joins the midpoints raises the total to 44.
Is there a formula for every figure?
No. Formulas exist for regular patterns like lines from one vertex. For any other figure, label the points and count by size or by vertex.