This set contains Geometrical Figures, Chessboard and Grid Questions with Solutions — Set 3 (Q21-Q30) covering a mix of question types and difficulty levels — from basic to advanced — exactly as asked in real competitive exams.
Solutions are written in a simple, step-by-step notebook style for easy self-study and quick understanding. Each solution is broken down step by step so even the toughest question feels easy. These questions are hand-picked for students preparing for SSC CGL, SSC CHSL, CAT, Bank PO, Bank Clerk, UPSC CSAT, Railway RRB, AMCAT, eLitmus, TCS NQT and all campus placement aptitude tests. International students preparing for GRE, GMAT, SAT, ACT, MAT and all Numerical Reasoning Tests will find these equally useful.
✏️ Attempt each question on your own first — then check the solution below.
Geometrical Figures, Chessboard and Grid Questions 21 to 30 with Solutions
21. There are 16 points in a plane out of which 7 of them are collinear. Find the number of quadrilaterals formed by joining these points as vertices?
22. A parallelogram is cut by two set of parallel lines parallel to sides of parallelogram. Each set has ‘n’ parallel lines. What is the total number of parallelogram thus formed?
23. Consider n points in a plane no three of which are collinear and the ratio of number of hexagon & octagon that can be formed from these n points is 7:69 then find the value of n.
24. The sides AB, BC, CA of a triangle ABC have 4, 5 and 6 interior points respectively on them. Find the number of triangles that can be constructed using given interior points as vertices?
25. Consider three set of parallel lines in a plane containing ‘x’, ‘y’, ‘z’ parallel lines respectively. What is the highest number of parallelograms that can be formed with these set of parallel lines?
26. Find the number of non-congruent rectangles on a chess board.
27. Consider ‘n’ points in a plane, if number of heptagon is equal to number of octagons then find the number of triangles that can be drawn from these n points.
28. Let Kₙ denotes the number of triangles which can b formedusing the vertices of a regular polygon of ‘n’ sides. Which one of the following could be the value of Pₙ₊₁ – Pₙ₋₁?
(a) 194 (b) 195
(c)196 (d) 197
29. Find the number of rectangles in a chess board of 11×11 grid instead of 8×8 grid.
30. Find the number of rectangles in a chess board of 12×14 grid instead of 8×8 grid.
Geometrical Figures, Chessboard and Grid Questions 21 to 30 — Step-by-Step Solutions
21. There are 16 points in a plane out of which 7 of them are collinear. Find the number of quadrilaterals formed by joining these points as vertices?
Solution:-
= 1820 – 35 – 35×9
= 1470 Answer
22. A parallelogram is cut by two set of parallel lines parallel to sides of parallelogram. Each set has ‘n’ parallel lines. What is the total number of parallelogram thus formed?
Solution:-
total horizontal parallel lines = n + 2
total vertical parallel lines = n + 2
Hence total number of parallelogram thus formed
= ⁿ⁺²C₂ × ⁿ⁺²C₂ Answer
23. Consider n points in a plane no three of which are collinear and the ratio of number of hexagon & octagon that can be formed from these n points is 7:69 then find the value of n.
Solution:-
24. The sides AB, BC, CA of a triangle ABC have 4, 5 and 6 interior points respectively on them. Find the number of triangles that can be constructed using given interior points as vertices?
Solution:-
Total number of points = 4 + 5 + 6 = 15
if none of them are collinear then number of triangle is = ¹⁵C₃
Reduce from this the number of triangle which will not be made due to collinear points but are being counted in ¹⁵C₃
Hence final number of triangle
= ¹⁵C₃ – ⁴C₃ – ⁵C₃ – ⁶C₃ Answer
25. Consider three set of parallel lines in a plane containing ‘x’, ‘y’, ‘z’ parallel lines respectively. What is the highest number of parallelograms that can be formed with these set of parallel lines?
Solution:-
Consider pair of parallel lines from ‘x’ and ‘y’ parallel lines. First we will select 2 lines from ‘x’ straight lines and 2 lines from ‘y’ straight lines, this can be done in ˣC₂×ʸC₂ ways.
Similarly, we can select pair from ‘y’ & ‘z’ straight lines & another pair from ‘z’ & ‘x’ straight lines. This can be done in ʸC₂×ᶻC₂ + ᶻC₂×ˣC₂ ways.
So total such parallelogram is
= ˣC₂×ʸC₂ + ʸC₂×ᶻC₂ + ᶻC₂×ˣC₂ Answer
26. Find the number of non-congruent rectangles on a chess board.
Solution:-
Here we need non-congruent rectangles means every rectangle should have unique pair of length and breadth i.e. no two rectangles should have same length and breadth.
Hence dimension (a, b) & (b, a) are congruent.
Number of non-congruent rectangles having length as 1 unit will be like
(1, 1) (1, 2) (1, 3), ………………….(1, 8) ⟹ total 8 rectangles
Now number of rectangles having length as 2 units will be like
(2, 2), (2, 3), (2, 4)…………….(2, 8) ⟹ total 7 rectangles
Similarly number of rectangles having length as 3 units will be like (3, 3) (3, 4) (3, 5) ………………………… (3, 8) ⟹ total 6 rectangles
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Number of rectangles having length as 8 units will be like (8, 8) ⟹ total 1 rectangle
Hence total number of rectangles
= 8 + 7 + 6 + ……….. + 1
= 36 Answer
27. Consider ‘n’ points in a plane, if number of heptagon is equal to number of octagons then find the number of triangles that can be drawn from these n points.
Solution:-
number of heptagon = number of octagon
⟹ ⁿC₇ = ⁿC₈
Hence n = 7 + 8 = 15
∴ number of triangles = ⁿC₃ = ¹⁵C₃ Answer
28. Let Kₙ denotes the number of triangles which can b formedusing the vertices of a regular polygon of ‘n’ sides. Which one of the following could be the value of Pₙ₊₁ – Pₙ₋₁?
(a) 194 (b) 195
(c)196 (d) 197
Solution:-
Pₙ₊₁ = ⁿ⁺¹C₃
= \(\frac{{(n – 1)(n)(n + 1)}}{6}\)
Pₙ₋₁ = ⁿ⁻¹C₃
= \(\frac{{(n – 3)(n – 2)(n – 1)}}{6}\)
∴ Pₙ₊₁ – Pₙ₋₁ = \(\frac{{(n – 1)(n)(n + 1) – (n – 3)(n – 2)(n – 1)}}{6}\)
= \(\frac{{(n – 1)\{ {n^2} + n – {n^2} + 5n – 6\} }}{6}\)
= (n-1)²
Hence this value must be a perfect square, from the given option (c) 196 Answer
29. Find the number of rectangles in a chess board of 11×11 grid instead of 8×8 grid.
Solution:-
Chess board has 11 + 1 = 12 set of parallel lines & another set of 11 + 1 = 12 set of parallel lines.
Hence number of rectangle = ¹²C₂ × ¹²C₂
= 66 × 66
= 4356 Answer
30. Find the number of rectangles in a chess board of 12×14 grid instead of 8×8 grid.
Solution:-
2 lines can be selected from 12 parallel lines in ¹²C₂ ways.
2 lines can be selected from 14 parallel lines in ¹⁴C₂ ways.
Hence number of rectangles = ¹²C₂×¹⁴C₂ Answer
Continue practising with Geometrical Figures, Chessboard and Grid Questions 31 to 40 → Set 4 or revisit the Geometrical Figures, Chessboard and Grid Concept Page to strengthen your formulas and tricks before moving ahead.
Consistent practice is the key to mastering Geometrical Figures, Chessboard and Grid for SSC CGL, SSC CHSL, CAT, Bank PO, Bank Clerk, UPSC CSAT, Railway RRB, AMCAT, eLitmus, TCS NQT and international exams including GRE, GMAT, SAT, ACT, MAT and all Numerical Reasoning Tests. Want to understand the concept better? Read about Shortest Path Problem on Wikipedia before attempting the next set.
This page is part of our complete series of Geometrical Figures, Chessboard and Grid questions with solutions for competitive exams — covering every question type from basic to advanced so you can build speed, accuracy and confidence. Practising these questions regularly will also strengthen your core Geometrical Figures, Chessboard and Grid concept before your exam day.