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Geometry Questions For CAT 2025
Geometry is an important topic in the CAT Quant section. In CAT 2025, geometry questions will check how well you understand basic rules, how accurately you can draw or imagine shapes, and how smartly you can use formulas.
This blog includes the most common types of geometry questions, key formulas, frequent mistakes to avoid, and practice questions to help you improve and score better.
Important Formulas for CAT Geometry Questions
If you're getting ready for Geometry questions in CAT 2025, learning some key formulas can make solving questions much easier and faster. We've also added a PDF with all the important Geometry formulas that you can download and use for quick revision anytime.
Topic | Concept | Formula |
Triangles | Sum of angles | 180∘180^\circ180∘ |
Pythagoras Theorem | a2+b2=c2a^2 + b^2 = c^2a2+b2=c2 | |
Basic Area | 12×base×height\frac{1}{2} \times \text{base} \times \text{height}21×base×height | |
Heron’s Area Formula | s(s−a)(s−b)(s−c)\sqrt{s(s-a)(s-b)(s-c)}s(s−a)(s−b)(s−c), s=a+b+c2s = \frac{a+b+c}{2}s=2a+b+c | |
Law of Sines | asinA=bsinB=csinC\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}sinAa=sinBb=sinCc | |
Law of Cosines | c2=a2+b2−2abcosCc^2 = a^2 + b^2 - 2ab \cos Cc2=a2+b2−2abcosC | |
Area with 2 sides & included angle | 12absinC\frac{1}{2} ab \sin C21absinC | |
Angle Bisector Theorem | Divides opposite side in ratio of adjacent sides | |
Centroid division | Divides median in ratio 2:1 from vertex | |
Circles | Circumference | 2πr2\pi r2πr |
Area | πr2\pi r^2πr2 | |
Arc Length | θ360×2πr\frac{\theta}{360} \times 2\pi r360θ×2πr | |
Sector Area | θ360×πr2\frac{\theta}{360} \times \pi r^2360θ×πr2 | |
Segment Area | Sector Area − Triangle Area | |
Equal tangents | Tangents from same external point are equal | |
Angle at center | 2×2 \times2× angle at circumference | |
Cyclic Quadrilateral | Opposite angles sum to 180∘180^\circ180∘ | |
Polygons & Quadrilaterals | Interior angle sum (n-gon) | (n−2)×180∘(n - 2) \times 180^\circ(n−2)×180∘ |
Each interior angle (regular polygon) | (n−2)×180∘n\frac{(n - 2) \times 180^\circ}{n}n(n−2)×180∘ | |
Exterior angles sum | 360∘360^\circ360∘ | |
Diagonals in n-gon | n(n−3)2\frac{n(n - 3)}{2}2n(n−3) | |
Parallelogram Area | b×hb \times hb×h | |
Trapezium Area | 12×(a+b)×h\frac{1}{2} \times (a + b) \times h21×(a+b)×h | |
Rhombus Area | 12×d1×d2\frac{1}{2} \times d_1 \times d_221×d1×d2 | |
Square Area | a2a^2a2 | |
Rectangle Area | l×bl \times bl×b | |
Coordinate Geometry | Distance | (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}(x2−x1)2+(y2−y1)2 |
Midpoint | (x1+x22,y1+y22)\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)(2x1+x2,2y1+y2) | |
Slope | y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1}x2−x1y2−y1 | |
Line equation (point-slope) | y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1=m(x−x1) | |
Line equation (two-point) | y−y1=y2−y1x2−x1(x−x1)y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)y−y1=x2−x1y2−y1(x−x1) | |
Circle equation | (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2 | |
Area of triangle | (\frac{1}{2} | |
3D Mensuration | Cube Volume | a3a^3a3 |
Cube Surface Area | 6a26a^26a2 | |
Cuboid Volume | l×b×hl \times b \times hl×b×h | |
Cuboid Surface Area | 2(lb+bh+hl)2(lb + bh + hl)2(lb+bh+hl) | |
Sphere Volume | 43πr3\frac{4}{3} \pi r^334πr3 | |
Sphere Surface Area | 4πr24\pi r^24πr2 | |
Hemisphere Volume | 23πr3\frac{2}{3} \pi r^332πr3 | |
Hemisphere Surface Area | 3πr23\pi r^23πr2 (total), 2πr22\pi r^22πr2 (curved) | |
Cylinder Volume | πr2h\pi r^2 hπr2h | |
Cylinder Surface Area | 2πr(h+r)2\pi r(h + r)2πr(h+r) | |
Cone Volume | 13πr2h\frac{1}{3} \pi r^2 h31πr2h | |
Cone Surface Area | πr(l+r)\pi r(l + r)πr(l+r), where l=r2+h2l = \sqrt{r^2 + h^2}l=r2+h2 |
Common Mistakes to Avoid in Geometry Questions
Avoiding common mistakes can help you solve geometry questions more accurately:
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Poor diagrams: Not drawing clean or labeled diagrams, or missing important marks like right angles or parallel lines.
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Not checking feasibility: Forgetting to see if a shape is even possible (like not checking the triangle inequality).
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Confusing circle rules: Mixing up angles at the center with angles on the edge of the circle.
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Sign or slope mistakes: Making errors with signs or slopes in coordinate geometry, or mixing up the points.
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Ignoring key properties: Missing special features like symmetry in regular shapes or angle rules in cyclic quadrilaterals.
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Not checking your answer: Forgetting to check if your final answer makes sense — like a negative length or incorrect area.
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Using complex methods: Trying difficult formulas when a simple trick like splitting the shape or using symmetry could work better.
List Of CAT Geometry Questions
Question 1
Let the consecutive vertices of a square S be A,B,C &D.; Let E,F & G be the mid-points of the sides AB, BC & AD respectively of the square. Then the ratio of the area of the quadrilateral EFDG to that of the square S is nearest to
correct answer:- 1
Question 2
Two identical circles intersect so that their centres, and the points at which they intersect, form a square of side 1 cm. The area in sq. cm of the portion that is common to the two circles is
correct answer:- 2
Question 3
In the given diagram, ABCD is a rectangle with AE = EF = FB. What is the ratio of the areas of CEF and that of the rectangle?
correct answer:- 1
Question 4
In triangleABC, ∠B is a right angle, AC = 6 cm, and D is the mid-point of AC. The length of BD is
correct answer:- 3
Question 5
Three circles, each of radius 20, have centres at P, Q and R. Further, AB = 5, CD = 10 and EF = 12. What is the perimeter of ΔPQR ?
correct answer:- 3
Question 6
Three circles of equal radii touch (but not cross) each other externally. Two other circles, X and Y, are drawn such that both touch (but not cross) each of the three previous circles. If the radius of X is more than that of Y, the ratio of the radii of X and Y is
correct answer:- 1
DIRECTIONS for the following two questions: These questions are based on the situation given below:
A rectangle PRSU, is divided into two smaller rectangles PQTU, and QRST by the line TQ. PQ=10cm, QR = 5 cm and RS = 10 cm. Points A, B, F are within rectangle PQTU, and points C, D, E are within the rectangle QRST. The closest pair of points among the pairs (A, C), (A, D), (A, E), (F, C), (F, D), (F, E), (B, C), (B, D), (B, E) are $$10 \sqrt{3}$$ cm apart.
Question 7
Which of the following statements is necessarily true?
correct answer:- 1
Question 8
ABCD is a trapezium in which AB is parallel to CD. The sides AD and BC when extended, intersect at point E. If AB = 2 cm, CD = 1 cm, and perimeter of ABCD is 6 cm, then the perimeter, in cm, of $$\triangle AEB$$ is
correct answer:- 1
Question 9
A circle with radius 2 is placed against a right angle. Another smaller circle is also placed as shown in the adjoining figure. What is the radius of the smaller circle?
correct answer:- 4
Question 10
On a semicircle with diameter AD, chord BC is parallel to the diameter. Further; AD and BC are separated by 2cm, while AD has length 8. What is the length of BC?
correct answer:- 2
Question 11
A right circular cone, of height 12 ft, stands on its base which has diameter 8 ft. The tip of the cone is cut off with a plane which is parallel to the base and 9 ft from the base. With $$\pi$$ = 22/7, the volume, in cubic ft, of the remaining part of the cone is
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correct answer:- 198
