100+ Most Important Geometry Questions for CAT 2025

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

The coordinates of the three vertices of a triangle are: (1, 2), (7, 2), and (1, 10). Then the radius of the incircle of the triangle is

The midpoints of sides AB, BC, and AC in ΔABC are M, N, and P, respectively. The medians drawn from A, B, and C intersect the line segments MP, MN and NP at X, Y, and Z, respectively. If the area of ΔABC is 1440 sq cm, then the area, in sq cm, of $$\triangle XYZ$$ is

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

A circular plot of land is divided into two regions by a chord of length $$10\sqrt{3}$$ meters such that the chord subtends an angle of 120° at the center. Then, the area, in square meters, of the smaller region is

ABCD is a rectangle with sides AB = 56 cm and BC = 45 cm, and E is the midpoint of side CD. Then, the length, in cm, of radius of incircle of $$\triangle ADE$$ is

The surface area of a closed rectangular box, which is inscribed in a sphere, is 846 sq cm, and the sum of the lengths of all its edges is 144 cm. The volume, in cubic cm, of the sphere is

A regular octagon ABCDEFGH has sides of length 6 cm each. Then the area, in sq. cm, of the square ACEG is

In the figure given below PK: KL = 5:4 and RL: LQ = 2:3, then PM: MQ is

A wedding stage is a square of side 10 metres each. At a corner, a triangular section of the stage is cordoned off using a rope in a straight line. If the length of the rope is 6 metres, what is the maximum area which can be cordoned off in the triangle?

One of the diameters of the circle $$x^2 + y^2 -2x -6y + 6 = 0$$ is a chord of the circle with center (2,1). What is the radius of the circle with center (2,1)?

From O, the incentre of $$\triangle$$ PQR, a perpendicular is drawn to the side QR such that it meets QR at A. If $$\angle$$ QOA = $$40^0$$, Then the measure of $$\angle POR$$ is

In a triangle, ABC, the length of the longest side is 58 cm. If the area of the triangle is 348 sq. cm and the length of one of the two remaining sides is 20 cm, then find out the length of the third side of the triangle?

A semicircle with centre O is drawn inside rectangle ABCD such that the longer side BC is the diameter of the circle. Point E divides AB in the ratio 5:4. Line DE is tangent to the semicircle. If the area of the triangle DEO is 65 square units, then the area (in sq. units) of the rectangle will be

A walking track AB is the diameter of a circular park of radius 10 m. A pole of height 6 m is standing on the circumference of the circular park and it subtends equal angles at A and B. A point R lies on the line AB and the pole subtends 30$$\degree$$ at R. What is the distance of the point R from the center?

An equilateral triangle is drawn inside an isosceles right-angled triangle such that both of them share a vertex and the height of both the triangles is the same. If the length of the side of the equilateral triangle is $$a$$, what is the perimeter of the right-angled triangle?

Three points lie at a distance of 6 cm from each other. A fourth point, which is not in the same plane as the other 3 points are, is a distance of 8 cm from each of the 3 points. What is the surface area of the packed figure formed by the four points?

If the measurements of two sides of a triangle are 34 cm and 42 cm, then the number of possible integral values of the third side is

The two circles shown are of radius 5 cm. AB and CD are tangents to them. If AB = 26 cm, what is the length of CD (in cm)?

In an obtuse-angled triangle, the difference between the obtuse angle and one of the acute angles is $$24^0$$. The difference between the two acute angles is $$54^0$$. Then which among the following cannot be the sum of the angles taken two at a time?

XYWZ is a trapezium with XY parallel to WZ. If XY is 10 cm; XZ is 8 cm, $$\angle XZW$$ is 45 degrees and YW is perpendicular to ZW, find the length of WZ?

In given figure parallelogram MNQR has area 74 sq. m. If S is the midpoint of RQ, find the area of NTQ?

In the figure given below PQ II RS II TU. If PQ:TU=3:2, then what is the value of SR:TU?

Despite x decrease in the length and 2x increase in the breadth of a rectangle, the area remains the same. If the initial ratio of the length and the breadth of the rectangle is 5:4, then find the percentage by which the length of the rectangle is decreased.

For example, enter 20 if the answer is 20%

Circles with radius of 6 units and 4 units are drawn in such a way that exactly 3 common tangents are possible. If a direct common tangent from point A touches the circles at points B and C, what is the length of BC?

A cone has a radius of 21 cm. 2 cuts parallel to the base are made such that a smaller cone and 2 frustums are formed. The cone and the frustums have the same height. What is the ratio of the lateral surface areas of the smaller frustum and the larger frustum?

Balu digs a cylindrical well in his backyard. Just the top surface of the ground (sand) is exposed to the air initially. He heaps the sand excavated in the form of a cone such that the diameter of the well is equal to the diameter of the heap of sand. If the amount of sand exposed to the air has increased by 25% and the diameter of the well is 'n' times the depth of the well, the value of 'n' is

x is the hypotenuse of an isosceles right-angled triangle. Taking the base of this isosceles triangle (the smaller side) as the hypotenuse, another isosceles right-angled triangle is drawn. This process is repeated an infinite number of times. What is the ratio of square of the length of the hypotenuse of the original right-angled triangle to the sum of the areas of all the isosceles right-angled triangles?

Consider the quadrilateral given below

If it is known that the area of Triangle QOR is 15, SOR is 18 and POS is 12 square units, then find the area of triangle POQ.

In a rectangle ABCD, AB = 9 cm and BC = 6 cm. P and Q are two points on BC such that the areas of the figures ABP, APQ, and AQCD are in geometric progression. If the area of the figure AQCD is four times the area of triangle ABP, then BP : PQ : QC is

A triangle is drawn with its vertices on the circle C such that one of its sides is a diameter of C and the other two sides have their lengths in the ratio a : b. If the radius of the circle is r, then the area of the triangle is

In a regular polygon, any interior angle exceeds the exterior angle by 120 degrees. Then, the number of diagonals of this polygon is

A rectangle with the largest possible area is drawn inside a semicircle of radius 2 cm. Then, the ratio of the lengths of the largest to the smallest side of this rectangle is

Let $$\triangle ABC$$ be an isosceles triangle such that AB and AC are of equal length. AD is the altitude from A on BC and BE is the altitude from B on AC. If AD and BE intersect at O such that $$\angle AOB = 105^\circ$$, then $$\frac{AD}{BE}$$ equals

In a right-angled triangle ∆ABC, the altitude AB is 5 cm, and the base BC is 12 cm. P and Q are two points on BC such that the areas of $$\triangle ABP, \triangle ABQ$$ and $$\triangle ABC$$ are in arithmetic progression. If the area of ∆ABC is 1.5 times the area of $$\triangle ABP$$, the length of PQ, in cm, is

A quadrilateral ABCD is inscribed in a circle such that AB : CD = 2 : 1 and BC : AD = 5 : 4. If AC and BD intersect at the point E, then AE : CE equals

Let C be the circle $$x^{2} + y^{2} + 4x - 6y - 3 = 0$$ and L be the locus of the point of intersection of a pair of tangents to C with the angle between the two tangents equal to $$60^{\circ}$$. Then, the point at which L touches the line $$x$$ = 6 is

In a triangle ABC, AB = AC = 8 cm. A circle drawn with BC as diameter passes through A. Another circle drawn with center at A passes through Band C. Then the area, in sq. cm, of the overlapping region between the two circles is

The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number of possible lengths of the fourth side is

Suppose the medians BD and CE of a triangle ABC intersect at a point O. If area of triangle ABC is 108 sq. cm., then, the area of the triangle EOD, in sq. cm., is

The length of each side of an equilateral triangle ABC is 3 cm. Let D be a point on BC such that the area of triangle ADC is half the area of triangle ABD. Then the length of AD, in cm, is

Regular polygons A and B have number of sides in the ratio 1 : 2 and interior angles in the ratio 3 : 4. Then the number of sides of B equals

In triangle ABC, altitudes AD and BE are drawn to the corresponding bases. If $$\angle BAC = 45^{\circ}$$ and $$\angle ABC=\theta\ $$, then $$\frac{AD}{BE}$$ equals

Let ABCD be a parallelogram such that the coordinates of its three vertices A, B, C are (1, 1), (3, 4) and (−2, 8), respectively. Then, the coordinates of the vertex D are

All the vertices of a rectangle lie on a circle of radius R. If the perimeter of the rectangle is P, then the area of the rectangle is

A trapezium $$ABCD$$ has side $$AD$$ parallel to $$BC, \angle BAD = 90^\circ, BC = 3$$ cm and $$AD= 8$$ cm. If the perimeter of this trapezium is 36 cm, then its area, in sq. cm, is

Let ABCD be a parallelogram. The lengths of the side AD and the diagonal AC are 10cm and 20cm, respectively. If the angle $$\angle ADC$$ is equal to $$30^{0}$$ then the area of the parallelogram, in sq.cm is

If a triangle ABC, $$\angle BCA=50^{0}$$. D and E are points on $$AB$$ and $$AC$$, respectively, such that $$AD=DE$$. If F is a point on $$BC$$ such that $$BD=DF$$, then $$\angle FDE$$, in degrees, is equal to

A park is shaped like a rhombus and has area 96 sq m. If 40 m of fencing is needed to enclose the park, the cost, in INR, of laying electric wires along its two diagonals, at the rate of ₹125 per m, is

The cost of fencing a rectangular plot is ₹ 200 per ft along one side, and ₹ 100 per ft along the three other sides. If the area of the rectangular plot is 60000 sq. ft, then the lowest possible cost of fencing all four sides, in INR, is

Let D and E be points on sides AB and AC, respectively, of a triangle ABC, such that AD : BD = 2 : 1 and AE : CE = 2 : 3. If the area of the triangle ADE is 8 sq cm, then the area of the triangle ABC, in sq cm, is

The sides AB and CD of a trapezium ABCD are parallel, with AB being the smaller side. P is the midpoint of CD and ABPD is a parallelogram. If the difference between the areas of the parallelogram ABPD and the triangle BPC is 10 sq cm, then the area, in sq cm, of the trapezium ABCD is

If a rhombus has area 12 sq cm and side length 5 cm, then the length, in cm, of its longer diagonal is

A circle of diameter 8 inches is inscribed in a triangle ABC where $$\angle ABC = 90^\circ$$. If BC = 10 inches then the area of the triangle in square inches is

Suppose the length of each side of a regular hexagon ABCDEF is 2 cm.It T is the mid point of CD,then the length of AT, in cm, is

If the area of a regular hexagon is equal to the area of an equilateral triangle of side 12 cm, then the length, in cm, of each side of the hexagon is

From an interior point of an equilateral triangle, perpendiculars are drawn on all three sides. The sum of the lengths of the three perpendiculars is s. Then the area of the triangle is

Let C1 and C2 be concentric circles such that the diameter of C1 is 2cm longer than that of C2. If a chord of C1 has length 6cm and is a tangent to C2, then the diameter, in cm, of C1 is

The sum of the perimeters of an equilateral triangle and a rectangle is 90cm. The area, T, of the triangle and the area, R, of the rectangle, both in sq cm, satisfying the relationship $$R=T^{2}$$. If the sides of the rectangle are in the ratio 1:3, then the length, in cm, of the longer side of the rectangle, is

Let C be a circle of radius 5 meters having center at O. Let PQ be a chord of C that passes through points A and B where A is located 4 meters north of O and B is located 3 meters east of O. Then, the length of PQ, in meters, is nearest to

The vertices of a triangle are (0,0), (4,0) and (3,9). The area of the circle passing through these three points is

In a trapezium $$ABCD$$, $$AB$$ is parallel to $$DC$$, $$BC$$ is perpendicular to $$DC$$ and $$\angle BAD=45^{0}$$. If $$DC$$ = 5cm, $$BC$$ = 4 cm,the area of the trapezium in sq cm is

The points (2,1) and (-3,-4) are opposite vertices of a parallelogram.If the other two vertices lie on the line $$x+9y+c=0$$, then c is

A circle is inscribed in a rhombus with diagonals 12 cm and 16 cm. The ratio of the area of circle to the area of rhombus is

On a rectangular metal sheet of area 135 sq in, a circle is painted such that the circle touches two opposite sides. If the area of the sheet left unpainted is two-thirds of the painted area then the perimeter of the rectangle in inches is

A solid right circular cone of height 27 cm is cut into two pieces along a plane parallel to its base at a height of 18 cm from the base. If the difference in volume of the two pieces is 225 cc, the volume, in cc, of the original cone is

Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L is

If the rectangular faces of a brick have their diagonals in the ratio $$3 : 2 \surd3 : \surd{15}$$, then the ratio of the length of the shortest edge of the brick to that of its longest edge is

Corners are cut off from an equilateral triangle T to produce a regular hexagon H. Then, the ratio of the area of H to the area of T is

In a circle of radius 11 cm, CD is a diameter and AB is a chord of length 20.5 cm. If AB and CD intersect at a point E inside the circle and CE has length 7 cm, then the difference of the lengths of BE and AE, in cm, is

AB is a diameter of a circle of radius 5 cm. Let P and Q be two points on the circle so that the length of PB is 6 cm, and the length of AP is twice that of AQ. Then the length, in cm, of QB is nearest to

Let A and B be two regular polygons having a and b sides, respectively. If b = 2a and each interior angle of B is $$\frac{3}{2}$$ times each interior angle of A, then each interior angle, in degrees, of a regular polygon with a + b sides is

In a triangle ABC, medians AD and BE are perpendicular to each other, and have lengths 12 cm and 9 cm, respectively. Then, the area of triangle ABC, in sq cm, is

A man makes complete use of 405 cc of iron, 783 cc of aluminium, and 351 cc of copper to make a number of solid right circular cylinders of each type of metal. These cylinders have the same volume and each of these has radius 3 cm. If the total number of cylinders is to be kept at a minimum, then the total surface area of all these cylinders, in sq cm, is

Let ABC be a right-angled triangle with hypotenuse BC of length 20 cm. If AP is perpendicular on BC, then the maximum possible length of AP, in cm, is

The base of a regular pyramid is a square and each of the other four sides is an equilateral triangle, length of each side being 20 cm. The vertical height of the pyramid, in cm, is

Two circles, each of radius 4 cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is

From a rectangle ABCD of area 768 sq cm, a semicircular part with diameter AB and area 72π sq cm is removed. The perimeter of the leftover portion, in cm, is

A triangle ABC has area 32 sq units and its side BC, of length 8 units, lies on the line x = 4. Then the shortest possible distance between A and the point (0,0) is

On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, is

The area of a rectangle and the square of its perimeter are in the ratio 1 ∶ 25. Then the lengths of the shorter and longer sides of the rectangle are in the ratio

A chord of length 5 cm subtends an angle of 60° at the centre of a circle. The length, in cm, of a chord that subtends an angle of 120° at the centre of the same circle is

A parallelogram ABCD has area 48 sqcm. If the length of CD is 8 cm and that of AD is s cm, then which one of the following is necessarily true?

In a circle with center O and radius 1 cm, an arc AB makes an angle 60 degrees at O. Let R be the region bounded by the radii OA, OB and the arc AB. If C and D are two points on OA and OB, respectively, such that OC = OD and the area of triangle OCD is half that of R, then the length of OC, in cm, is

In a parallelogram ABCD of area 72 sq cm, the sides CD and AD have lengths 9 cm and 16 cm, respectively. Let P be a point on CD such that AP is perpendicular to CD. Then the area, in sq cm, of triangle APD is

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

Given an equilateral triangle T1 with side 24 cm, a second triangle T2 is formed by joining the midpoints of the sides of T1. Then a third triangle T3 is formed by joining the midpoints of the sides of T2. If this process of forming triangles is continued, the sum of the areas, in sq cm, of infinitely many such triangles T1, T2, T3,... will be

Points E, F, G, H lie on the sides AB, BC, CD, and DA, respectively, of a square ABCD. If EFGH is also a square whose area is 62.5% of that of ABCD and CG is longer than EB, then the ratio of length of EB to that of CG is

Let ABCD be a rectangle inscribed in a circle of radius 13 cm. Which one of the following pairs can represent, in cm, the possible length and breadth of ABCD?

In a circle, two parallel chords on the same side of a diameter have lengths 4 cm and 6 cm. If the distance between these chords is 1 cm, then the radius of the circle, in cm, is

Let P be an interior point of a right-angled isosceles triangle ABC with hypotenuse AB. If the perpendicular distance of P from each of AB,BC,and CA is $$4(\sqrt{2}-1)$$ cm,then the area, in sq cm, of the triangle ABC is

If three sides of a rectangular park have a total length 400 ft, then the area of the park is maximum when the length (in ft) of its longer side is

ABCD is a quadrilateral inscribed in a circle with centre O such that O lies inside the quadrilateral. If $$\angle COD = 120$$ degrees and $$\angle BAC = 30$$ degrees, then the value of $$\angle BCD$$ (in degrees) is

The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y =3x+c,then c is

The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 cm and 20 cm while the other two sides are of equal length. The perpendicular distance between the parallel sides of the trapezium is 12 cm. If the height of the pillar is 20 cm, then the total area, in sq cm, of all six surfaces of the pillar is

Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is

The shortest distance of the point $$(\frac{1}{2},1)$$ from the curve y = I x -1I + I x + 1I is

Let ABC be a right-angled triangle with BC as the hypotenuse. Lengths of AB and AC are 15 km and 20 km, respectively. The minimum possible time, in minutes, required to reach the hypotenuse from A at a speed of 30 km per hour is

A ball of diameter 4 cm is kept on top of a hollow cylinder standing vertically. The height of the cylinder is 3 cm, while its volume is $$9\pi$$ cubic centimeters. Then the vertical distance, in cm, of the topmost point of the ball from the base of the cylinder is

A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8 : 27 : 27. The percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube is nearest to

Let ABC be a right-angled isosceles triangle with hypotenuse BC. Let BQC be a semi-circle, away from A, with diameter BC. Let BPC be an arc of a circle centered at A and lying between BC and BQC. If AB has length 6 cm then the area, in sq cm, of the region enclosed by BPC and BQC is

From a triangle ABC with sides of lengths 40 ft, 25 ft and 35 ft, a triangular portion GBC is cut off where G is the centroid of ABC. The area, in sq ft, of the remaining portion of triangle ABC is

The area of the closed region bounded by the equation I x I + I y I = 2 in the two-dimensional plane is