Geometry and Mensuration CAT PYQs , 153 Questions with Solutions

If the length of a side of a rhombus is 36 cm and the area of the rhombus is 396 sq. cm, then the absolute value of the difference between the lengths, in cm, of the diagonals of the rhombus is

In a circle with center $C$ and radius $6 \sqrt{2} \mathrm{~cm}, P Q$ and $S R$ are two parallel chords separated by one of the diameters. If $\angle P Q C=45^{\circ}$, and the ratio of the perpendicular distance of $P Q$ and $S R$ from $C$ is $3: 2$, then the area, in sq. cm , of the quadrilateral $P Q R S$ is

In a $\triangle A B C$, points D and E are on the sides BC and AC , respectively. BE and AD intersect at point T such that $\mathrm { AD } : A T = 4 : 3$, and $\mathrm { BE } : \mathrm { BT } = 5 : 4$. Point F lies on AC such that DF is parallel to BE . Then, $\mathrm { BD } : \mathrm { CD }$ is

In a triangle PQR, point X is on PQ and point Y is on PR such that XP = 1.5 units, XQ = 6 units, PY = 2 units and YR = 8 units. Consider the following statements:

Statement I: QR = 5 XY Statement II: Triangle PYX is similar to triangle PRQ

In the light of the above statements, choose the correct answer from the options given below.

Let ABCDEF be a regular hexagon and P and Q be the midpoints of AB and CD, respectively. Then, the ratio of the areas of trapezium PBCQ and hexagon ABCDEF is

A cylindrical box of radius 5 cm contains 10 solid spherical balls, each of radius 5 cm. If the topmost ball touches the upper cover of the box, then the volume of empty space in the box is:

Two tangents drawn from a point $P$ touch a circle with center $O$ at points $Q$ and $R$. Points $A$ and $B$ lie on $P Q$ and $P R$, respectively, such that $A B$ is also a tangent to the same circle. If $\angle A O B = 50 ^ { \circ }$, then $\angle A P B$, in degrees, equals

Given below are two statements:

Statement I: The ratio of the circumference of two circles is 3:4, then the ratio of their areas is 9:16. Statement II: If the circumference and the area of a circle are numerically equal, then the diameter is equal to 4.

In the light of the above statements, choose the correct answer from the options given below.

ABCD is a trapezium in which AB is parallel to $\mathrm { DC } , \mathrm { AD }$ is perpendicular to AB , and $\mathrm { AB } = 3 \mathrm { DC }$. If a circle inscribed in the trapezium touching all the sides has a radius of 3 cm , then the area, in sq. cm , of the trapezium is

In $\triangle A B C , A B = A C = 12 \mathrm {~cm}$ and $D$ is a point on side $B C$ such that $A D = 8 \mathrm {~cm}$. If $A D$ is extended to a point $E$ such that $\angle A C B = \angle A E B$, then the length, in cm , of $A E$ is

A triangle ABC is formed with AB = AC = 50 cm and BC = 80 cm. Then, the sum of the lengths, in cm, of all three altitudes of the triangle ABC is

A solid trophy, consisting of two parts, has been designed in the following manner: the bottom part is a frustum of a cone with the bottom radius 30 cm, the top radius 20 cm, and height 40 cm, while the top part is a hemisphere with radius 20 cm. Moreover, the flat surface of the hemisphere is the same as the top surface of the frustum. If the entire trophy is to be gold-plated at the cost of Rs. 40 per square cm, what would the cost for gold-plating be closest to?

Adu and Amu have bought two pieces of land on the Moon from an e-store. Both the pieces of land have the same perimeters, but Adu’s piece of land is in the shape of a square, while Amu’s piece of land is in the shape of a circle.

The ratio of the areas of Adu’s piece of land to Amu’s piece of land is:

A farmer has a quadrilateral parcel of land with a perimeter of 700 feet. Two opposite angles of that parcel of land are right angles, while the remaining two are not. The farmer wants to do organic farming on that parcel of land. The cost of organic farming is Rs. 400 per square foot.

Consider the following two additional pieces of information:

I. The length of one of the sides of that parcel of land is 110 feet. II. The distance between the two corner points where the non-perpendicular sides of that parcel of land intersect is 265 feet.

To determine the amount of money the farmer needs to spend to do organic farming on the entire parcel of land, which of the above additional pieces of information are MINIMALLY SUFFICIENT?

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 $\triangle A D E$ 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 cylindrical vessel of radius 4 cm contains water. A solid sphere of 3 cm radius is lowered into the water until it is completely immersed. The water level in the vessel will rise by :

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

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

$A B C D$ is a trapezium in which $A B$ is parallel to $C D$ . The sides $A D$ and $B C$ when extended, intersect at point $E$ . If $A B=2 \mathrm{~cm}, C D=1 \mathrm{~cm}$ , and perimeter of $A B C D$ is 6 cm , then the perimeter, in cm , of $\triangle A E B$ is

The midpoints of sides AB, BC, and AC in △ ABC 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 △ ABC is 1440 sq cm, then the area, in sq cm, of △ XYZ is

A regular octagon ABCDEFGH has sides of length 6 cm each. Then the area, in sq. cm , of the square ACEG 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 $120^{\circ}$ at the center. Then, the area, in square meters, of the smaller region is

The roots of the polynomial $P(x) = 2x^{3} - 11x^{2} + 17x - 6$ are the radii of three concentric circles. The ratio of their area, when arranged from the largest to the smallest, is:

A farmer has a triangular plot of land. One side of the plot, henceforth called the base, is 300 feet long and the other two sides are equal. The perpendicular distance, from the corner of the plot, where the two equal sides meet, to the base, is 200 feet. To counter the adverse effect of climate change, the farmer wants to dig a circular pond. He plans that half of the circular area will be inside the triangular plot and the other half will be outside, which he will purchase at the market rate from his neighbour. The diameter of the circular plot is entirely contained in the base and the circumference of the pond touches the two equal sides of the triangle from inside. If the market rate per square feet of land is Rs. 1400, how much does the farmer must pay to buy the land from his neighbour for the pond? (Choose the closest option.)

Ritu wants to make a trapezium such that AB is parallel to CD. Angle ABC and BAD are 90 and 45 degree respectively. Length of CD and BC are 5cm and 4cm respectively. Find the area of trapezium.

Consider a right-angled triangle ABC, right angled at B. Two circles, each of radius r, are drawn inside the triangle in such a way that one of them touches AB and BC, while the other one touches AC and BC. The two circles also touch each other (see the image). If AB = 18 cm and BC = 24 cm, then find the value of r.

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 ΔABP, ΔABQ and ΔABC are in arithmetic progression. If the area of ΔABC is 1.5 times the area of ΔABP, the length of PQ, in cm, is

Given below are two statements: Statement I: The perimeter of a triangle is greater than the sum of its three medians. Statement II: In any triangle ABC, if D is any point on BC, then AB + BC + CA > 2AD. In the light of the above statements, choose the correct answer from the options given below:

Given below are two statements: Statement I: If two chords XY and ZT of a circle intersects internally at point P, then PX − PY = PZ − PT Statement II: If two chords XY and ZT of a circle intersect internally at point P, then triangles PXZ and PTY are similar triangles. In the light of the above statements, choose the correct answer from the options given below:

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

Ice-cream, completely filled in a cylinder of diameter 35 cm and height 32 cm, is to be served by completely filling identical disposable cones of diameter 4 cm and height 7 cm. The maximum number of cones that can be used in this way is

Which two figures will have same area? (A) Circle with diameter of 14 cm. (B) Rectangle with length 12 cm and breadth 10 cm. (C) Rectangle with length 14 cm and breadth 11 cm. (D) Square with side 12 cm. Choose the correct answer from the options given below:

ABC is a triangle with $BC = 5$. $D$ is the foot of the perpendicular from $A$ on $BC$. $E$ is a point on $CD$ such that $BE = 3$. The value of $AB^{2} - AE^{2} + 6CD$ 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 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

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

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

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

From each vertex of an equilateral triangle with area of 240 square units, corners are cut to form a regular hexagon. Area of hexagon will be:

A non-flying ant wants to travel from the bottom corner to the diagonally opposite top corner of a cubical room. The side of the room is 2 meters. What will be the minimum distance that the ant needs to travel?

ABCD is a trapezoid where BC is parallel to AD and perpendicular to AB. Kindly note that $BC < AD$. P is a point on $AD$ such that $\triangle CPD$ is an equilateral triangle. Q is a point on $BC$ such that $AQ$ is parallel to $PC$. If the area of triangle $CPD$ is $4\sqrt{3}$, find the area of triangle $ABQ$.

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

Ramesh and Reena are playing with triangle ABC. Ramesh draws a line that bisects ∠BAC; this line cuts BC at D. Reena then extends AD to a point P. In response, Ramesh joins B and P. Reena then announces that BD bisects ∠PBA. Together, Ramesh and Reena find that BD = 6 cm, AC = 9 cm, DC = 5 cm, BP = 8 cm, and DP = 5 cm. How long is AP?

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

There are two containers of the same volume, first container half-filled with sugar syrup and the second container half-filled with milk. Half the content of the first container is transferred to the second container, and then the half of this mixture is transferred back to the first container. Next, half the content of the first container is transferred back to the second container. Then the ratio of sugar syrup and milk in the second container is

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

In the given diagram, ABCD is a square of area 300 sq. cm. Triangle ABE is an equilateral triangle sitting on side AB, with a circle inscribed inside the triangle. What is the radius of the circle shown in the figure ?

If a 30 meter ladder is placed against a wall such that it just reaches the top of the wall, if the horizontal distance between the wall and the base of the ladder is 1/3rd of the length of ladder, then the height of wall is :

A copper wire having length of 243 m and diameter 4 mm was melted to form a sphere. Find the diameter of the sphere:

If the radius and height of a solid cone are increased by 10% and 20% respectively. By what percentage the volume of the cone is increased?

The area of the circle is 616 m². What is the area of the rectangle? (the rectangle's top side passes through the circle's center, its bottom side is tangent to the circle, and its left/right sides touch the circle at the diameter's endpoints, as shown in the figure.)

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

A circle of diameter 8 inches is inscribed in a triangle ABC where ∠ABC = 90°. 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

In a triangle ABC, D, E and F are the mid-points of the sides AB, BC and CA respectively. Then the ratio of the area of a triangle DEF and the area of a triangle ABC is:

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

A roller is 4 m long and has a diameter of 0.7 m. It takes exactly 2000 rotations of the roller to level a road. If the cost of using the roller is Rs. 4 per square metres, then the total cost of levelling the road is: (Assume π = 22/7)

In triangle PQR, PS is perpendicular to QR and S divides QR in the ratio of 3:1 internally. If PQ = 21 and PR = 9, find QR.

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

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

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

In a triangle ABC , ∠ BCA =50°. 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 ∠ FDE, in degrees, is equal to

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

Area of a square natural lake is 50 (square miles). A diver wishing to cross the lake diagonally will have to swim a distance of:

The six faces of a wooden cube of side 6 cm are labelled A, B, C, D, E and F respectively. Three of these faces A, B, and C are each adjacent to the other two, and are painted red. The other three faces are not painted. Then, the wooden cube is neatly cut into 216 little cubes of equal size. How many of the little cubes have no sides painted?

Inside a triangular park, there is a flower bed, forming a similar triangle. Around the flower bed runs a uniform path so that the sides of the park are exactly double the corresponding sides of the flower bed. The ratio of areas of the path to the flower bed is:

A ladder is leaning against a wall which is 5 meters high. If the ladder's foot is 2 meters from the wall and the top touches the top edge of the wall, what is the length of the ladder?

At any point of time, let x be the smaller of the two angles made by the hour hand with the minute hand on an analogue clock (in degrees). During the time interval from 2:30 p.m. to 3:00 p.m., what is the minimum possible value of x?

Rajesh, a courier delivery agent, starts at point A and makes a delivery each at points B, C and D, in that order. He travels in a straight line between any two consecutive points. The following are known: (i) AB and CD intersect at a right angle at E, and (ii) BC, CE and ED are respectively 1.3 km, 0.5 km and 2.5 km long. If AD is parallel to BC, then what is the total distance (in km) that Rajesh covers in travelling from A to D?

Two circles P and Q, each of radius $2$ cm, pass through each other’s centres. They intersect at points A and B. A circle R is drawn with diameter AB. What is the area of overlap (in square cm) between the circles R and P?

ABC is a triangle with integer-valued sides AB = 1, BC > 1, and CA > 1. If D is the mid-point of AB, then, which of the following options is the closest to the maximum possible value of the angle ACD (in degrees)?

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

A solid right circular cone of height 27 cm is cut into 2 pieces along a plane parallel to it's base at a height of 18 cm from the base. If the difference in the volume of the two pieces is 225 cc, the volume, in cc, of the original cone is

On a rectangular metal sheet of area 135 sq in, a circle is painted such that the circle touches opposite two 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

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

The sum of perimeters of an equilateral triangle and a rectangle is 90 cm. The area, T, of the triangle and the area , R, of the rectangle, both in sq cm, satisfy 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

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 6 cm and is a tangent to C2, then the diameter, in cm of C1 is

In a trepezium ABCD, AB is parallel to DC, BC is perpendicular to DC and ∠BAD = 45°. If DC = 5 cm, BC = 4 cm, the area of the trepezium in sq. cm is

A rectangular swimming pool is $50$ meters long and $25$ meters wide. Its depth is always the same along its width but linearly increases along its length from $1$ meter at one end to $4$ meters at the other end. How much water (in cubic meters) is needed to completely fill the pool?

Mohanlal, a prosperous farmer, has a square land of side 2 km. For the current season, he decides to have some fun. He marks two distinct points on one of the diagonals of the land. Using these points as centers, he constructs two circles. Each of these circles falls completely within the land, and touches at least two sides of the land. To his surprise, the radii of both the circles are exactly equal to $\frac{2}{3}$ km. Mohanlal plants potatoes on the overlapping portion of these circles. Calculate the area on which Mohanlal planted potatoes (in sq. km).

Two lighthouses, located at points A and B on the earth, are $60$ feet and $40$ feet tall respectively. Each lighthouse is perfectly vertical and the land connecting A and B is perfectly flat. The topmost point of the lighthouse at A is A’ and of the lighthouse at B is B’. Draw line segments A’B and B’A, and let them intersect at point C’. Drop a perpendicular from C’ to touch the earth at point C. What is the length of CC’ in feet?

In the given question, two quantities are given, one as 'Quantity I' and another as 'Quantity II'. Determine the relationship between the two quantities and choose the appropriate option:

A circle with centre 'O' is circumscribed about an equilateral triangle ABC of side $24\sqrt{3}$ cm.

Quantity I: Find the height of the triangle (in cm). Quantity II: Find the length of a rectangle (in cm) having perimeter 120 cm and breadth equal to the radius of the circle.

In the given question, two quantities are given, one as 'Quantity I' and another as 'Quantity II'. Determine the relationship between the two quantities and choose the appropriate option:

The length of the body diagonal of a cube is $60\sqrt{3}$ cm. The cube is melted and formed into smaller cubes with side 4 cm each, or alternatively melted and formed into cuboids each having length 8 cm, width 5 cm & height 2 cm.

Quantity I: Find the difference between the increase in total surface area of all smaller cubes with respect to the original cube, and the increase in total surface area of all cuboids formed with respect to the original cube (in $m^2$). Quantity II: 3.16 $m^2$

XYZ is an equilateral triangle, inscribed in a circle. P is a point on the arc YZ such that X and P are on opposite sides of the chord YZ. Which of the following MUST always be true? A XZ + YP = XY + PZ B XP = XY C XP + PZ = XY + YP D XP = YP + PZ E XP = XY + YZ

Find the ratio of the area of a circle to the area of a square, if the perimeter of the circle and the square are equal.

In the given question, two quantities are given, one as 'Quantity I' and another as 'Quantity II'. Determine the relationship between the two quantities and choose the appropriate option:

Quantity I: A man has two solid balls. The ratio between the radius of the first ball and the second ball is 4 : 3. If the man cuts the second ball from the middle, then the difference between the total surface area of the first ball and the total surface area of a part (hemisphere) of the second ball is 1424.5 $cm^2$. Find the radius of the bigger ball. Quantity II: The height of a cylindrical vessel is equal to the side of a square having area 256 $cm^2$. If the volume of the cylindrical vessel is 22176 $cm^3$, find the radius of the cylindrical vessel.

In the figure given below, the circle has a chord AB of length 12 cm, which makes an angle of 60° at the center of the circle, O. ABCD, as shown in the diagram, is a rectangle. OQ is the perpendicular bisector of AB, intersecting the chord AB at P, the arc AB at M and CD at Q. OM = MQ. The area of the region enclosed by the line segments AQ and QB, and the arc BMA, is closest to (in cm²):

A rectangular field is 40 meters long and 30 meters wide. Draw diagonals on this field and then draw circles of radius 1.25 meters, with centers only on the diagonals. Each circle must fall completely within the field. Any two circles can touch each other but should not overlap. What is the maximum number of such circles that can be drawn in the field?

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 S be the set of all points (x,y) in the x-y plane such that |x| + |y| ≤ 2 and |x| ≥ 1. Then, the area, in square units, of the region represented by S equals [TITA]

If the rectangular faces of a brick have their diagonals in the ratio $3 : 2 \sqrt{3} : \sqrt{15}$, 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

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

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

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

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

In the picture below, $EFGH$, $ABCD$ are squares, and $ABE$, $BCF$, $CDG$, $DAH$ are equilateral triangles. What is the ratio of the area of the square $EFGH$ to that of $ABCD$?

Square, Circle, Hexagon and Octagon have equal perimeter. Which has maximum area?

A gold ingot in the shape of a cylinder is melted and the resulting molten metal molded into a few identical conical ingots. If the height of each cone is half the height of the original cylinder and the area of the circular base of each cone is one fifth that of the circular base of the cylinder, then how many conical ingots can be made?

Three circles of radius 1cm each (labeled A, B, C) are mutually tangent, forming a small curved-triangle gap between them. What is the area of the shaded portion (the gap)?

In a closed wooden box, length = 20 cm, breadth = 14 cm, height = 10 cm, thickness = 5 mm. If the weight of the empty box is 3.462 kg, what is the weight per cm³ of wood?

What is the maximum number of points that can be placed on a circular disk of radius $1$ metre (some of the points could be placed on the bounding circle of the disk) such that no two points are at a distance of less than $1$ metre from each other?

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:

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 π = $\frac{22}{7}$ , the volume, in cubic ft, of the remaining part of the cone is:[TITA]

Given an equilateral triangle $T_1$ with side 24 cm, a second triangle $T_2$ is formed by joining the midpoints of the sides of $T_1$. Then a third triangle $T_3$ is formed by joining the midpoints of the sides of $T_2$. If this process of forming triangles is continued, the sum of the areas, in sq cm, of infinitely many such triangles $T_1, T_2, T_3, \ldots$ will be

In a circle with centre 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 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

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

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?

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?

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

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

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 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

In the given figure, the boundary of shaded region consists of four semi-circular arcs, three smallest being equal. If the diameter of the largest is 14 cm and that of the smallest is 3.5 cm. Calculate the area of shaded region.

It takes $2$ liters to paint the surface of a solid sphere. If this solid sphere is sliced into $4$ identical pieces, how many liters will be required to paint all the surfaces of these $4$ pieces.

If the diagonals of a rhombus of side $15$ cm are in the ratio $3:4$, nd the area of the rhombus.

If 'h', 'c' and 'v' represents height of cone, curved surface area of cone and volume of cone respectively, then the value of 3πvh³ is

A cylinder of diameter 8√3 units is taken out from a cone of base diameter 16√3 units and height 30 units. If the height of cylinder is 2/3rd of the height of cone, then find the volume of the remaining part of cone.

250 boxes of each type of dimensions 25 × 20 × 5 and 15 × 12 × 5 (in metres) are to be covered with wood. An additional 5% of total surface area is required to cover the box with wood. If the cost of wood is ₹4/m², then total cost is

Find the area of the given figure: a trapezium with parallel sides 60 units (top) and 77 units (bottom), and slant sides 25 units (left) and 26 units (right).

Two circles with radius $2R$ and $\sqrt{2}R$ intersect each other at points $A$ and $B$. The centers of both the circles are on the same side of $AB$. $O$ is the centre of the bigger circle and $\angle AOB$ is $60^\circ$. Find the area of the common region between the two circles.

Given, l1∥l2 and ∠XAB=140°, also OA and OB are angle bisectors of ∠XAB and ∠ZBA respectively, then find the value of x (the angle marked at O).

An umbrella is made by stitching 10 triangular pieces of cloth of two different colours, each piece measuring 20 cm, 50 cm and 50 cm. How much cloth of each colour is required for the umbrella?

A cone of radius $4$ cm with a slant height of $12$ cm was sliced horizontally, resulting into a smaller cone (upper portion) and a frustum (lower portion). If the ratio of the curved surface area of the upper smaller cone and the lower frustum is $1:2$, what will be the slant height of the frustum?

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:

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 π cm3. Then the vertical distance, in cm, of the topmost point of the ball from the base of the cylinder is: (TITA)

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 :

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 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:

Let $AB$, $CD$, $EF$, $GH$, and $JK$ be five diameters of a circle with center $O$.
How many ways can three points be chosen from $A, B, C, D, E, F, G, H, J, K, O$ to form a triangle?

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 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(√2 - 1) cm, then the area, in sq. cm, of the triangle ABC is [TITA

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 [TITA]

ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120° and ∠BAC = 30°, then the value of ∠BCD (in degrees) is [TITA]

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 numbers 1, 2,..., 9 are arranged in a 3 X 3 square grid in such a way that each number occurs once and the entries along each column, each row, and each of the two diagonals add up to the same value. If the top left and the top right entries of the grid are 6 and 2, respectively, then the bottom middle entry is: [TITA]

In the given figure, O is the centre of a circle with OP = OQ = 10 cm (two radii) and angle POQ = 120°. PQ is a chord shown as approximately 12 cm. The area of minor segment PQ is

In the given figure CD = 10 cm. CT is the tangent at C, making an angle of π/4 with CD. O is the centre of the circle. The perimeter of the shaded region AOC is

AB is a chord of a circle. The length of AB is 24 cm. P is the midpoint of AB. Perpendiculars from P on either side of the chord meet the circle at M and N respectively. If PM < PN and PM = 8 cm, then what will be the length of PN?

In the given figure, the semi-circle with centre O has a diameter AB = 2r. PQRS is a square of maximum possible area with P, Q on the diameter AB and R, S on the arc. Two more squares of maximum possible area EFGP and CDQH are constructed alongside. Then RC + FS = …….

The volume of a pyramid with a square base is 200 cubic cm. The height of the pyramid is 13 cm. What will be the length of the slant edges (i.e. the distance between the apex and any other vertex), rounded to the nearest integer?

Two congruent triangles △ABC and △DEF, each with area 198, are placed over each other with AB parallel to DE and the centres of both triangles coinciding. The area of region PQRSTU (the hexagonal overlap) is:

$ABCD$ is a rectangle. $P, Q$ and $R$ are the midpoint of $BC, CD$ and $DA$. The point $S$ lies on the line $QR$ in such a way that $SR:QS=1:3$. The ratio of the area of triangle $APS$ to area of rectangle $ABCD$ is