Geometry Questions

The circumference of a circle exceeds its diameter by 30 cm. The area (in cm
2
) of the circle is: (Take π=
7
22
​
)

What is the total surface area of a cone whose radius is $\frac{r}{4}$ and slant height is $4l$?

In a rhombus, one diagonal is $40\text{cm}$ and the other diagonal is $22\text{cm}$ longer than half the first diagonal. Find the length of each side of the rhombus.

Starting from a point, Karan moves $6\text{km}$ in one direction and then $8\text{km}$ at a right angle to his initial path. What is the direct distance between his starting point and his final position?

If you have a regular pentagonal prism and you slice it parallel to its bases, what shape is the cross-section?

A question along with a set of statements is given. Find which of the given statements is/are sufficient to answer the question.
Question: Find the perimeter of a triangle given that:
Statements:
I. Area of the triangle is $24\text{sq. units}$.
II. Height of the triangle is $4\text{units}$.

Atul walks 5 km toward North and then turns to the right. After walking 3 km he turns to the right and walks 5 km. How far away is he from his starting position?

Choose the figure which is different from the rest based on the relationship between the polygon sides and internal dots:

Choose the figure which is different from the rest based on its boundary properties:

In ΔABC, D and E are points on sides AB and BC, respectively, such that DE∥AC. If BD=4 cm and AB=12 cm, then the ratio of the area of ΔBDE to that of quadrilateral ACED is:

Find out the number of triangles in the given figure.

The geometry and magnetic behaviour of the complex $[{\text{Ni(CO)}}_4]$ are:

When a cone is sliced by a plane parallel to its base, but not passing through its apex, what is the shape of the resulting cross-section?

When a cone is sliced by a plane parallel to its base, but not passing through its apex, what is the shape of the resulting cross-section?

Let A,B,C be the mid-points of sides XY,YZ and XZ, respectively of ΔXYZ. If the area of ΔXYZ is 8464 cm
2
, then find the area (in cm
2
) of ΔABC.

In ΔABC, P and Q are the middle points of the sides AB and AC, respectively. R is a point on the segment PQ such that PR:RQ=1:4. If PR=5 cm, then BC= ?

If the surface areas of two spheres are in the ratio of 9: 49, then the ratio of their volumes is:

In $\Delta ABC$, $BD\perp AC$ at $D$. $E$ is a point on $BC$ such that $\angle BEA=x^{\circ }$. If $\angle EAC=36^{\circ }$ and $\angle EBD=40^{\circ }$, then the value of $x$ is:

Find the related word.
$$\pi :\text{Circle}::\varphi :?$$

For a regular polygon, let $r$ and $R$ be the radii of the inscribed and the circumscribed circles. A false statement among the following is:

How many rectangles are there in the given figure?

Find the area of a right angle whose base is 15 cm and hypotenuse is 17 cm.

Read the question and statements provided and decide whether the data provided in the statements are sufficient to answer the question.
Question: Find the area of a triangle.
Statements:
I. One of its sides is $6\text{ units}$.
II. It is an equilateral triangle.

The base of a parallelogram is twice its height. If the area of the parallelogram is $338{\text{cm}}^2$, then find its height (in $\text{cm}$).

In a parallelogram, one of the parallel sides is $16\text{cm}$ and the other side is $12\text{cm}$. If the perpendicular distance between the two parallel sides of dimension $16\text{cm}$ is $24\text{cm}$, then the perpendicular distance between its other two parallel sides is:

How many quadrilaterals are there in the given figure?

If the volume of a cube is given as 12167cm
3
, then the surface area of the cube will be:

In △ABC and △PQR, AB=7m, BC=8m, AC=9m, PQ=7m, QR=8m and PR=9m. Which of the following is true for these triangles?

The difference between supplementary and complementary angles of any given acute angle is:

Count the total number of squares in a standard $4\times 4$ grid matrix configuration:

Determine the number of pentagons in the following figure.

The diagonals $AC$ and $BD$ of a parallelogram $ABCD$ intersect each other at the point $O$. If $\angle DAC=32^{\circ }$ and $\angle AOB=70^{\circ }$, then $\angle DBC$ is equal to

In ΔABC, AB and AC are produced to points D and E, respectively. If the bisectors of ∠CBD and ∠BCE meet at point O such that ∠BOC=76
∘
, then ∠A= ?

Match List-I with List-II:

The lines $L_1:y-x=0$ and $L_2:2x+y=0$ intersect the line $L_3:y+2=0$ at $P$ and $Q$ respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersect $L_3$ at $R$.Statement-1 : The ratio $PR:RQ$ equals $2\sqrt{2}:\sqrt{5}$.Statement-2 : In any triangle, bisector of an angle divides the opposite side in the ratio of the sides containing the angle.

Two adjacent sides of a parallelogram are $74\text{cm}$ and $40\text{cm}$. If one of its diagonals is $102\text{cm}$, then the area of the parallelogram is

△ABC is inscribed in a circle with centre O. If AB=35 cm, BC=12 cm and AC=37 cm, then what is the circumradius of the triangle?

If each exterior angle of a regular polygon is $18^{\circ }$, then the number of sides of the polygon is

The ratio of length and breadth of a rectangle is $21:20$ and its perimeter is $164\text{cm}$. What is the length of its diagonal (in cm)?

Count the number of triangles in the given figure.

In $\Delta ABC$, $AD$ is the bisector of $\angle A$ and intersects $BC$ at $D$. If $BC=a$, $AC=b$ and $AB=c$, then $BD$ is equal to:

Three solid iron cubes of edges 8 cm, 10 cm and 12 cm are melted together to make a new cube. It is observed that 496 cm³ of the melted material is lost due to improper handling. The area (in cm²) of the whole surface of the newly formed cube is:

If the area of an equilateral triangle is $a$ and its height is $b$, then $\frac{b^2}{a}$ is

In a triangle $ABC$, medians $AD$ and $BE$ are drawn. If $AD=4$, $\angle DAB=\frac{\pi }{6}$ and $\angle ABE=\frac{\pi }{3}$, then the area of the $\Delta ABC$ is:

A horse is tethered to a stake by a rope $30\text{meters}$ long. If the horse moves along the circumference of a circle keeping the rope tight, find how far the horse has gone from the initial point if the rope has traced an angle of $105^{\circ }$ at the centre?

In ΔXYZ, R and S are points on the sides XY and XZ respectively, such that RS is parallel to YZ. XR=15cm, XY=25cm, XS=12cm and XZ=20cm. RS is equal to:

If the angles of a triangle are in the ratio of 9:11:16, then the difference between the greatest angle and the smallest angle is:

In ΔABC, D and F are the midpoints of the sides AB and AC, respectively. E is a point on the segment DF such that DE:EF=1:2. If DE=4cm, then BC is equal to:

What is the area of a triangle with base $8\text{meters}$ and height $16\text{meters}$?

If the distance between the centres of two circles is less than the sum of radii of the two, then they:

In a parallelogram ABCD, if vertices are in respective order and diagonals AC and BD intersect at O, then which of the following is NOT always correct?

Which of the following pairs of compounds is isoelectronic and isostructural?

In a pair of adjacent angles: (i) Vertex is always common, (ii) one arm is always common and (iii) uncommon arms are always opposite rays. Then

The length and breadth of a rectangle are $12\text{cm}$ and $5\text{cm}$ respectively. What must be the length of its diagonal (in $\text{cm}$)?

The sum of the radii of inscribed and circumscribed circles for an $n$ -sided regular polygon of side $a$, is:

If the angle between the line $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z-3}{\lambda }$ and the plane $x+2y+3z=4$ is ${\cos }^{-1}\sqrt{\frac{5}{14}}$, then $\lambda$ equals:

The radius of a hemisphere is 7cm, find its volume.

The area of a circle is $38.5{\text{cm}}^2$. Find its radius (in $\text{cm}$).

A pentagonal prism has 15 edges. How many vertices does it have?

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point $(-3,1)$ and has eccentricity $\sqrt{2/5}$ is:

△ABC is similar to △PQR. Length of AB is 18cm and length of the corresponding side PQ is 12cm. If area of △ABC is 324sq cm, what is the area of △PQR?

What is the area (in cm
2
) of a sector of a circle, with central angle 75
∘
, when the diameter of the circle is 9cm?

Points P and Q lie on sides AB and AC of triangle ABC respectively such that segment PQ is parallel to side BC. If the ratio of AP:PB is 2:5, and the area of △APQ is 4sq cm, what is the area of trapezium PQCB?

The diameter of a circle is equal to the side of the square. What is the area of the square if the area of the circle is 49πsq cm?

In a circle with centre 'O', if the angle subtended by an arc at the circumference ∠BCA=50
∘
, then what is the value of ∠BDA where D is another point on the same major arc segment?

Given that triangle ABC is congruent to triangle DEF. If AC=11 m, ED=6 m, EF=15 m, ∠FDE=85
∘
and ∠ABC=55
∘
, then the perimeter of triangle ABC and ∠EFD are, respectively:

The perimeter of an isosceles right-angled triangle having an area of 200cm
2
is:

The volume of a cube is four times the volume of a cuboid. If the sides of the cuboid are $32\text{cm}$, $8\text{cm}$ and $4\text{cm}$, then find the ratio of the total surface area of the cube to that of the cuboid.

Find the area of a minor sector of a circle whose circumference is $88\text{cm}$ and the length of its minor arc is $22\text{cm}$ (use $\pi =\frac{22}{7}$).

The two circles $x^2+y^2=ax$ and $x^2+y^2=c^2$ ($c>0$) touch each other if:

What is the length of the arc of a circle whose radius is $35\text{cm}$ and whose arc subtends an angle of $72^{\circ }$ at the centre of the circle?

How many triangles are there in the given figure?

In triangle $PQR$, the length of side $QR$ is less than twice the length of side $PQ$ by $3\text{cm}$. The length of side $PR$ exceeds the length of side $PQ$ by $14\text{cm}$. If the perimeter is $55\text{cm}$, find the length of the smallest side of the triangle $PQR$.

Two circles with centers $B$ and $D$ have radii $DA=8\text{cm}$ and $BC=x\text{cm}$, respectively. $AC$ is tangent to both circles. If $DB$ and $AC$ intersect at the point $E$, $AE=12\text{cm}$ and $EC=18\text{cm}$, then find the value of $x$ (in $\text{cm}$).

A regular dodecahedron is sliced by a plane passing through two opposite edges and not through any vertices. What is the shape of the cross-section?

A cuboid with dimensions 20cm×12cm×10cm is cut into 8 identical pieces by 3 cuts. What will be the total surface area of all the pieces?

A cone is cut through a plane parallel to its base and then the cone that is formed on one side of that plane is moved. The new part that is left over on the other side of the plane is called_______________.

The volumes of two spheres are in the ratio of $512:3375$. The ratio of their surface areas is:

When a cylinder is sliced by a plane parallel to its height and passing through the center, what is the shape of the cross-section ?

What is the measure of an interior angle of a regular dodecagon?

Let $z_1$ and $z_2$ be two roots of the equation $z^2+az+b=0$, $z$ being complex. Further, assume that the origin, $z_1$ and $z_2$ form an equilateral triangle, then:

Statement-1 : The point $A(1,0,7)$ is the mirror image of the point $B(1,6,3)$ in the line $\frac{x-1}{0}=\frac{y-1}{2}=\frac{z-2}{3}$.Statement-2 : The line $\frac{x-1}{0}=\frac{y-1}{2}=\frac{z-2}{3}$ bisects the line segment joining $A(1,0,7)$ and $B(1,6,3)$.

Let ABCD be a quadrilateral and AC,BD be diagonals. The length of BD=20cm and the heights of the triangles ABD and BCD are, respectively, 6cm and 8cm. Find the area of the quadrilateral ABCD (in cm
2
).

Find the total surface area of a hemisphere with a radius of $11\text{cm}$?

Chords AB and CD of a circle, when produced, meet at a point P outside the circle. If AB=6cm, CD=3cm and PD=5cm, then PB is equal to:

How many triangles are there in the given figure?

If $L$ is the mid-point of the side $YZ$ of $△XYZ$, and the area of $△XYL$ is $13{\text{cm}}^2$, then the area (in ${\text{cm}}^2$) of $△XYZ$ is:

Find the sum of the measures of all the exterior angles of a decagon.

The centres of two circles of radii 20cm and 32cm are 60cm apart. What is the ratio of the length of the direct common tangent to the length of the transverse common tangent to these circles?

The perimeter of the rectangle is 280m and the difference between its two sides is 40m. Find the side of a square whose area is equal to the area of this rectangle.

In a triangle PQR, RS intersects PQ at point S. The sides of the triangle QR=36cm, SQ=27cm, RS=18cm and ∠QRS=∠QPR. What is the ratio of the perimeter of △PRS to that of △QSR?

In a $\Delta PQR$, if $3\sin P+4\cos Q=6$ and $4\sin Q+3\cos P=1$, then the angle $R$ is equal to:

A hemisphere has 14cm diameter. Find its volume.

The sides AB and AC of △ABC are produced to points D and E, respectively. The bisectors of ∠CBD and ∠BCE meet at P. If ∠A=72
∘
, then the measure of ∠P is:

If $z\ne 1$ and $\frac{z^2}{z-1}$ is real, then the point represented by the complex number $z$ lies:

In a triangle ABC, P is the midpoint of BC. If AB = (2x+4) cm, AC = 6 cm and AP ⊥ BC, then the value of x is:

If $\angle A=\angle D$ and $\frac{AB}{DE}=\frac{AC}{DF}$, then both triangles $ABC$ and $DEF$ are similar by which of the following criteria?

In △ABC, D is the mid-point of BC and G is the centroid. If GD = 10 cm, then the length of AD is ____________.

$\text{ABCD}$ is a trapezium such that $\text{AB}$ and $\text{CD}$ are parallel and $\text{BC}\perp \text{CD}$. If $\angle \text{ADB}=\theta$, $\text{BC}=p$ and $\text{CD}=q$, then $\text{AB}$ is equal to:

Find the area (in m
2
) of the rectangular field, one of whose sides is 35m and a diagonal is 37m.