Geometry Questions

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

In 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:

The chord of a circle is $10\text{cm}$ in length and at a distance of $\frac{10}{\sqrt{3}}\text{cm}$ from the centre. Calculate the angle subtended by the chord at the centre of the circle.

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 $\text{cm}$)?

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

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

△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?

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

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:

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

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?

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

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?

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

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

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

In an isosceles triangle ABC, the angle bisectors of ∠B and ∠C intersect at point O. Find the angle ∠BOC (in degrees), when ∠ABC=∠ACB=75
∘



.

Find the in-radius (in cm) of an equilateral triangle whose sides are 6 cm each.

The area of a square is 60.5cm
2
. Find the length of its diagonal (in cm).

$\text{ABCD}$ 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:

What is the sum of the measures of all the interior angles of a regular polygon of 6 sides?

The total surface area of a cube is 181.5cm
2
. Find its side (in cm).

What is the angle between the hour and minute hands of a clock at 3 o'clock?

What is the shape of the cross-section, when a cone is cut by a plane parallel to its slant height?

A series is given with one term missing. Select the correct alternative from the given ones that will complete the series.
Acute angle, right angle, obtuse angle, ?

The length of the diagonal of a rectangle is $26\text{cm}$ and that of one side is $10\text{cm}$. What is the area of this rectangle?

Find the measure of the central angle of the arc whose length is $22\text{cm}$ and the radius of the circle is $28\text{cm}$.

The area of an equilateral triangle is $9\sqrt{3}{\text{cm}}^2$, find its perimeter.

If the vectors $\overrightarrow{AB}=3\hat{i}+4\hat{k}$ and $\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}$ are the sides of a triangle $ABC$, then the length of the median through $A$ is:

In △ABC, AB=9cm and AC=13cm, then the length of BC could be:

In an isosceles triangle, the angle bisectors of ∠B and ∠C intersect at point O. Find the angle ∠BOC (in degrees), when ∠ABC=∠ACB=75



∘
.

Find the in-radius (in cm) of an equilateral triangle whose sides are 6cm each.

How many balls of radius 10cm can be made by melting a bigger ball of diameter 80cm?

Which of the following best describes a straight angle?

ΔGHI is similar to ΔKLM. If the ratio of Perimeter of ΔGHI:Perimeter of ΔKLM=4:9 and the length of GH is 8cm, what is the length of the corresponding side KL?

What is the measure of an exterior angle of a regular nonagon?

In △ABC and △PQR, we are given ∠A=∠P and AC=PR.
Which of the following options needs to be satisfied for △PQR and △ABC to be congruent?

In a given square figure $\text{ABCD}$, both main diagonals $\text{AC}$ and $\text{BD}$ are drawn, intersecting each other at point $\text{O}$. Count the total number of triangles formed inside the figure.

If the radius of a circle is increased by 12% then its area increases by:

Let $T_n$ be the number of all possible triangles formed by joining vertices of an $n$ -sided regular polygon. If $T_{n+1}-T_n=10$, then the value of $n$ is:

A sphere has a radius of $9.21\text{cm}$. What is its volume? (Use $\pi =3.14$)

Find the number of triangles in the given matrix grid figure.

A question along with a set of statements is given below. We have to find which of the given statements is/are sufficient to answer the given question.

Question : Find the area of the triangle PQR.
Statements :
I. PQ = 5 m., PR = 5 m.
II. Length of PO is 4 m.

Two circles of radii $8\text{cm}$ and $17\text{cm}$ are externally tangent to each other. Find the distance between their centres.

Points P and Q lie on side AB and AC of triangle ABC respectively such that segment PQ is parallel to side BC. If the ratio of AP:PB is 2:3, and area of △APQ is 8sq cm, what is the area of trapezium PQCB?

PQ is a diameter of a circle and PS is a chord. If PQ is 82cm and PS is 80cm, the distance (in cm) of PS from the center of the circle is:

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

From the given image_7f89a2.png, out of the given figures, find the 3 figures that will form a square, when fitted together.

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 total surface area of a hemisphere is 166.32 sq cm, find its radius?

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 $\text{cm}$).

Find the sum of interior angles of a dodecagon?

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:

If the circumradius of an equilateral triangle is 12cm, then what is the value of the inradius (in cm) of the triangle?

Find the area (in cm
2
) of a circle with a maximum radius that can be inscribed in a rectangle of length 18cm and breadth 12cm.

If the side of an equilateral triangle is 6
3

​
cm, then what is the length (in cm) of the circumradius of that triangle?

The length of one side and the diagonal of a rectangle are 5cm and 13cm respectively. Find its area (in cm
2
).

The median AD of a triangle ABC is produced and a perpendicular CF is dropped on it. BE is perpendicular to AD. If BC=34cm and DF=8cm, what is the length (in cm) of BE?

Find the area (in cm
2
) of a semi-circle of radius 28cm.

If the sum of the measures of all the interior angles of a polygon is 1440
∘
, find the number of sides of the polygon.

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 $4\text{units}$.

Find the sum of the measures of all the interior angles of a polygon having 10 sides.

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

Find the volume (in cm
3
) of a cuboid of length, breadth, and height of 8.5cm, 6cm, and 7cm respectively.

If x
2
−1, 2x and x
2
+1 are the three sides of a right-angled triangle, then which of the following can be its hypotenuse?

Calculate the length (in $\text{cm}$) of $PQ$, if the radius of the circle is $7\text{cm}$ and the line through the centre $O$ meets the tangent at point $Q$ such that $OP=25\text{cm}$.

Consider two similar triangles $△PQR$ and $△DEF$ where vertices correspond precisely as $P\leftrightarrow D$, $Q\leftrightarrow E$, and $R\leftrightarrow F$. Choose which of the following equations correctly represents the similarity of both triangles?

The square of the diagonal of a cube is 2346 cm
2
. What is the total surface area (in cm
2
) of the cube?

What is the perpendicular distance (in cm) between the parallel sides of a trapezium whose area is 108sq cm and the lengths of the parallel sides are 9cm and 36cm?

If the lengths of the two parallel sides of a trapezium are $7\text{cm}$ and $9\text{cm}$ and its area is $80{\text{cm}}^2$. Find the distance between its parallel sides (in $\text{cm}$).

The length of a radius of a circle is $13\text{cm}$ and the length of a chord of the circle is $10\text{cm}$, the distance of the chord from the center of the circle is

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= ?

In a trapezium $ABCD$, $AD\parallel BC$, $AD=10\text{cm}$, $BC=5\text{cm}$, and $CD=10\text{cm}$. If $ABCD$ represents an isosceles right-height variant configuration where the non-parallel perpendicular height standard provides direct bounds, let's establish the calculation step using standard geometric parameters. Given structural norms for this platform textbook layout, evaluating the area directly via standard isosceles right drop properties gives $\text{Area}=75\text{sq.cm}$.

If area of circular field is $X$ sq. unit, perimeter is $Y$ unit and length of diameter is $Z$ unit then the value of $\frac{X}{YZ}$ is

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?

The length, breadth and height of a cuboid are 10cm, 6cm and 4cm respectively. What is the total surface area?

A sector is cut from a circle of diameter $42\text{ cm}$. If the angle of the sector is $150^{\circ }$, then what is its area in ${\text{cm}}^2$? (Take $\pi =\frac{22}{7}$)

One internal angle of a rhombus of side 12cm is 120
∘
. What is the length of its longer diagonal?

In a triangle $\Delta PQR$, points $A$, $B$, and $C$ are the midpoints of the sides $PQ$, $QR$, and $RP$ respectively. If the area of the triangle $\Delta ABC$ is $16\text{sq.units}$, then the area of the triangle $\Delta PQR$ is:

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= ?

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)?

$ABCD$ is a cyclic quadrilateral and the sides $DC$ and $AB$ are produced to meet at $P$. If $\angle APD=25^{\circ }$ and $\angle PBC=70^{\circ }$, then $\angle BAD$ is equal to:

If the radius of a circle is increased by 25%, its area increases by:

The chord length of a chord made on an arc of a circle is equal to the radius of the circle. Find the length of the arc (in units), if the radius of the circle is 21 units. (Take $\pi =\frac{22}{7}$)

If the length of each of the two equal sides of an isosceles triangle is 15 cm and the adjacent angle is 30°, then the area of the triangle is:

An angle is thrice its complementary angle. What is the measure of the angle?

In a right-angled triangle $\Delta ABC$, $\angle B=90^{\circ }$, $AB=6\text{cm}$, and $BC=8\text{cm}$. The radius $r$ of the inscribed circle is:

Count the number of triangles in the given figure.

The areas of two similar triangles are $36\text{sq.cm}$ and $81\text{sq.cm}$ respectively. If the median of the smaller triangle is $12\text{cm}$ then the corresponding median of the larger triangle is

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:

The area of an equilateral triangle is $400\sqrt{3}{\text{ m}}^2$. Its perimeter is:

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:

From an external point $A$, tangents $AP$ and $AQ$ are drawn to a circle with centre $O$. If $\angle APQ=40^{\circ }$, then find $\angle POQ$.

What is the length of the side of an equilateral triangle, if its area is 36
3

​
sq cm?

A circle is inscribed in a square. If the length of the diagonal of the square is $14\sqrt{2}\text{ cm}$, what is the area of the circle?

Find the sum of the interior angles of a decagon.

Determine the quadrant in which the point $(\sqrt{5}-3,\sqrt{5}-2)$ lies.

If the length of the side of an equilateral triangle is 8cm, what is its area?