EM-SSLC Worksheets - Questions - John P A

2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts

⋆ Two tangents from an outer point and two radii to the point where
the tangents touch the circle form a cyclic quadrilateral.

1) P is a point at the distance 7cm from the centre of a circle of radius 3 cm
and centre O.
a) Draw two tangents fro P to the circle.
b) Measure the length of radius and write aside.
c) Write the geometric concept of this construction.

2) A circle touches two sides of an equilateral triangle, the centre of the circle
is a point on the third side.
a) If the side of the triangle is 4cm , complete the construction.
b) What is the radius of the semicircle?
c) What is the distance from base vertex of the triangle to the point where
the side touches the circle?

3) In the figure P A and P B are the tangents to the circle.O is the centre of
the circle.

1

a) If ∠ACB : ∠AP B = 2 : 5then find these angles.
b) What is the measure of ∠AOB?
c) What is the measure of ∠ADB?
4) Manju is drawing a circle touches the arms of an angle . The angle taken
for this construction is 40◦.
a) Draw the bisector of the angle.
b) Mark a point on the bisector of the angle.
c) Draw a perpendicular from this point to an arm
d) Draw a circle with the point marked on the bisector as the centre and

the perpendicular distance to the arm as the radius
e) Measure the length of tangent from the vertex to the point where the

circle touces the arm
5) Prove that the length of tangents from an exterior point to a circle are equal.

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2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts
Prove that the length of tangents from an exterior point to a circle are equal.

⋆ In the figure P A, P B are tangents from P to the circle. OA, OB radii.
⋆ △OAP, △OBP are the right triangles.
⋆ P A2 = OP 2 − OA2 → OP 2 − OB2 = P B2

P A2 = P B2, P A = P B

1) In the figure AB = AC, the circle touches the sides at P, Q, R.

a) AP = AQwhy?
b) Prove that BR = CR

1

2) In the figure ∠B = 90◦,AB = 15cm , BC = 8 cm .

a) Draw a rough figure , mark O as the centre . Suggest a suitable name to P ORB
b) If P B = x then fin the length AP, AQ, CR, C
c) What is the radius of the circle.
3) In the figure P Q and P R are the tangents from P outside the circle. P Q = 24cm AQ = 10cm
, BR = 8cm then

a) What is the length of P R
b) What is the length of AB?
c) What is the perimetre of △P AB
d) Prove that P Q + P R = △P AB.
4) In the figure P A, QB are the parallel tangents.P Q touches the circle at R

2

a) Prove that △P AO and △P ROare equal triangles
b) Prove that △QBO and △QROare equal triangles
c) Find ∠P OQ

5) XP, XQare the tangents to the circle from X outside the circle. The line AB touches the circle
at R

Prove that XA + AR = XB + BR

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2020-21 Academic year Worksheets

Mathematics X
െതാ വരകൾ

I
Concepts

1. b the right triangle △ABC, ∠C = 90◦. a, b, care the sides opposite to A, B and C.A circle touches
sides of the triangle.

a) If the radius of the circle is r write the lengths P B and AP

b) Prove that r = a+b−c
2

c) If the perpendicular sides are 6cm and 8 cm then find the length of the hypotenuse

d) If the perpendicular sides are 6 and 8 cm mthen find the radius of the circle .

1

In the figure P M, P N are the tangents to the circle.The distance from P to the centre of the circle is
13cm, radius of the circle is 5cm . The line AB touches the circle at C

a) Find the length of P M and P N
b) If AM = x then find AC and AP
c) Find x
d) What is the length of AB
The sides of ABCD touches the circle at P, Q, R, S

a) Prove that AB + CD = AD + BC
b) If AB = 12cm CD = 8cm , AD = 14cm then find BC.
In the figure ,the line ABtouches a circle.CP is the common tangent .

a) Prove that P A = P B
b) Prove that △ABC is a right triangle.
c) If AC = BC = 10cm then find the length AB
Draw 60◦angle , construct two circles touches the arms of the angle and touch eachother.

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2020-21 Academic year Worksheets

Mathematics X
െതാ വരകൾ

Concepts
⋆ Tangents from an outer point to a circle are equal

⋆ The circle which touches the sides of a quadrilateral inside is called incircle.All triangles will
have an incircle.

⋆ In the quadrialterals having incircle , sum of the opposite sides are equal.

1) ചി ിൽABCDഒ സാമാ രീകമാണ്.ഒ ം വശ െളP, Q, R, Sഎ ീ ബി ളിൽ െതാ .

a) AD + BC = AB + CDഎ ് െതളിയി ക ളിൽ െതാ .BC = 38
b) ABCDഒ സമ ജസാമാ രീകമാെണ ് ാപി ക

2) ABCDഎ ച ർ ജ ിൽ ∠D = 90◦
AB, BC, CD, DAഎ ീ വശ ൾ െ P, Q, R, Sഎ ീ ബി
െസ ീമീ ർ, CD = 25 െസ ീമീ ർ, BP = 27െസ ീമീ ർ

1

a) Prove that ORDS is a square
b) Find the length of CQ
c) What is the side of ORDS ?
d) What is the radius of the circle which touches the sides?
3) In the figure P Qis the common tangent to the circles . Radius of the big circle is 6cm , radius of
the small circle is 3cm . The diatance between the centres is 15cm.

a) Are the traingles AP C and BQC similar ?
b) What is the length AC and BC?
c) What is the length of P Q?
4) In the traingel ABC, ∠B = 90◦, area of the triangle 30cm, sum of the perpendicular sides is
17cm
a) What is the length of AC?
b) What is the radius of the circle ?
5) In △ABC , a, b, c are the sides opposite to A, B and C .
r is the radius of the circle touches the sides, area of the triangle is A,half of its perimetre is s
a) Prove that A = rs
b) Is this relation true in the case of all quarilaterals having incircle?

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2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts
⋆ Tangents from an outer point to a circle are equal
⋆ The circle which touches the sides of a quadrilateral inside is called incircle.All triangles will

have an incircle.
⋆ In the quadrialterals having incircle , sum of the opposite sides are equal.

1) Sides of a triangular metal sheets are 26cm, 24cm and 10cm
a) What kind of triangle is this ?
b) What is the perimetre of this triangle?
c) What is the area of this triangle?
d) Can this metal sheet is used to cover the upper open face of a cyclindrical vessel of radius
5cm?

2) Side of an equilateral triangle is 10cm
a) What is the altitude of this triangle?
b) Find the perimetre and area of the triangle
c) Find the radius of the incircle of this triangle.

3) Draw the incircle of an equilateral triangle using the instructions given below .
a) Draw an equilateral triangle of side 4cm
b) Draw the bisector of two angles
c) Draw a perpendicular to the side of the triangle from the point of intersection of the bisectors.
d) Draw a circle with centre at the intersecting point and the perpendicular distance to the side
as the radius

4) In triangle ABC, BC = 15cm , AB = 12cm , ∠B = 30◦
a) What is the perpendicular distance from A to BC?
b) Calculate the area of the triangle?
c) What is the length of AC?
d) Find the radius of the incircle.

1

5) There are quadrilaterals having incircle and circumcircle.Construct such a quadrilateral as follows
a) Draw a circle and two perpendicular chords.
b) Draw tangents at the ends of the chords to the circle.
c) Mark the quadrilaetarl formed by the tangents .
d) This will be a cyclic quadrilaetral. Draw the circle passing through the vertices.
e) Draw this figure using geogebra software .
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2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts
Angle between a tangent to a circle and the chord from the point where the tangent touches the
circle is equal to angle in the other segment of the circle.
We can prove it · · ·

⋆ If ∠BAY = x, ∠ACB = y then we have to prove x = y
⋆ AP is a diametre of the circle.Draw line P B, ∠AP B = ∠ABC = y , ∠P BA =

90◦.∠P AB = 90 − y
⋆ Line AP is perpendicular to XY
⋆ x = 90 − ∠P AB = 90 − (90 − y) = y

x=y
1) In the figure AB is a chord , line XY is a tangent at A. If ∠Y AB = 40◦then

1

a) Find ∠ACB
b) Find ∠AOB?
c) Find ∠ADB?
2) ABCD is a square .The vertices of the square are on the circle. Tangent at A meet CBproduced
at P .

a) What is ∠BAP ?
b) What is ∠ABP ?
c) What is ∠AP B?
d) If AP = 20 then what is the area of the square ?
3) AP is the tangent of a circle with centre O. The angle between AB and tangent is 140◦

a) What is the measure of ∠ACB
b) What is the central angle of arc ADB?
c) What is the measure of ∠ADB
d) Name an angle in the figure equal to∠ADB
4) In the figure AB is the diametre of the circle, AP is a chord. Draw tangent at P using the steps
given below P

2

a) Draw a circle, chord AP and the diametre AB
b) BP വര ക.
c) Draw an angle eqaul to ∠P AB with P as the vertex and P B an arm
d) Why the other arm become tangent to the circle?
5) Draw a circle , mark a point P on it
a) Using the centre of the circle draw a tangent at P
b) Without using centre draw the tangent at P
c) Write the geometric principle of these two constructions.

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2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts
⋆ The angle between the tangent and the chord to the point of tangency is equal to the chord

in the other segment of the circle.
⋆ Thangents from an outer point to a circle are equal in length.
⋆ Two tangents from an outer point to a circle and two radii to the touching point make a

cyclic quadrilateral.
⋆ If A is the area of the circle , r is the radius of the incircle and s is the semiperimetre then

A = rs
⋆ The sum of the opposite sides of a quadrilateral having incircle are equal.

1) In the figure ABCD is a rectangle.A circle touches the triangle formed by two sides and diagonal
at P, Q, Rand S.If AP = 2cm, DQ = 3cm then

a) What is AD?
b) What is the length of the side AB?
c) What is the length of the diagonal of the rectangle?
d) What is the radius of the circle?

1

2) ABC is an equilateral triangle.Tangents are drawn at the vertices to the circumcircle.The tangents
form another triangle P QR.

a) Prove that P QR is an equilateral triangle.
b) If the perimetre of ABC is 12cm then what is the perimetre of △P QR.
c) How many times the area of P QR is that of ABC?

√
3) In the figure AP is the diametre of the circle.AB = 6 3cm P B = 6 cm

a) What is the radius of the circle?
b) What are the angles of △AP B?
c) What is the measure of ∠ACB?
d) What is the measure of ∠BAQ?
4) ⋆⋆⋆ Three measurments are needed to construct a triangle.Only two measurments are given below
In △ABC AB = 6cm , ∠C = 60◦.
a) Draw a triangle using the concepts we discuss in tangents .
b) How many triangles are possible?
c) If AB = 6cm , ∠C = 60◦, the altitude to AB is 4cm, construct the triangle.
5) In the figure QRis the diametre of the circle, P A is the tangent ,∠RP A = 30◦.

2

a) What is the measure of ∠P QR?
b) What is the measure of ∠P RQ?
c) What is the acute angle formed by P A with P Q?

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2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts
⋆ The angle between the tangent and the chord to the point of tangency is equal to the chord

in the other segment of the circle.
⋆ Thangents from an outer point to a circle are equal in length.
⋆ Two tangents from an outer point to a circle and two radii to the touching point make a

cyclic quadrilateral.
⋆ If A is the area of the circle , r is the radius of the incircle and s is the semiperimetre then

A = rs
⋆ The sum of the opposite sides of a quadrilateral having incircle are equal.
1) In △ABC AB = AC, a tangent P Q is drawn through A to its circumcircle.Prove that P Q is
parallel to BC.

2) In △ABC a tangentP Q is drawn through A to the circumcircle of the triangle.If BC is parallel to
P Q then prove that AB = AC

1

3) In the figure P Q is the diametre of the circle , M N is the tangent to the circle at P .
If ∠RP N = 50◦

a) What is the measure of ∠P QR?
b) What is the measure of ∠P RQ?
c) What is the measure of ∠QP N ?
4) In the figure BC is the diametre of the circle, P A is a tangent .If ∠AP B = x, ∠P AB = y then

a) What is the measure of ∠BCA and ∠CAQ
b) Whatn is the measure of ∠ABC?
c) Find x + 2y
5) ABCD is a cyclic quadrilaterl. The diagonal AC bisects ∠C. Prove that the diagonal BD is
parallel to the tangent at A

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2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts
⋆ The angle between the tangent and the chord to the point of tangency is equal to the chord

in the other segment of the circle.
⋆ Thangents from an outer point to a circle are equal in length.
⋆ Two tangents from an outer point to a circle and two radii to the touching point make a

cyclic quadrilateral.
⋆ If A is the area of the circle , r is the radius of the incircle and s is the semiperimetre then

A = rs
⋆ The sum of the opposite sides of a quadrilateral having incircle are equal.

1) ABCD is a cyclic quadrilaeral . P Q is a tangent atC.BD is the diametre of the circle.
∠DCP = 40◦, ∠ABD = 60◦

a) What is the measure of angle DBC?
b) What is the measure of angle BCQ?
c) What is the measure of angle BDC?
d) What is the measure of ADB?

1

2) Tangent from an outer point T to the circle is AT . B and C are the points a line from T cut the
circle.In triangle ACB, AD is the bisector of ∠A, ∠A = 70◦

a) What is the measure of ∠ACD?
b) What is the measure of ∠BAT ?
c) Find the angles of △DAT
3) AB is the diametre of the circle, P A is a tangent . The line P B cut the circle at C, also CQ is
the tangent at C

a) If is drawn AC whatn is the measure of ∠ACB?
b) If ∠ACQ = x then what are the acute angles of △ABC?
c) Is AQ = QC? Why?
d) Prove that the line CQ bisects AP .
4) Two circles intersect at P, C.AB is the common tangent.

2

Prove that ∠AP C + ∠ACB = 180◦

5) In the figure AB is the diameter of the circle. P is a point on ABproduced.The line from P
touches the circle at C.If ∠CAB = 30◦ and the radius of the circle is 6cm

a) Find the lengths AC and BC
b) Prove that BP = BC.

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2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts
Let P be a point outside the circle. P T is a tangent to the circle and another line from P cut the
circle at A and B.

PA×PB = PT2
1) Let P be a point outside the circle. P T is a tangent to the circle and another line from P cut the

circle at A and B.

a) What is the relation between ∠P T A, ∠P BT ?
b) Are △P T A, △P BT similar
c) Prove that P A × P B = P T 2

1

2) P T is a tangent from an outer point P to the circle.
Another line from P intersect the circle at A and B. If the length of the chord P B is 16cm and
AB = 7 cm then

a) What is the length P A?
b) What is the relation between P A, P B, P T ?
c) What is the length of the tangent P T ?
d) What is the length of the other tangent from P to the circle.
3) BCis the diametre of the circle.P is a point on BC produced.
TangentP A is drawn from P to the circle. If P A = 6cm and P C = 3cm then

a) What is the length P B?
b) Find the radius of the circle.
4) In △ABC, AB = AC, A circle passing through B intersect AB at P . The circle touches AC
at its mid point D

Prove that 4AP = AB

2

5) In the figure BC is the diametre of the circle and AB is a tangent.

a) Write the relation between AC, AD and AB
b) Prove that AC × CD = BC2

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2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts

If P T is a tangent to a circle from the outer point P and another line from P intersect the cxircle

at A and B then

PA×PB = PT2

This can be proved by Pythagorous theorem also.

⋆ P A × P B = (P M − AM )(P M + BM ) = (P M − AM )(P M + AM )
= (P M 2 − AM 2) = (P M 2 − (OA2 − OM 2)) = P M 2 + OM 2 − OA2
= OP 2 − OA2 = OP 2 − OT 2 = P T 2

1) In the figure AB = BD, also the line AD is a tangent from A.

a) What is the relation between AB, AC and AD
b) Prove that AB × AC = CD2
c) What kind of triangle is △ACD?
d) If ∠BAD = 30◦ and perpendicular distance from D to AB is 12 then what is the length of

tangent AD?
1

2) In the figure ACis the diametre and BA is a tangent to the circle. The line BC intersect the circle
at P
If the radius of the circle is 2.5cm and the length of tangent is 12cm

a) What is the length BC?
b) What is the length P C?
c) What is the length AP ?
3) P A is a tangent from the outer point to a circle of diametre AB.The line P B intersect the circle
at C.If the radius of the circle is 5cm and AC = 6cm then

a) What is the length BC?
b) Find P C
4) Ois the centre of a circle of diametre AB. P A is a tangent from P to the circle, line P B intersect
the circle atC.If ∠AOC = 60◦, AC = 6 cm then

2

a) What is the measure of ∠ABC?
b) What is the diametre of the circle?
c) What is the length BC?
d) What is the length P C?
5) Draw a square of side 4cm , construct a rectangle having area equal to the area of the square and
one side 6cm . Write the principle of this construction.

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2020-21 Academic year Worksheets

Mathematics X
Tangents

Concepts

If P T is a tangent to a circle from the outer point P and another line from P intersect the cxircle

at A and B then

PA×PB = PT2

This can be proved by Pythagorous theorem also.

PA×PB = PT2

⋆ P A × P B = (P M − AM )(P M + BM ) = (P M − AM )(P M + AM )
= (P M 2 − AM 2) = (P M 2 − (OA2 − OM 2)) = P M 2 + OM 2 − OA2
= OP 2 − OA2 = OP 2 − OT 2 = P T 2

1) ABCDE is a regular pentagon and its circumcircle. The tangents to the circumcircle at A and B
intersect at P .

a) Draw AD, what are the angles of △ADE?
b) What is the measure of ∠ADB?
c) Tangents at A, Bintersect at P . What is the measure of ∠BAP ?
d) What is the measure of ∠AP B? 1

2) In △ABC the sides AB, BC and AC touches a circle at D, E, F .
If AB = 12cm ,BC = 8 cm ,AC = 10cm then find AD, BE and CF .

3) M is the mid point of AB.Semicircles are drawn with AM ,BM and AB as the diametre .Another
circle of radius r touches the semicircles.

a) Prov that r = AB
6

b) If AB = 12then what is the radius of the circle?

c) If AB = 12 what is the perimetre of the triangle formed by joining the centre of the circle
and small semicircles?

4) A semicircle is drawn with AB as the diametre in the square ABCD. DE touches the
semicircle at P . If the side of the square is of length 1unit

a) What is the length DP ?
b) If P E = x then find the equation connecting DE, CD and CE
c) Find the length of the line DE.
5) In the figure AB = AC, BC = 10 cm , altitude from A to BC is 12 cm.The centre of the
semicircle is on BC and the semicircle touches the sides AB and AC.

2

a) What is the perimetre of △ABC?
b) What is the area of triangle ABC?
c) What is the radius of semicircle?

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08

Solids

ഘനരൂപങ്ങൾ

2020-21 Academic year Worksheets

Mathematics X
Solids

Concepts

⋆ There are 5 faces in a square pyramid. Four lateral faces and a base. Lateral faces are triangles
and base is a square.

⋆ There are 8 edges in a square pyramid. Four base edges and four lateral edges .Base edge is
denoted by a and lateral edges by e.

⋆ പാദ ിെ വികർ ംd, കളിെല ശീർഷ ിൽ നി ം പാദ ിേല ഉ തി hഎെ .

⋆ The lattitude of lateral face to the base edge is called slant height of the pyramid.

1) Manju has drawn an outline in a square card board for making a square pyramid as given below.

a) What is the total length of its edges?
b) What is the slant height of the square pyramid?
c) What is length of the side of the square paper in which outline is drawn.
2) A wire of length 96cm is cut into eight equal parts . The ends of the pieces are joined to make the pyramid.
a) What is the length of the edge of the pyramid?
b) What kind of triangle is its lateral edge?
c) What is its slant height?
3) The base perimetre of a square pyramid is 40 cm, height 12 cm.
a) What is the base edge of the pyramid?
b) what is the slant height of the pyramid?
c) What is the lateral edge of the pyramid?

1

4) There is a square pyramid having its lateral faces equilateral triangles.Length of one leteral edge is 32cm
a) What is its base edge?
b) What is its slant height?
c) What is the area of one lateral face?
d) Calculate the total area of its lateral faces

5) The base diagonal of a square pyramid is 12cm , height 8cm
a) What is its base edge?
b) What is its base area?
c) What is the length of its lateral edge?
d) Calculate the total length of its edges.
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2020-21 Academic year Worksheets

Mathematics X
Solids

Concepts

⋆ There are 5 faces in a square pyramid. Four lateral faces and a base. Lateral faces are triangles
and base is a square.

⋆ There are 8 edges in a square pyramid. Four base edges and four lateral edges .Base edge is
denoted by a and lateral edges by e.

1) The height of a square pyramid is 2 more than its base edge. Slant height is 13 cm
a) If the base edge is a what will be its height?
b) Write an equation connecting base edge , height and slant height
c) Calculate the length of base edge?
d) Calculate the total lateral surface area of the pyramid.

2) Base area of a square pyramid is 100 sq.cm , lateral surface area 480 sq.cm
a) What is the length of base edge?
b) What is the slant height?
c) Find the height of the pyramid.
d) Calculate the total surface area of the pyramid.

3) A sectoral sheet of central angle 240◦ is taken from a circular sheet of radius 10cm. Four equal triangles
are made from the sector as in the figure.They are joined in such a way as to get a square pyramid.

a) What is the length of its edge?
b) What is the slant height of the pyramid ?
c) Find the height of the pyramid.

1

4) A steel wire of length 120cm is cut into 8 equal parts, the ends are joined in such a way as to get a square
pyramid.
a) What is the length of its edge?
b) What is its slant height?
c) Calculate the area of paper used to cover the pyramid.

5) Base edge of a square pyramid is a and slant height l.
a) Write a formula to find the lateral surface area of the pyramid.
b) Is it possible to make a square pyramid having base area and lateral surface area equal.
c) If the base edge is 10 and lateral surface area of a square pyramid is two times its base area. What
is its height?
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2020-21 Academic year Worksheets

Mathematics X
ഘന പ ൾ

Concepts

⋆ If base edge a,height h, and slant height lthen
l2 = h2 + ( a )2
2

If base edge a, slant height l, and lateral edge e then
e2 = l2 + ( a )2
2

Height h, base diagonal d, lateral edge e then

e2 = h2 + ( d )2
2

⋆ Lateral face area of the square pyramid = 2al

total surface area = a2 + 2al

volume = 1 × a2 ×h
3
√
⋆ If the lateral faces are equilateral triangles , total surface area = a2 + 3a2

1) There is a square pyramid of height 12cm , slant height 13cm.

a) What is the length of its base edge ?
b) What is the base area of the pyramid ?
c) Find the lateral face area of the pyramid.
d) Calculate the total surface area of the pyramid.

2) Height of the square pyramid is h , slant height l and lateral edge e .

a) If the base edge is a, write the relations between h, l and e
b) Prove that h2, l2, e2are in an arithmetic sequence
c) If the slant height is 13 , base edge 10 find height and lateral edge

3) Base perimetre of a square pyramid is 40cm, total length of the edges is 92cm

a) What is the length of base edge?
b) What is base diagonal?
c) Find the height of the pyramid
d) Calculate the total surface area.

1

4) Base perimetre of a square pyramid is 40cm , height 12cm
a) What is the lenghth of base edge ?
b) Find the volume of the pyramid.
c) What is the volume of the square prism having same base area and height ?

5) Ratio of the base edges of two square pyramids is 1 : 2. Heights are in the ratio 2 : 3
a) If the base edge of the first pyramid is a then what is the ratio of their base area?
b) If the height of the first pyramid is h then what is the height of second pyramid?
c) What is the ratio of the volume ?
d) If the volume of the first pyramid is 10 cubic cm then what is the volume of the second pyramid?
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2020-21 Academic year Worksheets

Mathematics X
Solids

Concepts

⋆ If base edge a,height h, and slant height lthen
l2 = h2 + ( a )2
2

If base edge a, slant height l, and lateral edge e then
e2 = l2 + ( a )2
2

Height h, base diagonal d, lateral edge e then

e2 = h2 + ( d )2
2

⋆ Lateral face area of the square pyramid = 2al

total surface area = a2 + 2al

volume = 1 × a2 ×h
3
√
⋆ If the lateral faces are equilateral triangles , total surface area = a2 + 3a2

1) Angle between base edge and lateral edge of a square pyramid is 60◦, base edge is 12 cm

a) What is the length of its lateral edge?
b) What is its slant height?
c) Find the total surfsce area of the pyramid
d) Find the height of the pyramid
e) Calculate the volume of the pyramid

2) The base edge and height of a square pyramid are equal in length .Volume of the pyramid is 576 cubic cm

a) What is the base edge of the pyramid?
b) What is the slant height of the pyramid ?
c) Find the volume of the pyramid
d) calculate the total surface area of the pyramid.

3) Can isosceles right triangles be the lateral surfaces of a square pyramid. Justify your answer.

1

4) Base edge of a square pyramid is a and height h. The volume of the pyramid is 1 a2h.
3

a) What is the change of its volume if the base edge and height are doubled.

b) If the initial volume is 10 cubic cm then what will be the volume if height and base edge are doubled?

5) Base perimetre of a square pyramid is 64cm , volume is 1280 cubic cm

a) What is its base edge ?
b) What is the height of the pyramid?
c) What is the slant height of the pyramid ?
d) Calculate the total surface area of the pyramid.

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2020-21 Academic year Worksheets

Mathematics X
Solids

Concepts

a) Cone can be made by rolling a sectorol sheet.While doing so the arc length of the sector becomes
the base perimetre of the cons.Area of the sector becomes lateral surface area of the cone. Lateral
surface area is also known as curved surface area .

b) The radius of the sector becomes slant height of the cone . It can be denoted by l for between
convenience.

c) Since arc length of the sector is equal to base perimetre of the cone we can make the relation given

below 2πl
x = 2πr

360

l radius of the sector , x central angle of the sector and r is the radius of the cone , lx = 360r.

d) Area of the sector becomes the lateral surface area of the cone.We can make a formula to calculate

lateral surface area of the cone

Curved surface area= Area of the sector
πr2
Curved surface area = 360 x = π×l×l×x
360

lx = 360r

Curved surface area = π×l×360r = πrl
360

1) A sectoral sheet of central angle 120◦is cut off from a circular sheet of radius 12cm . It is rolled in such a
way as to get a cone.

a) What is the slant height of the cone?
b) What is the radius of the cone ?
c) Find the curved surface area of the cone.

2) A cone is made by rolling a semicircular metal sheet of radius 10cm

a) What is the slant height of the cone.
b) What is the radius of the cone.
c) Find the curved surface area of the cone.
d) Base is made by a suitable circular sheet. What is its total surface area ?

1

3) A circular sheet of card board of radius 12cm .It is cut off into two sectors of central angle 120◦ and
240◦.Both of them are rolled into cones.
a) Name the measure coomon to both comes
b) What is the radius of small cone ?
c) What is the radius of the big cone.
d) How radii of the cones are related to the radius of the circular sheet.

4) A sector of central angle 90◦ is cut off from a circular sheet of radius 16cm .It is rolled in such a way as to
get a cone.
a) What is the lateral surface are of the cone?
b) What is the radius of the cone?
c) The remaining part of the circular sheet is also rolled to get a cone . What is its base radius?
d) Which cone has more height ? Explain

5) A cone is made by a sectoral sheet taken from a circular sheet.The slant height of the cone is two times its
radius.
a) What is the relation between lateral surface area and base area?
b) If the base perimetre is 20πcm then what will be its lateral surface area ?
c) What is the central angle of this sector?
d) The remaining part is also rolled to get a cone. What is the ratio of the heights of cones so formed
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2020-21 Academic year Worksheets

Mathematics X
Solids

Concepts

a) Cone can be made by rolling a sectorol sheet.While doing so the arc length of the sector becomes
the base perimetre of the cons.Area of the sector becomes lateral surface area of the cone. Lateral
surface area is also known as curved surface area .

b) The radius of the sector becomes slant height of the cone . It can be denoted by l for between
convenience.

c) Since arc length of the sector is equal to base perimetre of the cone we can make the relation given

below 2πl
x = 2πr

360

l radius of the sector , x central angle of the sector and r is the radius of the cone , lx = 360r.

d) Area of the sector becomes the lateral surface area of the cone.We can make a formula to calculate

lateral surface area of the cone

Curved surface area= Area of the sector
πr2
Curved surface area = 360 x = π×l×l×x
360

lx = 360r

Curved surface area = π×l×360r = πrl
360

1) A cone of radius r1 is made by using a sector of a circular sheet of radius R.The remaining part of the sheet
is rolled in such a way as to get another cone of radius r2

a) Which measure is common in both cones?
b) Write the relation between the radius , slant height and central angle of the sector in the case of first

cone.
c) Write the relation between the radius , slant height and central angle of the sector in the case of

second cone.
d) prove that R = r1 + r2

2) A cone is made by taking a sector from a circular sheet.The slant height of the cone is 25cm and its radius
110cm

a) What is the radius of the circular sheet?
b) What is the central angle of the sector?
c) What is the central angle of the remaining part?
d) What is the radius of the cone made by rolling the remaining part?

1

3) The base perimetre of a cone is 20π cm, slant height 18cm . It is rolled to get a cone.
a) What is the radius of the sector?
b) What is the radius of the cone?
c) What is the central angle of the sector?
d) Find the lateral surface area of the cone?

4) A sector of central angle 288◦ and radius 25cm is taken from a circulat sheet .
a) What is the radius of the cone?
b) What is the height of the cone ?
c) Find the lateral surface area of the cone?
d) What is the radius of the cone made by rolling the remaining part?

5) A cone of maximum size is carved from a square prism of base edge 10cm and height 12cm.
a) What is the radius of the cone?
b) What is the slant height of the cone?
c) What is the lateral surface area of the cone?
d) Find the total surface area of the cone?
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2020-21 Academic year Worksheets

Mathematics X
Solids

Concepts

a) There are three basic neasurements in a cone.Radius r, height h and slant height l. These
measurements form a right triangle. l2 = r2 + h2

b) Base perimetre of the cone= 2πr, Base area = πr2

c) Lateral surface area of the cone = πrl,Total surface area = πr2 + πrl

d) Volume of the cone= 1 πr2h
3

1) Radius of a cone is 5cm, height 12cm

a) what is the slant height of the cone?
b) What is the total surface area of the cone?
c) What is the volume of the cone ?
d) In a cone , radius and height are equal. If the volume and curved surface area are equal then what is

its radius ? What is its slant height?

2) The base perimetre of a cone is 30πcm, height 20 cm

a) What is the radius and slant height of the cone ?
b) What is the total surface area?
c) Find the volume of the cone?
d) What is the vaolume of a cyclidrical vessel of radius and height equal to that of the cone.

3) Diametre and height of a cone are equal.

a) What is the relation between radius and slant height?
b) What is the ratio of radius , height and slant height?
c) If the radius is 6 cm then what is its volume?
d) If the radius is 6 cm then what is the total surface area ?

1

4) Radius of a cone is r and height h.
a) What will be the change in volume if radius and height are doubled?
b) What will be the change in volume if radius is doubled and height is halved?
c) How many solid cones can be made by melting a solid cone of radius 10cm and height 6cm with half
the radius and height of the melted cone?

5) A conical measuring vessel is made by rolling a sectoral sheet of central angle 288◦ and radius 10cm.
a) What is the radius of the vessel?
b) What is the height of the vessel?
c) What is the capacity of the vessel in litres ?
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2020-21 Academic year Worksheets

Mathematics X
Solids

Concepts

a) There are three basic neasurements in a cone.Radius r, height h and slant height l. These
measurements form a right triangle. l2 = r2 + h2

b) Base perimetre of the cone= 2πr, Base area = πr2

c) Lateral surface area of the cone = πrl,Total surface area = πr2 + πrl

d) Volume of the cone= 1 πr2h
3

1) Radius of a cone is 21cm , height 28cm.

a) Calculate slant height.
b) Find the leteral surface area .
c) Calculate the total surface area
d) Calcualte the volume of the cone..

2) Ratio of radius and height of a cone are 3 : 4.Volume of the cone is 301.44 cubic cm

a) Find the radius of the cone.
b) Find the height of the cone.
c) Calculate the slant height of the cone.
d) Calculate the lateral surface area of the cone.

3) Lateral surface area of a cone is 4070 sq.cm , diametre 70 cm

a) Find the slant height of the cone .
b) Find the height of the cone?
c) Calcualte the volume of the cone.

4) The height of a cone is 24cm , its lateral surface area 550 sq.cm

a) What is the radius of the cone?
b) Find the slant height of the cone?
c) Calcualte the volume of the cone?

5) A semicircular sheet of radius 28cm is rolled in such a way as to get a cone.

a) What is the slant height of the cone?
b) Find the radius of the cone?
c) Find the height of the cone?
d) Calcualte the volume of the cone

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2020-21 Academic year Worksheets

Mathematics X
ഘന പ ൾ

Concepts

a) The basic measurement of a sphere is its radius. Surface area and volume can be calculated using
radius of the sphere.

b) If r is the radius of the sphere , its surface area is 4πr2

c) Volume of the sphere with radius r is 4 πr3
3

d) Volume of the hemisphere is 2 πr3
3

e) Curved surface area of the hemisphere = 2πr2. Total surface area of the hemisphere = 3πr2

1) Calculate the following measures of sphere with radius 3cm

a) Find the total surface area of the sphere.
b) Calculate the volume of the sphere
c) Calculate the curved surface area of the hemisphere taken from this sphere .
d) If it is solid sphere find the total surface area of the hemisphere.

2) Volume and surface area of a sphere are equal in number.

a) What is its radius
b) Calculate the volume or total surface area
c) How many spheres of radius 1cm can be made by melting this solid sphere ?

3) Find the volume of the sphere accoring the changes of radius as given below

a) What will be the change in volume if radius is doubled?
b) What will be the change in volume if radius is halved?
c) The volume of a sphere is 10 cubic cm. What will be the volume of the sphere of diametre two times

the first one.

4) A sphere of maximum side is carved from a wooden cube of side 6cm

a) What is the radius of the sphere?
b) Calculate the surface area of the sphere.
c) calculate the volume of the sphere .

1

5) A sphere is fixed inside a conical vessel of diametre 10cm and height 12cm. It touches the curved surface
of the cone and its base.

a) What is the slant height of the cone ?
b) Find the radius of the sphere?
c) Calculate the volume of the sphere ?
d) What fraction of the inner volume of the cone is occupied by the sphere?

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2020-21 Academic year Worksheets

Mathematics X
Solids

Concepts

a) The basic measurement of a sphere is its radius. Surface area and volume can be calculated using
radius of the sphere.

b) If r is the radius of the sphere , its surface area is 4πr2

c) Volume of the sphere with radius r is 4 πr3
3

d) Volume of the hemisphere is 2 πr3
3

e) Curved surface area of the hemisphere = 2πr2. Total surface area of the hemisphere = 3πr2

1) There is a sphere of radius 1cm.It is melted an recast into small spheres of radius 1 cm .
2

a) What part of the volume of the melted sphere is the volume of the small sphere ?

b) What part of the surface area of the melted sphere is the surface area of the small sphere ?

c) How many spheres of radius 1 can be made ?
2

d) What is the difference between the surface area of big sphere and sum of the surface area of small

spheres.

2) Total surface area of a solid sphere is 64 sq.cm

a) What is the radius of the sphere?
b) What is the volume of the sphere ?
c) The sphere is split up into two hemispheres . What is the total surface area of a hemisphere?

3) Total surface area of a solid hemisphere is 27π sq.cm

a) What is the radius of the hemisphere ?
b) What is the curved surface area of this hemisphere?
c) What is the total surface area of the sphere made by joining two such hemispheres?

4) A cone of maximum size is carved from a hemisphere of radius 10cm

a) What is the height of the cone?
b) What is the slant height of the cone?
c) Find the total surface area of the cone?
d) Find the volume of the cone?

1