34 area, volume & nets of solids

l

b h

l

l

l

h

r

l

MENSURATION – 2Area and Volume of Prisms

General formulae

Volume = area of cross-section × length or heightLateral Surface Area = perimeter of the cross-section × length or heightTotal Surface Area = lateral surface area + 2 × cross-section area

CUBOID

Volume = l × b × hTotal surface area 2 (lb + bh + hl)

CUBEVolume of the cube = l3

Surface area = 6l2

CYLINDER

Volume = π r2hL.S.A (or) C.S.A = 2πrh T.S.A. = 2πrh+2πr2 = 2πr (h+r)

PRISM

Volume = area of cross section × lengthL.S.A = perimeter of cross section × lengthT.S.A = L.S.A + 2 cross sectional area

MS/Dept. Of Maths/33 Mensuration (S.A, Volume & Nets)/2006 of 21 C.T Firoz Babu

12cm

8.5cm

Find a) the cross sectional area b) Volume c) Lateral surface area d) Total surface area

10cmA C

B

D

E

F

42cm13 cm

1. The side of the cube is 6 cm. calculatea) the volume of the cubeb) the surface area of the cube

2. A metallic cube of side 8 cm is melted and formed into smaller metallic cubes of side 2 cm. how many small cubes are obtained?

3. The surface area of the cube is 384 cm2. Find the volume?

4. a) Find the volume of the cuboid whose dimensions are10 cm by 8 cm by 7 cmb) Find the surface area of this cuboid?

5. Calculate the height of the cuboid if l = 5 cm, b = 4 cm and the surface area =104cm2.

6. Find the edge of a cube whose surface area is numerically equal to volume.

7. Calculate the volume and surface area of the cylinder having radius 4 cm and height 9 cm.

8. A cube and the cylinder have the same surface area. If the cube has a side of length 6 cm and the cylinder a radius of 2 cm. calculate:

a) The surface area of the cube b) The height of the cylinder

9.

10. ABC is triangle where AC = 10cm,AB = 13cm.

A triangular prism whose cross-section is ABC is shown in the given figure. If the length of the prism is 42 cm, find:

a) the area of the triangle ABC b) the volume of the prism c) the lateral surface area of the prism d) the total surface area of the prism.

11.


8cm

4cm

E

O

B

C

4cm

G

F D

A

C

G

H

I

The cross section of a toy is in the shape of the given figure. Calculate

a) i) the area of the rectangle ABCG ii) the area of the triangle GOC iii) the area of the circle DEF with center O and the radius OF = 4cm iv) the area of the sector DOF v) the area of the crossection of the toy vi) the perimeter of the toy

If the length of the toy is 38cm, find b) the volume of the toy. c) the lateral surface area of the toy d) the total surface area of the toy.

12. A solid cylinder of height 37cm and diameter 18cm is cut along its cross section through its diameter. Find the volume and T.S.A of the solid so formed

13.

Figure shows a prism of cross-section sector OAB. Calculate:i) the arc length ABii) Volumeiii) T.S.A

14.


26 cm

18 cm

6 cm

1 m

A B

CD

Figure shows a solid prism whose cross-section ABCDE is a combination of square and semi - circle. One side of square is 20 cm and length of prism is 55 cm.

Calculate:i. the area of cross-section ABCDE, correct to 1 decimal place. [ 4 ]

ii. the perimeter of cross-section ABCDE. [ 3 ]iii. Lateral Surface Area of prism. [ 2 ]iv. Total Surface Area of prism. Write your answer in standard form. [ 3 ]v. volume of solid prism in cm3. [ 2 ]

vi. the mass of this prism if the mass of 1cm3 = 7.5 g. [ 2 ]

15. Figure shows a metallic prism.

ABCD is a trapezium. AB = 26 cm, DC = 18cm, AD = 6 cm , angle DAB = 900 and length of prism is 1 metre.


A B

CD 17 cm

10 cm

2cm

B

C

DE

FO

a) Calculate:i) the length BCii) L.S.A. of the prismiii) T.S.A of the prismiv) Volume of the prismv) The cost of painting the surface of this prism at the rate of $0.75 per cm2.

b) this prism melted and formed a cube. Find the length of the edge of that cube to the nearest cm.

16. a) ABCD is a rectangle whose measurements are 17cm &10cm.The uniform width of the shaded region is 2 cm.

Find the area of shaded region

b)

17.a)


A hollow prism has formed by using part (a) shaded figure as cross-section.i) Find the length of this prism if the volume is 5520 cm3.ii) Calculate the out side Lateral Surface Area

ABCDEF is a regular polygon. AB = 8cm.a) Name the polygon,b) Show that angle OAB = 600.c) Find the area of triangle OAB to3 s.f.d) Find the area of the polygon.

A

B C

10 cm

15 cm

25 cm

720

b)

18.

MENSURATION – 3Area and Volume of Pyramids


Cross- section of this prism is the same polygon of part (a) and volume of prism is 3324 cm3.

a) Show that the length of prism is 20 cm.

b) Find the L.S.A. of the prism.c) Find the T.S.A. of the prism.

AB = 10 cm, BC = 15 cm,Angle ABC = 720 andThe length of the prism is 25 cm.Calculate:

i) the area of triangle ABC,ii) the volume of prism,iii) the length AC,iv) T.S.A.

O

A B

C D

E

G

F

O

A B

C D

E

G

F

* OAB, OBC, OCD and OAD are congruent triangles *OAB and ODC are congruent.*OAD and OBC are congruent.

Lateral Surface Area = Sum of areas of these triangles

* The following are right angle triangles:OEF, OEA, OGB, OFC

*OA, OB, OC, OD are known as lateral edges.

*OG & OF are known as lateral heights.

l – slant heighth- perpendicular heightr- radius of the base.

)( rlr h l

General formulae

Volume =

13×¿ ¿

Base Area × Perpendicular HeightTotal Surface Area = Lateral Surface Area + Base Area

Right Pyramids

CONE


r

r

C

D

E

SPHERE

HEMISPHERE

1. A B C D E is a pyramid with a square base A B C D. ‘F’ is the mid point of BC. G is the centre of the square ABCD. And E is vertically above G. AB = BC = 30cm and GE = 42 cm. calculate to 3 s.f


Volume =

43πr3

Surface Area = 4 πr2

Volume =

23πr3

Curved Surface Area = 2πr2

Surface Area = 3 πr2

O

A B

CD

E

G

F

O

AB

CD

9 cm

6 cm

8 cm

O

15 cm

a). the length in cm, of FEb) The area in cm2, of triangular face CEBc) the total area in cm2, of the triangular faces of the pyramidd) the total surface area in cm2, of the pyramid e) the volume in cm3, of the pyramid

2.

3.

4.


OABCD is a pyramid with rectangular base. AB=10cm, AD = 6cm, OF=13cma) Calculate :

i) The perpendicular height OE,ii) Volume of the prism,iii) Show that OG = 10.44cmiv) Area of the triangle OABv) L.S.A.vi) T.S.A.

b) Calculate the cost of painting the surface of this pyramid at the rate of $1.25 per cm2.

Figure shows a solid pyramid with rectangular base ABCD.O is vertically above C.AB = 9 cm, BC = 6 cm and OC = 8 cm.a) Calculate:

i) the volume,ii) the length OB,iii) the length OD,iv) the total area of the triangular faces of

pyramid &v) T.S.A.

b) This pyramid melted and formed a cylinder with base radius 3cm. calculate the length of that cylinder to the nearest cm.

XYZ is the base of tetrahedron. O is vertically above X.OX = 15 cm, YZ = 12 cm and XZ = 10 cm.a) Calculate :

i) the area of triangle XYZ.ii) Volume of the tetrahedron.

b) i) Show that the length OZ = 18 cm to the nearest whole number.

ii) Write down the length OY.

O

BA

D C

E F

5.

6. The height of the right circular cone is 12 cm and its slant height is 13 cm. Find :

i) the base radius,ii) the volume,iii) C.S.A.iv) T.S.A.

7. Calculate the volume and surface area of the sphere whose diameter is 25 cm.8. Base radius of a hemi-sphere is 8cm.

Calculate:i) the volume,ii) C.S.A. andiii) Surface area.

9. Base area of a hemisphere is 314 cm2.a) Calculate :

i) the radius to the nearest integer,


XYZ is the base of tetrahedron. O is vertically above X.OX = 15 cm, YZ = 12 cm and XZ = 10 cm.a) Calculate :

i) the area of triangle XYZ.ii) Volume of the tetrahedron.

b) i) Show that the length OZ = 18 cm to the nearest whole number.

ii) Write down the length OY.

OABCD is a square pyramid.AB = 10 cm, OC = 26 cm.a) Find:

i) the lateral height OF.ii) the perpendicular height OE.iii) the volume of pramid.iv) the area of triangle OBC.v) the total surface area of the pyramid.

8cm

12cmm

O

A B

CD

E

G

F

ii) the volume andiii) T.S.A. of the hemisphere.

b) Find the radius of a sphere which has formed by using this hemisphere.

10. Volume of a sphere is 4000 cm3.a) Find the radius of the sphere.b) This sphere cut equally in to two parts. What will be surface area of

one part?

11. The figure shown is a hemi-sphere over a cone. The radius of the cone is 8cm and the perpendicular height of the cone is 12cm.

a. Calculate the volume of the figure

b. Calculate the slant height of the cone

c. Calculate the surface area of the figure

12.

13.


Figure shows a rectangular pyramid.AB= 2x +1, BC= x-1 , OE = 24cm.Volume is this pyramid is 200cm3.

i) Form an equation in terms ox and simplify to 2x2 – x – 26=0,

ii) Solve this equation by using quadratic formula,

iii) Write down the lengths AB & BC ,iv) Find the length OF,v) Find the length of OG vi) L.S.A. vii) T.S.A.

x cm11. 3

15 cm

x +5

The above fig. shows a well to draw water from the well with the help of a bucket and a rope.a) If the depth of the well is 10m, find the volume of the well b) The bucket is the shape of a frustum( a part of a cone- a small cone cut off from a bigger cone), as shown in the figureThe height of the bucket is x cm and the radius of the bigger cone is 5 cm more than the height of the cone. The radius of the smaller cone is 11.3 cm.The height of the smaller cone is 15 cm

i) Use Pythagoras theorem to form an equation if the the slant height of the bigger cone is 50 cm.

ii) Show that this equation simplifies to x2 + 20x – 1125 = 0

iii) Solve x2 + 20x – 1125 = 0 iv)Find the volume of the bigger cone and the smaller cone

v) Hence find the volume of the bucket.

Nets


r

Cube: Net of Cube:

Cuboid: Net of cuboid:

Cylinder: Net of Cylinder


O

A B

CD

O

O

O

O

BA

D C

O OXO

Square Pyramid: Net of square pyramid

Tetrahedron Net of tetrahedron


A B

The length AB will be equal to the circumference of the circle

r

l l

O

AB

h l

r

5 cm

Cone: Net of cone:

1.


NOTE:i) Total Surface Area of any solid = Total area of net.ii) Volume of any prism = Area of cross-section X height.iii) Volume of any Pyramid = Base area X perpendicular height.

Arc length AB will be equal to the circumference of the circle. Angle AOB can be found by using arc length AB.

a) Name the prism which can be formed by using this net.

b) Calculate :i) the total area of the net.

ii) the total surface area of the prism.

A B

C

E

D

GH

9 cmF

4.2 cm

2.9 cm

14 cm

2.

Figure shows a cuboid. AB = 4.2 cm, AD = 2.9 cm and DH = 9 cm.a) Construct an accurate net of this prism.b) Label every corner of this net with the appropriate letter.c) Calculate

i) the total surface area of this cuboid.ii) The volume of this cuboid.

3.


a) Name the prism which can be formed by using this net.

b) Calculate :i) the total area of the net.

ii) the total surface area of the prism.

Diagram shows the net of a solid prism.a) Name the prism.b) Circumference of the circle in the net is

13cm.Find the radius to the nearest whole number.

c) Construct an accurate net of this prism.d) Calculate T.S.A. of the prism.e) Show that the volume of the prism is

A

B C

D

E F

5 cm

6 cm 7 cm

A B C

D E F

4 cm 5 cm 6 cm

O

A B

CD

O

O

O4cm

4 cm

2cm

BA

CD

4.

The diagrams show a triangular prism ABCDEF and a rough sketch of one of its possible nets.a) Calculate, correct to 1 d.p., the size of angle ABC.b) Construct accurately the net of prism shown in Diagram 2.

Label every corner of your net with the appropriate letter A, B, C, D, E or F.c) Calculate:

i) the T.S.A. of the prism.ii) The volume of the prism.

5.

6.


Diagram shows the net of a solid prism.a) Name the prism.b) Circumference of the circle in the net is

13cm.Find the radius to the nearest whole number.

c) Construct an accurate net of this prism.d) Calculate T.S.A. of the prism.e) Show that the volume of the prism is

4 cm

M

ABCD is a square and all 4 triangles are congruent. AB= 10 cm , OM = 12 cm & OC = OB.

a) What special name is given to each of the 4 triangle in the net?

b) Find the length OCc) Calculate the perimeter of the net.d) Draw a rough sketch of the solid which can be

formed by using this net, name the solid.e) Find the perpendicular height of this solid.f) Calculate the total surface area of the solid.g) Calculate the volume of the solid.h) Calculate the sum of the lengths of the edges

of the solid.

ABCD is a trapezium. AB = BC = 4 cm and DC = 2cm.

2cm Not to scaleF

G

AB

CD E

H

4cm

4cm

6cm

CD 2cm

3.5cm

FE

GH

Not to scale

A B4cm 4cm

6cm

i. Show by calculation the height AD of the trapezium is 3.5 cm , correct to 1 decimal place.

ii. Find the area, in cm2, of the trapezium ABCD

The same trapezium ABCD forms the cross section of a prism of length 6cm. The figure below represents the prism.

iii. Find the volume, in cm3, of the prism.iv. Write the number of plane(s) of symmetry of the above prism

The following is the net diagram of the above prism.

v. Construct an accurate net of the prism by taking instruction from the above given net and label it completely.vi. Find the perimeter, in cm, of the net.vii. Find the area, in cm2, of the net


A B

CD

P

Q

R

S

5cmA B

C

D

7cm440

390

viii. Find the shortest distance of H from A, in cm, when the net is folded into a prism.

7.

8.

9.


XY

25 cm

20 cm

26 cm

Figure shows the net of a right rectangular pyramid.AB = 30 cm, BC = 24 cm and QC = 26 cm.

i) Write down the lengths SD & DR.ii) Find SX, the height of triangle SAD.iii) Find RY, the height of triangle RDC, to

the nearest 1 d.p..iv) Calculate the T.S.A. of the pyramid.v) Show that the perpendicular height of

the pyramid is 20 cm to the nearest integer.

vi) Calculate the volume of the pyramid

The figure shows one of the possible nets of a solid

with a triangular base ABC. Given AB = 5cm ,

BC = 7cm , angle ACB = 440 and angle DBA = 390.

a) What is the geometrical name of the solid formed when the net is folded?

b) Calculate:i. to the nearest degree, the angle CABii. the area of the triangle ABCiii. to the nearest cm, the length of AD

c) Find the volume of the solid formed by folding the net.

d) Given AC =6cm, correct to the nearest cm, find the perimeter of the net

D A

BCK

L

M

N

9 cm

5 cm

12 cm

Figure shows a rectangular pyramid with base ABCD.AB = 5 cm, AD = 9 cm, BK = 12 cm and <ABC = <BCD = <BAD = <ABK = <CBL = <MCD = <DAN = 900.

a) Write down the length BL.b) Find the length AK.c) Show that the length CL = 15 cm.d) Write down the lengths CM and AN.e) Calculate the volume of the pyramid.f) Calculate the surface area of the pyramid.

10.


34 area, volume & nets of solids

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