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Class: Subject: Topic: VIII Mathematics Mensuration

I. Areas of square, rectangle, triangle and circle.You should know:

What is a triangle?

What is a square?

What is a rectangle?

What is a circle?

You will learn:

The list of Subtopics Why these? Why should I care? Common Mistakes

Areas of square, rectangle,triangle and circle.

Area of a trapezium.

Area of a quadrilateral andspecial quadrilaterals:rhombus & parallelogram.

Area of cube, cuboidscylinder

Volume of cube cuboidand cylinder.

This topic involves a lot ofcalculation - so many errorsare possible in calculating.Inability to correlate thegiven data to arrive at ananswer - for instance “Whatis given” “What can Iassume”, “What do I needfind”, etc.Confusion between varioustypes of quadrilaterls - theirproperties and formulaes.

This topic is very useful incalculation of areas andvolumes in mathematics aswell as in physics.Important for board examsas well as competativeexams in future.Study of mensurationexpands our understandingof other irregular shapestoo.Useful in making modelsetc.

All around us there areobjects - some havingregular shape someirregular shape, is it notwonderful to be able tomeasure the surface area orvolume and compare.For instance you can planout wonderful interiors,furnishings etc. bymeasuring the size of yourroom, walls etc.Look at the clothes thatyou wear or the bags thatyou carry, they have beenmade by using the conceptsof mensuration.

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Fill up the blanks:

Figure Perimeter Area

1)

2)

3)

4)

4 x side

2 x ( ) x

Sum of 12 x x

x

The area of a square whose side is 4 cm is .

The length of a recangle is twice the breadth. If its area is 200 cm2,

then its perimeter is .

If base and height of a triangle are 13cm and 6cm, its area is .

The area and perimeter of a circle happen to be equal in magnitude. Then the diameter

of this circle is .

The sides of a triangle are 6cm, 8cm and 10cm. Its area is .

Now consider this question:

A B

D C

P Q

S R

45

35

80

60

In this given figure PQRS is a rectangle of sides80cm and 60cm.

ABCD is a rectangle inside PQRS of sides 45cmand 35cm.

Find area of the shaded portion. Also findperimeter of the shaded portion.

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Let us solve:

Area of the shaded portion is the difference in areas between outer rectangle and innerrectangle.

is area of shaded = (Area of PQRS) - (Area of ABCD)

Portion = (60 x 80) - (35 x 45)

= 4800 - 1575

= 3225 sqcm.

The perimeter of the shaded portion is the sum of the perimeters of the 2 rectangles.

= 2 (60 + 80) + 2 (35 + 45)

= 440cm.

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Now you solve these questions:

1) Find areas of the figures below.

D

A

C

B8cm

8cm

a)

b)

Find the area of ABC

A

BC

12cm

20cm

Worksheet

d) A B

D C

ABCD is a rectangle of sides

AB = 49 cm

AD = 14 cm

The semi circles are drawn using the length andbreadth as diameter.

P

Q

R

S

T

UX

Y

UQRT is rectangle

UQ = 25 cm

UT = 10 cm

PS = 32 cm

PX = YS

c)

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Find areas of the shaded portions below:

b)

a)7cm

7cm

25

25cm3cm

5cm5cm

5cm

c) 20

10

4

4

d)

35cm

3) A floor is in the shape of a rectangle of sides 15m and 12m. Square tiles of side

1.5m are to be used for flooring. If the cost of a tile is Rs.50/-, find the totalcost of flooring.

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4)

D 25cm

A

CB

12cm

30cm

In ABC,

AD is altitude on BC.

Find the magnitude of the altitude on AC.

5) There is a triangular park which has sides AB = 12m, BC = 20m and AC =

16m. Find the least distance a person has to travel from A to reach the roadconnecting B and C.

6) The area of a square is equal to five times the area of a rectangle of dimensions

20cm by 64cm. Find the perimeter of this square.

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TILES

If this tile costs ` 50, how much would it cost to tile these areas?

Tiles can be cut to fit the shape. Give the most practical answer.

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1) A landscaper has 20 sections of fence, each 1m long, to enclose a rectangulargarden. The diagram models one of the possible rectangular shapes with aperimeter of 20m. Complete the chart below and answer the following questions:

Optimizing Perimeter and Area

a) What dimensions give the maximum area? (Think: What dimensions give me the largest area value?)

b) What is the maximum area?

Rectangle Width (m) Length (m) Perimeter (m) Area (m2 )

1 20

2 20

3 20

4 20

5 20

2) A farmer wants to fence a rectangular animal pen with the minimum amountof fencing, so that the pen has an area of 24m2. The rectangle shown has anarea of 24 m2. Complete the chart below and answer the following questions.

a) What dimensions uses the minimum amount of fencing? (Think: What values give me the smallest perimeter value?)

b) What is the minumum perimeter?

Rectangle Width (m) Length (m) Area (m2) Perimeter (m)

1 24

2 24

3 24

4 24

2m

8m

2m

12m

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II. Area of a Trapezium

You should know:

What is a trapezium?

You will learn:

How to calculate the area of a trapezium.

A

D C

Ba cm

b cm

hcm

ABCD is a trapezium with AB DC

AB = a cm

DC = b cm

and the distance between AB & DC is h cm

Area = 12 x ( a + b ) x h

Write in words: Area of trapezium equals

____________________

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1) Find the area of the following trapeziums.

A

D C

B

6 cm

6 cm12

cm

A B

CD

7cm

20cm

( a )

Solution: 12 x ( AD + BC ) x DC

= 12 x (12 + 6) x 6

= 54 sqcm.

( b )

2) The area of a trapezium whose parallel sides measure 25cm and 35cm is 300

sq.cm. Find the distance between the parallel sides.

3) There is a trapezium in which the longer of the two parallel sides is twice the

shorter side and the distance between the two parallel sides is half the shorterside. Find the area of this trapezium.

4) The area of a trapezium is 126 sq.cm. The height of the trapezium is 7cm. If

one of the bases is longer than the other by 6cm, find the lengths of thebases.

5) In the adjoining figure AB || DC and DA is perpendicular to AB. Further DC =

7cm, CB = 10 cm and AB = 13 cm. Find the area of the quadrilateral ABCD.

6) The area of a field in the shape of a trapezium measures 1440 m2. The

perpendicular distance between its parallel sides is 24 m. If the ratio of the

A B

CD

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7) The lengths of the parallel sides of a trapezium are in the ratio 5 : 3 and the

distance between them is 12.5cm. If the area of the trapezium is 450 cm2, findthe lengths of its parallel sides.

8) The lengths of parallel sides of a trapezium are x cm and y cm and area of the

trapezium is 12 (x2 - y2) cm2. Find the distance between the parallel sides (in

cm) in terms of x and y.

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III. Areas of quadrilaterals and special quadrilaterals - rhombus & parallelogram.

You should know:

What is a quadrilateral?

What is a parallelogram?

What is a rhombus?

Identify the similarities and differences between a parallelogram and a rhombus.

You will learn:

Area of a parallelogram = base x height

Not ice t h at ar ea of le = 12 base x height. That is because a parallelogram really comprises

of two congruent triangles of base & height as the parallelogram.

height

base

height

base

1

2height

base

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Area of a quadrilateral is 12 x diagonal x (sum of the two offsets)

12 x DB x (AX + CY)

A

B

C

DX

Y

Area of a Rhombus:

The formula for a parallelogram is also applicable to the rhombus

Area = base x height

base

height

Additionally area = 12 x d1 x d2

Where d1 and d2 are length of the two diagonals.

d1

d2

Note: A square is also a rhombus

area of square = side x side

and also = 12 (diagonal)2

For example:

Find area of a square whose diagonal is 8 cm.

Ans: 12 8 8 = 32 cm2.

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AB

C

D

L

K

Find area of ABCD if

BD = 15cm

AK = 10cm

CL = 12cm

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2)C

B

A D

E

F

PQ

RS

Find area of this polygon ABCDEF if

AD = 18 cm

AS = 14 cm

AR = 12 cm

AQ = 8 cm

AP = 4 cm

also BP = 5 cm

CR = 6 cm

FQ = 5 cm

ES = 5 cm

Hint: Find areas of triangles ABP, CRD, SED, AQF and areas of trapeziums BCRP & QFES.Add them up.

Worksheet

3)A

B

C

D

E

F X

Y

Find area of the given figure ABCDEFif AD = 36 cm

BC = 12 YD

FE AD BC

AB = 20 cmYB = 12 cmFX = 8 cmFE = AY

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4) A

B

CD

h1

h2

In this quadrilateral,

area = 351 sq.cm

DB = 27 cm

h1 = 12 cm

Find h2.

5) Find areas of

7cm

10 cm(a) (b)

8cm

7cm

6

3

6

10

(c)

6) Find the area of a rhombus whose side measures 13cm and one of its diagonals

is10cm.

(Hint: Use the pythogoras theorem)

7) Find the perimeter of a rhombus whose diagonals measure 12cm and 16cm.

8) The diagonals of two square are in the ratio of 2 : 5. Find the ratio of their

areas.

9) What is the perimeter of a square, if the length of its diagonals is 12 2cm?

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10) One of the diagonals of a rhombus is double the other diagonal. Its area is 25

sq.cm. Find the sum of the diagonals.

11) The perimeter of a rhombus is 56 m and its height is 5 m. Its area is:

12) Find the perimeter, diagonal and area of a square having each side 8m

long. Take 2 1.41

13) A field in the form of a parallelogram has one of its diagonals 42m long and the

perpendicular distance of this diagonal form either of the outlying vertices is10.8m. Find the area of the field.

14) A parallelogram has sides of 15 cm and 12 cm. If the distance between its

shorter sides is 7.5 cm, find the distance between its longer sides.

15) One diagonal of a parallelogram is 70 cm and the perpendicular distance of

this diagonal from either of the outlying vertices is 27 cm. Find the area of theparallelogram.

16) Matrix Matching:

Column - A Column - B1) Area of a square a) Sum of length of three sides

2) Perimeter of a square b)12 d2

3) Area of a rectangle c) r2

4) Perimeter of a rectangle d) length breadth

5) Area of triangle e)12 d1 d2

6) Perimeter of a triangle f)12 d (h1 + h2)

7) Area of a circle g)12 h (a + b)

8) Perimeter of a circle h) twice the sum of length and breadth9) Area of a quadrilateral i) 2 r

10) Area of a parallelogram j)12 base height

11) Area of a trapezium k) base height12) Area of a rhombus l) four times the side13) Area of a kite

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Some HOTS questions:

1) There is a rhombus whose area is 21sqcm and the perimeter is 40cm. Find thesum of lengths of diagonals.

2) If the length and breadth of a rectangle are increased by 3cm, its area increasesby 72sqcm. If the length alone is increased by 1cm the area increases by 9sqcm.Find length and breadth.

3) A square shed of side 7m is in the middle of a huge grass field. A cow is tied toone ofits corners outside the shed, with a rope of length 14m. What is the areathat the cow can graze assuming that it cannot enter the shed?

4) A wire is shaped into a square, it enclosed an area of 100sqcm. If this wire isremodelled to form a semicircle, find area of the semicircle.

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IV. Areas of cube, cuboid, cylinder.

You should know:

Cube, cuboid, cylinders belong to the family of prisms. Prisms are three dimensional objectswhose bases are parallel to the tops and the base & top are congruent shapes.

All prisms have two types of areas.

a) Lateral surface area - the area of the walls (L.S.A)

b) Total surface area is the sum of lateral surface area with twice the area of the base (T.S.A)

Explain the following with appropriate figures:

a) cube b) cuboid c) cylinder

You will learn:

L.S.A of cube = 4 (side)2

T.S.A of cube = 6 (side)2

L.S.A of cuboid = 2 ( l + b ) h

T.S.A of cuboid = 2 (lh + bh + lh)

L.S.A of cylinder = 2 r h

T.S.A of cylinder = 2 r (r + h)

- S - l

h

b

Cube Cuboid

h

- r -

Cylinder

You may also understand that the

L.S.A = Per im et er of t h e base height.

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For example:

a) Find L.S.A and T.S.A of a cube of side 8cm.

Ans: L.S.A = 4 S2 = 4 8 8 = 256 sqcm.

T.S.A = 6 S2 = 6 8 8 = 384 sqcm.

b) Find L.S.A and T.S.A of a cuboid whose height is twice the breadth and length istwice the height. And the breadth = 10 cm.

Solution:

b = 10cm

h = 2 10 = 20cm.

l = 2 h = 2 20= 40cm.

L.S.A = 2 ( l + b ) h

= 2 (40 + 10 ) 20 = 2000 sqcm.

T.S.A = 2 ( l b + bh + l h )

= 2 ( 40 10 + 10 20 + 40 20 )

= 2 ( 400 + 200 + 800 )

= 2800 sqcm.

c) A rectangular sheet of dimensions 88cm by 20cm is rolled along its length so that thetwo breadths are joined to form a cylinder.

Find L.S.A & T.S.A of this cylinder.

Solution:

If one can imagine in this cylinder.

88

2020

Perimeter = 88cm

20

Perimeter of base = 2 r = 88cm.

2 227 r = 88

r = 7 x 882 x 22 = 14cm.

L.S.A = 2 r h

= 2 227 14 20 = 88 20

= 1760 sqcm.

(Note: The area of the rect.sheet is the same as L.S.A.of this cylinder)

T.S.A = 2 r (r + h)

= 2 227 14 (14 + 20)

= 2992 sqcm.

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Now solve the following questions yourself:

1) A road roller is in the shape of a cylinder. The radius of the cross section is

14cm and length is 1m. What is the area covered by it in making 200 revolutions.

2) The breadth and height of a cuboid are4cm and 2cm respectively. If the total

surface area of the cuboid is 88 sqcm, find its length.

3) A metallic trunk is to be made from a rectangular sheet of metal bearing

dimensions

l = 80 cm

b = 60 cm

h = 50 cm.

If the metal costs Rs. 250/- per 100 sqcm and making charges for the trunk isRs. 500/-, find the total cost incurred.

4) Two cubes of side 7cm are joined along a face to form a cuboid. Find L.S.A &

T.S.A of this cuboid.

5) The T.S.A of a cuboid is 214 sqcm. The areas of 2 of its faces are 42 sqcm and

35 sqcm. Find the length of the cuboid if it is the greatest of the dimensions.

6) There is a square sheet of side 30cm. From each of its four corners 4 squares

of side 5cm are cut away. The rest of the sheet is folded to form an opencuboidal box. Find L.S.A of this box.

5cm

5cm

5cm5cm

Fold along the alloted line to get a open box.

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V. Volume of Cube, Cuboid, Cylinder.

You should know:

What is cube,

What is cuboid,

What is cylinder.

You will learn:

Formula for volume of

cube, cuboid, cylinder

Volume of a cube = (Side)3

Volume of a cuboid = l b h

Volume of cylinder = r2h

Note: a) Volume has its units as cubic units

b) Volume of a prism = area of base height.

1) Find the volume of

a) a cube of side 5cm

b) a cuboid of l = 10cm, b = 6cm, h = 4cm.

c) a cylinder of base area 154 sqcm and height 10cm.

2) There is a reservoir of the shape of a cylinder whose base radius is 21m and

height is 15m. How many hours will it take to fill it with water. Which flows atthe rate of 45m3 per hour?

3) The ratio of volumes of two cubes is 729 : 1331. Find ratio of their T.S.A.s.

4) There is a cubical wooden block of side 7cm. A cylinder which has a height of

7cm and base resting on one of the faces of the cube is carved out. Find thevolume of wood that is left.

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5) A square sheet has a side of 60cm. Four small squares of side 6cm are cut

from the four corners. The rest is then folded to form an open cuboid box. Findthe volume of this box.

6) A wall has dimensions 15m x 10m x 8m. 10% of this wall is occupied by

mortar. The rest 90% is occupied by bricks. The cost of 1000 bricks ofdimensions 10cm x 8cm x 4cm is 400/-. Find total cost of making this wall.

7) A cylindrical vessel of diameter 48cm has water to a height of 10cm. A metal

cube of 14cm edge is immersed in it. Calculate the height to which the water inthe vessel rises.

8) A swimming pool 150m long and 50m wide, is 1 m deep at the shallow end and

6m deep at the deep end. Find the volume of the pool.

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Using Math to Build a Swimming Pool

Imagine that you have decided to build a swimming pool. You have some requirements

for the pool, and need to know how much it is going to cost. You can use math to

answer your questions.

Calculating the design for the pool.

Question 1: Pools are either measured in yards or meters. How many feet

are in a yard? About how many feet are in a meter? (You can round to the nearest

foot.) Which is longer, a yard or a meter?

The design for the pool is:

• It should be rectangular, 25 meters long and 15 meters wide.

• The bottom of the first five meters of the pool is flat and is 1 meter deep.

• In the next 10 meters, the pool gets deeper at a rate of 1 meter of depth for every

one meter of length, so that at 15 meters from the shallow end of the pool, the pool is

6 meters deep.

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• The last 10 meters of the pool also have a flat bottom, and are 6 meters deep.

Question 2: What is the area of the surface of the pool?

Question 3: What is the area of the wall at the shallow end of the pool?

Question 4: What is the area of the wall at the deep end of the pool?

These next two questions are tricky. Think how you could divide up the shape you have

to measure so that it will be easier.

Question 5: What is the area of one of the sides of the pool?

Question 6: What is the area of the bottom of the pool?

Question 7: How many cubic meters of water are you going to need to fill the pool? (Inother words, what is the volume of the pool?)

(Hint: Use the areas you just figured out to answer this question.)

Bonus: Explain how to divide up the pool (in your head) to most easily figure out thevolume of the pool.

Question 8: You have picked out a blue tile to cover the bottom and sides of the pool. Eachtile is 25 centimeters on a side. How many of them does it take to cover a square meter?

Question 9: How many tiles will it take to cover the sides and bottom of the pool?

Question 10: Each tile costs `1.25. How much will it cost to cover the pool in tile?

Question 11: The local digging company charges by the cubic meter. They charge `100 foreach cubic meter of dirt they remove. How much will it cost to dig the pool?

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CONTAINER DESIGN

In industry one of the consierations when designing a container is the cost of the

materials. There needs to be a balance between the amount a container holds (itsvolume) and the amount of material needed (its surface area).

Calculate the volume of this tin to the nearest millilitre (ml).

Explain why the surface area (S) of the tin can be given by the formula:

S = 2 r2 + 2 rh

Find a design which minimises the cost of the tin (and keeps it’s volume the same).You could start by copying and completing a table like the one below.

h (cm) r (cm) s (cm2)

For the optimal tin, (where surface area is minimal), what is the relationship betweenthe height and the radius?

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Volume and Surface Area word problems

1. If both of these cans of pizza sauce are cylinders, which is the better buy?

2. Cans of soup are often packed in boxes as shown below. Calculate the area that

is wasted in between all of the cans.

3. A cubic tank with 4.6m edges is filled with water. How much water will be left in

the tank if some is drained off to fill a cylindrical tank with a radius of 2.2m anda height of 4.6m?

4. How much cat food would fit into a can that has a height of 14.5cm and a

diameter of 9cm?

5. A birthday gift is 55cm, 40cm wide, and 5cm high. The sheet of paper you want

to use to wrap it measures 75cm by 100cm is the paper large enough to wrapthe gift? Explain.

6. What happens to the surface area of a rectangular prism if all three of its

dimensions are doubled? Tripled?

7. The volume of a rectangular prism is 24cm3. Find two other shapes that have

the same volume. What are their dimensions?

` 300

` 100

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8. The label of a soup can has 2cm of overlap. Find the surface area of the label.

9. A square pyramid has a base with an area of 40cm and a volume of 100cm3.

What is height of the pyramid?

10. Three identical tennis balls with an 8cm diameter are s tacked in a cylindrical

container. For this container, calculate

a) volume b) surface area

11. What volume of concrete is required to build this footbridge?

12. A ramp shaped like a triangular prism is 2.5m wide, and reaches a loading dock

1.2m high. The ramp starts 3m from the dock. All sides of the ramp are to becovered with pressure-treated plywood. Calculate the amount of plywoodsheathing required to cover the ramp.

13. A donut shop has two different sizes of donuts. The regular donuts have a diameter

of 12cm with a 3cm hole, and the minidonuts have a 6cm diameter with a 1.5cmhole. The price of a regular donut is ` 70, and minidonuts sell for ` 30 each.Each donut is 3cm thick. Which donut is the better buy?

(Hint: The label covers the LSA)

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14. Ace pool company sells portable spas as shown. The inside diameter of the spa

is 8’ and the depth is 3’ 6”. The sides and the bottom are each4” thick.

a) Calculate the total amount of vinyl required for the spa (onboth the interior and the exterior).

b) The sides and the bottom of the spa are filled with adense foam material. Calculate the volume of foam required.

c) If the spa is to be filled with water to within 8” of the top,how much water does the spa hold?

15. Create 2 different cylinders from a 812 ” by 11” sheet peice of paper, one rolled

lengthwise, the other widthwise (do not worry about/include making thebase). Will the containers hold the same amount of popcorn?

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Name these figures:

( a ) ( b ) ( c )

Is this a cylinder?

If not, then find out its name.

Fill up the blanks:

1) A cube has edges.

2) Amount of region occuped by a solid is called .

3) 1 litre = cm3

1 m3 = litres.

4) Sum of areas of faces of a solid is called its .

5) Cube, cuboid & cylinder belong to the family of .