3. Additional questions
1. For what size paper would the two cylinders
hold the same amount?
2. Using any non–square rectangular shape of
paper, does the shorter cylinder always holds
more?
3. Suppose you had two equal lengths of wire.
Fold the wires to make rectangles. Do you
think the two rectangles will always have the
same area?
4.
5. • Area = ½ (a x h1) = ½(b x h2) = ½(c x
h3)
• Semi-Perimeter s = (a + b + c)/2
• Area = √(s x (s – a) x (s - b) x (s - c))
• Perimeter = a + b + c
EQUILATERAL TRIANGLE
(Sides a)
• Area = √3/4 x a2
• Perimeter = 3 x a
• Height = √3/2 x a
TRIANGLE
(Sides a, b, c; Opp. Angles A, B, C respectively;
heights h1, h2 and h3 respectively)
a a
a
A
B C
A
aB
b
h1
h2
h3
c
C
6. RECTANGLE
(length l; breadth b)
• Area = l x b
• Perimeter = 2(l + b)
SQUARE
(side a)
• Area = a2
• Perimeter = 4a
l
b
a
a
7. PARALLELOGRAM
(height h, base b, other side a)
• Area = b x h
• Perimeter = 2 x (a + b)
• Area = ½ (d1 x d2)
• Perimeter = 4 x a
h d1
d2
b
a
a
RHOMBUS
(side a; diagonals of length d1 and d2 )
a
8. • Area = ½ (a + b) x h
• Perimeter = a + b + c + d
• Area = ½ (d1 x d2)
• Perimeter = a + b + c + d
a
b
h
d1
d2
TRAPEZIUM
(distance between parallel sides h, length
of parallel sides a & b)
KITE
(diagonals of length d1 and d2 )
c d
dc
b a
9. • Area = πr2/2
• Perimeter= πr + 2r
• Area = πr2
• Circumference = 2πr
r
rO
O
CIRCLE
(radius r)
SEMI-CIRCLE
(radius r)
10. • Area = /360 x (πr2)
• Length of arc = ( /360 x 2πr)
• Perimeter = ( /360 x 2πr) + 2r
SECTOR
(centre O, radius r, angle of sector )
SEGMENT
(centre O, radius r, angle of sector )
• Area = 1/2r2 x {( /180) x π –Sin }
r
O
r
O
11. • Volume = a3
• Total Surface Area = 6a2
• Longest diagonal = a x √3
• Volume = l x b x h
• Total Surface Area = 2{(l x b) + (b x h) + (l
x h)}
• Longest diagonal = √(l2 + b2 + h2)
a
a
a
l
b
h
CUBE
(length = breadth = height = a)
CUBOID
(length l; breadth b; height h)
12. • Volume = π x r2 x h
• Total Surface Area = 2π x r x (h + r)
• Curved (Lateral) Surface Area = 2π x r x h
• Volume = 1/3 π x r2 xh
• Total Surface Area = π x r x (l + r)
• Curved (Lateral) Surface Area = π x r x l
r
r
h
h
r
l
CYLINDER
(height h; radius of base r)
CONE
(height h; radius of base r; slant height l )
13. • Volume = 4/3 π x r3
• Total Surface Area = 4π x r2
• Volume = 2/3 π x r3
• Total Surface Area = 3π x r2
• Curved (Lateral) Surface Area = 2π x r2
r
r
HEMISPHERE
(radius r)
SPHERE
(radius r)
14. PRISM
(length of side of base a; number of sides of
base n; height h)
• For a right prism, base can be any shape
like square, triangle, pentagon or hexagon
as shown
• Volume = area of base x h
• Total Surface Area = 2 x area of base + n x
(h x a)
• Lateral surface area = n x (a x h)
PYRAMID
(length of side of base a; number of sides
of base n; height h; slant height l)
• For a right pyramid, base can be any
shape like square, triangle, pentagon or
hexagon as shown
• Volume = 1/3 x area of base x h
• Lateral surface area = ½ x (n x a) x l
• Total Surface Area = area of base +
lateral surface area
ha
l
n = 6 for
hexagon
a
n = 6 for
hexagon
h
15. FRUSTUM OF CONE
Slant height l; height h, radius of small
base r; radius of large base R
• Volume = 1/3π x h (R2 + r2 + Rr)
• Total Surface Area = π x (R2 + r2 + Rl +
rl)
• Lateral Surface Area = π x l (R + r)
• Slant height l = √{(R-r)2 + h2}
FRUSTUM OF PYRAMID
Slant height l; height h, side of base a
• Volume = 1/3 x h x (A1 + A2 +
√A1A2)
• Lateral Surface Area = ½ x
perimeter of base x l
• Total Surface Area = LSA + A1 + A2
h l
R
r
l
A2 a
h
A1
16. • A boat having a length 3m and breadth 2m
floating on a lake. The boat sinks by 1 cm
when a man gets on it. The mass of the man is
• Volume of water displaced = 3 x 2 x 0.01 m3
• = 0.06 m3
• Mass of man = volume of water displaced x
density of water
• = 0.06 x 1000 kg = 60kg.
