GR 11 MATHS
MEASUREMENT FORMULAE :
Total Surface Area & Volume
CONTENTS:
Right Prisms vs Right Pyramids
Page 1
Cylinders vs Cones
Page 1
Spheres and Hemispheres (Half-spheres)
Page 2
Compliments of
Volume
Right Prisms vs Right Pyramids
Compare right prisms to right pyramids
Right Prisms
Volume of the prism
A Prism
with
a square base
Special case: a cube
h
rectangular
base
square
base
x
x
triangular
base
Volume of the pyramid
h s
A Pyramid
with
a square base
Right Pyramids
= x2 h
x
x
x
= area of base × height
=
x
1
3
area of base × perpendicular height
= 1 x2 . h
3
x
Cylinders vs Cones
square
base
triangular
base
Cylinders
Surface Area
Cones
The base is
a circle in
both cases
Practically speaking, the total surface area (TSA) of any 3D solid is equal
to the sum of the areas of all the faces
Surface Area
e.g.
A Prism
with
a square base
h
x
x
TSA of the prism (6 faces)
Cylinder
= x2 + x2 + x h + x h + x h + x h
= 2 x2 + 4xh
h
TSA = 2(area of base) + circumf. of base % height
r
r
= 2(πr 2 ) + (2πr)h
circumference
= 2πr(r + h)
same idea as
for prisms
r
h s
A Pyramid
with
a square base
x
TSA of the pyramid (5 faces)
. . . xs = 1 % x % s
= x 2 + 4 ⎛⎜ xs ⎞⎟
⎝ 2 ⎠
2
s
2
= x2 + 2 x s
Cylinder
h
. . . same as above
r
Pyramid : TSA = area of the base + 1 perimeter of base % slant height
= x 2 + 1 (4 x ) . s
2
= x2 + 2 x s
Copyright © The Answer
same idea as
for pyramids
Volume
TSA = 2(area of the base) + (perimeter of base × height)
= 2( x ) + (4 x )(h)
h
= πr + πrs
Using the formulae for prisms and pyramids in general :
2
2
= πr + 1 (2πr).s
2
2
2
x
Prism :
TSA = area of base + 1 circumf. of base % slant height
r
Cone
2
3
h
1
= πr2.h
Volume = 1 area of base % perpendicular height
Cone
. . . same as above
Volume = area of base % height
r
= 1 π r2 . h
3
Spheres and Hemispheres (Half-Spheres)
The effect on surface area and volume of a prism when
multiplying the dimensions by a constant factor k.
Surface Area
TSA of a sphere = 4 × πr2
= 4πr2
. . . 4 × the area of a circle
(also with radius, r )
Observe the effect on the total surface area (TSA) and volume of a cube of
multiplying the dimensions by 2, and then, by 3.
TSA of a hemisphere = 1 (TSA of a sphere) + the area of a circle
2
= 1 (4πr2) +
2
πr
If a hemisphere is attached to a cylinder (see alongside),
then TSA of this object = area of base of cylinder (πr2)
+ area of hemisphere (2πr2)
Volume
Volume of a hemisphere . . .
= 4
3
= 1 × 4 (πr3) = 2 πr3
2
3
3
πr
2
2x
2
TSA = 4πr
A hemisphere :
TSA = area of base (πr2) + curved surface area (2πr2) = 3πr
3
= 23 . x
2x
= 24 x
2
3
= 8x
Or, all dimensions x3 :
TSA = 6 (3x × 3x)
3x
3x
. . . 4 % the area of a circle
Vol = 3x × 3x × 3x
2
3
= 6.32 . x
= 54 x
TSA = area of base + 1 perimeter of base × slant height
2
TSA = area of base + 1 circumference of base × slant height
2
A sphere :
Vol = 2x × 2x × 2x
2
= 6.2 . x
Total Surface Area (TSA)
A right cone :
3
= x
TSA = 6(2x × 2x)
2x
SUMMARY OF FORMULAE : Pyramids /Cones/(Half)Spheres
A right pyramid :
Vol = x × x × x
2
All dimensions x2 :
Note : The area of the
base of the hemisphere
is ignored.
Volume of a sphere . . .
3
r
= 6x
x
x
h
+ area of curved surface of cylinder (2πrh)
TSA = 6(x × x)
x
r
= 3πr2
= 3πr2 + 2πrh
A Cube :
2
= 33 . x
2
3
= 27x
3x
Summary of the effects of scale factors 2, 3 and k:
2
All dimensions × 2
TSA × 4
Volume × 8
A right pyramid : Volume = 1 area of base × perpendicular height
All dimensions × 3
TSA × 9
Volume × 27
A cone :
In general :
TSA × k2
Volume × k3
Volume
A sphere :
A hemisphere :
Copyright © The Answer
3
Volume = 1 area of base × perpendicular height
3
3
Volume = 4 πr
3
3
Volume = 2 πr
3
All dimensions × k
2