3-D Measurements
Surface Area & Volume of a Rectangular Prism and a Cube:
Shape
Rectangular Prism
Surface Area (cm2, m2)
Volume (cm3, m3)
SA = 2(lw + wh + lh)
V = lwh
SA = 6s2
V = s3
Cube
Cubes: A cube is 3 - D figure with six matching square sides. The figure above shows a cube. The dotted lines
indicate edges hidden from your view. If s is the length of one of its sides, the Volume of the cube = s3
Since the cube has six square-shape sides, the Surface area of a cube = 6s2
Rectangular Prisms: In a rectangular solid, the length, width and height may be of different lengths. The volume
of the above rectangular solid would be the product of the length, width and height that is
Volume of rectangular solid V = lwh
Surface area of rectangular solid SA = 2(lw + wh + lh)
Example : Find the Surface Area and volume of the following shapes. Don’t forget the formulas and the units.
Surface Area
Volume
SA = 2 ( lw + wh + lh )
Volume = l × w × h
a)
= 2 (6 x 3 + 3 x 4 + 6 x 4)
= 6 cm × 3 cm × 4 cm
= 2 (18 + 12 + 24)
= 72 cm3
2
= 2 x 54 = 108 cm
b)
3 cm
3 cm
3 cm
SA = 6 s2
= 6 x 32
=6x9
= 54 cm2
V = S3
= 33
= 27 cm3
Problem Solving Questions:
1.
A box has a volume of 308 cm3. Its height is 4 cm greater than its length. Its length is 3 cm greater than its
width. What is the width of the box?
W (cm)
2
3
4
L=w+3
5
6
7
H=l+4
9
10
11
V = lwh (in cm3) Just multiply to get the answers
90
180
308 (that’s your given volume. Therefore, W = 4 cm. The answer)
2.
A box has a volume of 195 cm3. Its height is 8 cm greater than its length. Its length is 2 cm greater than its
width. What is the width of the box?
W (cm)
1
2
3
L=w+2
3
4
5
H = l +8
11
12
13
V = lwh (in cm3) Just multiply to get the answers
33
96
195 (that’s your given volume. Therefore, W = 3 cm. The answer)
3.
A box has a volume of 570 cm3 and a surface area of 478 cm2. What are the length, width, and height of the
box?
To start with, try to get different combinations for the same volume (in this case 570 cm3). When you find the
given surface area (in this case 478 cm2), this is your answer.
l (cm)
5
5
10
15
6
4.
w (cm)
2
3
3
2
5
H (cm)
57
38
19
19
19
V = lwh ( in cm 3)
570
=
=
=
=
SA = SA = 2 ( lw + wh + lh ) (in cm2)
818
638
554
706
478  Therefore, L = 6 cm, w = 5 cm, and h = 19 cm
What is the capacity of the aquarium in the example (a) above? Hint: 1 cm3 = 1 mL
Since 1 cm3 = 1 mL  Volume (in cm 3) = Capacity (in mL)  Capacity = 72 mL
5.
Karen is painting the floor and four walls of a basement. The basement is 12 m long, 4 m wide, and 2.5 m
high. Paint comes in 4L cans. One litre of paint covers 10 m2.
a) What area does Karen need to paint?
SA = 2 ( lw + wh + lh )
Since we have no ceiling, we have to find only 5 areas:
SA = lw(floor) + 2wh + 2lh
= 12 x 4 + 2 (4 x 2.5) + 2 (12 x 2.5)
= 48 + 20 + 60 = 128 m2
b) How many cans of paint does Karen need?
1 L covers 10 m2  4L = 4 x 10 = 40 m2
128 40 = 3.2
i.e., Karen needs a little more than 3 cans.
So the answer is 4 Cans
6.
Mercury Courier charges $0.10/cm2 to deliver a package. How much will the courier service charge to
deliver each of this package?
30 cm
V = lwh
= 27 x 4 x 30
= 3240 cm3
27 cm
Cost = $0.10/cm2 x 3240 cm2
= $ 324
4 cm
Volume, Capacity, and Mass
A capacity of a container is the greatest volume it will hold. This means that the capacity (of liquids) in mL, L, KL
= the volume of the container in cm3, dm3, m3
There is a special relationship between the volume and mass of water. The following table will explain the
relationship between the capacity, the volume, and the mass.
Basic Units
Multiply by 1000
This means 
Multiply by 1000
This means 
Volume (of container)
1 cm3
1 cm3 x 1000 = 1000 cm3
= 1 dm3
1 dm3
3
1 dm x 1000 = 1000 dm3
= 1m3
1 m3
Capacity (of water)
1 mL
1 mL x 1000 = 1000 mL =
1L
1L
1L x 1000 = 1000 L
= 1 KL
1 KL
Mass (of water)
1g
1 g x 1000 = 1000 g =1
kg
1 kg
1 kg x 1000 = 1000 kg
=1t
1t
Length:
king
henry drank milk during christmas mass
Km
m
dm
cm
mm
From Larger Units to Smaller Units: You Multiply (each time by 10 …) {ignore hecta and deca meters)
1 km = 1000 m
1m = 10 dm
1 dm = 10 cm
1 cm = 10 mm
From Smaller Units to Larger Units: You Divide (each time by 10 …) {ignore hecta and deca meters)
1 m = 0.001
1 dm = 0.1 m
1 cm = 0.1 dm
1 mm = 0.1 cm
From larger to smaller, you multiply. From smaller to larger, you divide.
large to small (mult.)
Small to large (divide)
large to small (mult.)
Small to large (divide)
1 m = 100 cm
1 cm = 0.001 m
1 m = 1000 mm
1 mm = 0.001 m
1 dm = 100 mm
1 mm = 0.001 dm
1 dm = 100 mm
1 mm = 0.01 dm
1 km = 10 000 dm
1 dm = 0.00001 km
1 km = 100 000 cm
1 cm = 0.000001 km
1 km = 1000 000 mm
1 mm = 0.0000001 km
Capacity:
king
henry drank lime during christmas mass
From Smaller Units to Larger Units: You Divide (each time by 10 …) {ONLY KL, L, and mL}
1 KL = 1000 L
1 L = 1000 mL
1 KL = 1000 000 mL
1L = 0.001 KL
1 mL = 0.001 L
1 mL = 0.000000 KL
Mass:
king
henry drank grape during christmas mass
From Smaller Units to Larger Units: You Divide (each time by 10 …) {ONLY: t, Kg, g, and mg}
large to small (mult.)
Small to large (divide)
large to small (mult.)
Small to large (divide)
1 t = 1000 kg
1 kg = 0.001 t
1 t = 1000 000 g
1 g = 0.000001 t
1 kg = 1000 g
1 g = 0.001 kg
1 kg = 1000 000 mg
1 mg = 0.000001 kg
1 g = 1000 mg
1 mg = 0.001 g