SME
Grade 7 Math · Geometry
Volume and surface area of prisms
Use this for “Find the volume and surface area of a box 6 cm long, 4 cm wide and 5 cm tall.”
Diagram
The cut is a rectangle: B = 6 × 4 = 24 cm².
Every cut parallel to the base is the same shape as the base.
V = B × h = 24 × 5 = 120 cm³
This is an example. Type your own values to start a new problem.
l
Length
example
w
Width
example
h
Height
example
a
Cut height
example
B
Base area
worked out (not typed)
V
Volume
calculated
S
Surface area
worked out (not typed)
Formulas
B
=
l
×
w
(base
area
=
length
×
width)
24 = 6 × 4
V
=
B
×
h
(volume
=
base
area
×
height)
120 = 24 × 5
S
=
2
×
B
+
h
×
(2
×
l
+
2
×
w
)
(surface
area
=
2
×
base
area
+
height
×
(2
×
length
+
2
×
width))
148 = 2 × 24 + 5 × (2 × 6 + 2 × 4)
Assumptions
•
A prism has the same cross-section all the way up, so volume = base area × height.
•
The cut is parallel to the base.
•
Surface area adds every face: two bases plus the sides, which unroll to a rectangle, height × perimeter.
•
Volume is in cubic units, surface area in square units.
Step-by-step
WE KNOW
l = 6 cm (length), w = 4 cm (width), h = 5 cm (height), a = 2 cm (cut height)
FIND
base area (B), volume (V), surface area (S)
Step 1 · Find base area (B)
Use B = l × w (base area = length × width)
The cut is the base: a rectangle, length times width.
B
=
6
×
4
B
=
24
cm²
Step 2 · Find volume (V)
Use V = B × h (volume = base area × height)
Stack the base area up the height: base area times height.
V
=
24
×
5
V
=
120
cm³
Step 3 · Find surface area (S)
Use S = 2 × B + h × (2 × l + 2 × w) (surface area = 2 × base area + height × (2 × length + 2 × width))
Two bases, and the four sides unrolled: the height times the distance around the base.
S
=
2
×
24
+
5
×
(2
×
6
+
2
×
4)
S
=
2
×
24
+
5
×
(12
+
8)
S
=
2
×
24
+
5
×
20
S
=
48
+
100
S
=
148
cm²
Check
Put the numbers back in. Both sides should match.
24 = 6 × 4 ✓
120 = 24 × 5 ✓
148 = 2 × 24 + 5 × (2 × 6 + 2 × 4) ✓
Topic check
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More problem types
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Use this for “The base is an L shape, 8 by 6 with a 3 by 2 corner cut out. The prism is 4 cm tall. Find its volume and surface area.”
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Use this for “What shape is the cross-section when the pyramid is cut parallel to its base?”
Prism or not?
Use this for “Which solids are prisms?” before using V = Bh.