SME
Grade 7 Math · Problem type
A base made of rectangles
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.”
Diagram
48 − 6 = 42 cm²
h = 4 cm (prism height)
V = 168 cm³ (volume)
S = 196 cm² (surface area)
This is an example. Type your own values to start a new problem.
l
Length of the whole rectangle
example
w
Width of the whole rectangle
example
c
Length of the cut-out
example
d
Width of the cut-out
example
h
Prism height
example
B
Base area
worked out (not typed)
V
Volume
worked out (not typed)
S
Surface area
worked out (not typed)
Formulas
B
=
l
×
w
−
c
×
d
(base
area
=
length
of
the
whole
rectangle
×
width
of
the
whole
rectangle
−
length
of
the
cut-out
×
width
of
the
cut-out)
42 = 8 × 6 − 3 × 2
V
=
B
×
h
(volume
=
base
area
×
prism
height)
168 = 42 × 4
S
=
2
×
B
+
(2
×
l
+
2
×
w
)
×
h
(surface
area
=
2
×
base
area
+
(2
×
length
of
the
whole
rectangle
+
2
×
width
of
the
whole
rectangle)
×
prism
height)
196 = 2 × 42 + (2 × 8 + 2 × 6) × 4
Assumptions
•
The base is a rectangle with a rectangle cut from one corner: an L.
•
Base area = the whole rectangle − the cut-out. Volume = base area × height.
•
Cutting a corner keeps the perimeter: the two new sides are as long as the two lost.
•
Surface area = 2 bases + perimeter × height, since the sides unroll to one rectangle.
Step-by-step
WE KNOW
l = 8 cm (length of the whole rectangle), w = 6 cm (width of the whole rectangle), c = 3 cm (length of the cut-out), d = 2 cm (width of the cut-out), h = 4 cm (prism height)
FIND
base area (B), volume (V), surface area (S)
Step 1 · Find base area (B)
Use B = l × w − c × d (base area = length of the whole rectangle × width of the whole rectangle − length of the cut-out × width of the cut-out)
The whole rectangle, take away the corner cut out.
B
=
8
×
6
−
3
×
2
B
=
48
−
6
B
=
42
cm²
Step 2 · Find volume (V)
Use V = B × h (volume = base area × prism height)
Stack the base area up the prism’s height.
V
=
42
×
4
V
=
168
cm³
Step 3 · Find surface area (S)
Use S = 2 × B + (2 × l + 2 × w) × h (surface area = 2 × base area + (2 × length of the whole rectangle + 2 × width of the whole rectangle) × prism height)
Two L-shaped bases, and the sides: the perimeter times the height.
S
=
2
×
42
+
(2
×
8
+
2
×
6)
×
4
Perimeter of the L = 2 × 8 + 2 × 6 = 28
S
=
2
×
42
+
28
×
4
S
=
84
+
112
S
=
196
cm²
Check
Put the numbers back in. Both sides should match.
42 = 8 × 6 − 3 × 2 ✓
168 = 42 × 4 ✓
196 = 2 × 42 + (2 × 8 + 2 × 6) × 4 ✓
Topic check
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