SME
Grade 10 Math · Problem type
Surface area of a pyramid
Use this for “A square pyramid has base side 6 cm and height 4 cm. Find its slant height and surface area.”
Diagram
Base and four triangles add to 96 cm².
Height: 4 cm
This is an example. Type your own values to start a new problem.
ℓ
2
=
h
2
+
(
÷
2)
2
S (cm²)
=
b
2
+
2
×
b
×
ℓ
Formulas
ℓ
2
=
h
2
+
(
b
2
)
2
5
2
=
4
2
+
(
6
2
)
2
S
=
b
2
+
2
×
b
×
ℓ
96
=
6
2
+
2
×
6
×
5
Assumptions
•
The base is a square with side b; the four side faces are equal triangles.
•
The slant height ℓ is each triangle’s height: the hypotenuse of legs h and b ÷ 2.
•
So S = b² + 4 × ½ × b × ℓ = b² + 2bℓ.
Step-by-step
WE KNOW
b = 6 cm, h = 4 cm
FIND
slant height (ℓ), surface area (S)
Step 1 · Find slant height (ℓ)
Use ℓ² = h² + (b ÷ 2)²
The slant height is the hypotenuse over the middle of a base edge.
ℓ
=
√(
h
2
+
(
b
2
)
2
)
ℓ
=
√(
4
2
+
(
6
2
)
2
)
ℓ
=
√
4
2
+
3
2
ℓ
=
√
16 + 9
ℓ
=
√
25
ℓ
=
5
cm
Step 2 · Find surface area (S)
Use S = b² + 2 × b × ℓ
The
square
base
and
four
triangles,
each
1
2
×
b
×
ℓ
.
S
=
6
2
+
2
×
6
×
5
S
=
36
+
2
×
6
×
5
S
=
36
+
12
×
5
S
=
36
+
60
S
=
96
cm²
Check
Put the numbers back in. Both sides should match.
5
2
=
4
2
+
(
6
2
)
2
✓
96
=
6
2
+
2
×
6
×
5
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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