A square-based pyramid is shown.
A) Calculate the area of one of its triangular faces.
B) Calculate the total surface area of the pyramid.
Answers
Answer:
Solution
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(a) The perimeter of the base is P=4s, since it is a square, therefore,
P=4×6=24 cm
The general formula for the lateral surface area of a regular pyramid is LSA=
2
1
Pl where P represents the perimeter of the base and l is the slant height.
Since the perimeter of the pyramid is P=24 cm and the slant height is l=14 cm, therefore, the lateral surface area is:
LSA=
2
1
Pl=
2
1
×24×14=168 cm
2
Now, the area of the base B=s
2
with s=6 cm is:
B=s
2
=6
2
=36 cm
2
The general formula for the total surface area of a regular pyramid is TSA=
2
1
Pl+B where P represents the perimeter of the base, l is the slant height and B is the area of the base.
Since LSA=
2
1
Pl=168 cm
2
and area of the base is B=36 cm
2
, therefore, the total surface area is:
TSA=
2
1
Pl+B=168+36=204 cm
2
Hence, lateral surface area of the pyramid is 168 cm
2
and total surface area is 204 cm
2
.
(b) The perimeter of the base is P=4s, since it is a triangle, therefore,
P=3×12=36 cm
The general formula for the lateral surface area of a regular pyramid is LSA=
2
1
Pl where P represents the perimeter of the base and l is the slant height.
Since the perimeter of the pyramid is P=36 cm and the slant height is l=20 cm, therefore, the lateral surface area is:
LSA=
2
1
Pl=
2
1
×36×20=360 cm
2
Now, the area of the base B=
4
3
s
2
with s=12 cm is:
B=
4
3
s
2
=
4
3
×12×12=36
3
cm
2
The general formula for the total surface area of a regular pyramid is TSA=
2
1
Pl+B where P represents the perimeter of the base, l is the slant height and B is the area of the base.
Since LSA=
2
1
Pl=360 cm
2
and area of the base is B=36
3
cm
2
, therefore, the total surface area is:
TSA=
2
1
Pl+B=360+36
3
=36(10+
3
) cm
2
Hence, lateral surface area of the pyramid is 360 cm
2
and total surface area is 36(10+
3
) cm
2
.
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