When we think of pyramids we think of the Great Pyramids of Egypt. They are actually Square Pyramids, because their base is a Square.
A pyramid is made by connecting a base to an apex
There are many types of Pyramids, and they are named after the shape of their base.
When all side faces are the same:
When side faces are different:
Notes: The Surface Area has two parts: the area of the base (the Base Area), and the area of the side faces (the Lateral Area).
For Base Area :
It depends on the shape, there are different formulas for triangle, square, etc.
For Lateral Area :
When all the side faces are the same:
But when the side faces are different (such as an “irregular” pyramid) we must add up the area of each triangle to find the total lateral area.
The diagram below shows a pyramid whose base is a regular pentagon of area 42 cm2 and whose height is 7 cm
What is the volume of the pyramid?
SOLUTION
Volume = 1/3 x base area x height = 1/3 X 42CM2 X 7CM = 294/3CM3 = 98 CM3.
SURFACE AREA OF A PYRAMID
EXAMPLE
The diagram shows a rectangular-based pyramid with base length 15 cm and width 8cm. The height of the pyramid is 20 cm
What is the volume of the pyramid?
SOLUTION
We will use the formula
Surface Area of a Pyramid = 1/2 × Perimeter × [Side Length] + [Base Area]
First find the Perimeter:
Perimeter = 4 × 10 in = 40 in
Now find the Base Area:
Base Area = 10 in × 10 in = 100 in2
Next find the side length:
If V is the vertex of the pyramid, O is the center point of the base ABCD and M is the midpoint of AB, then triangle VOM is a right triangle with base 5 in and height 12 in
Therefore we can use Pythagoras’ Theorem in triangle VOM:
l 2 = 52 + 122 = 25 + 144 = 169
⇒ l = √169 = 13
Now substitute into the formula
Surface Area of a Pyramid = 1/2 × Perimeter × [Side Length] + [Base Area]
= 1/2 × 40 in × [13 in] + [100 in2]
= 260 in2 + 100 in2= 360in2
EXAMPLE
The diagram shows a pyramid with vertex V and a rectangular base ABCD. M is the midpoint of AB, N is the midpoint of BC and O is the point at the center of the base.
AB = 10 ft
BC = 18 ft
VO = 12 ft
VM = 15 ft
VN = 13 ft
What is the total surface area of the pyramid?
Solution
The surface area consists of one rectangle (ABCD) and four triangles (VAB, VBC, VCD and VDA)
Area of the base ABCD = 18 ft × 10 ft = 180 ft2
Area of triangle VAB = ½ × 10 ft × 15 ft = 75 ft2
Area of triangle VBC = ½ × 18 ft × 13 ft = 117 ft2
Area of triangle VCD = ½ × 10 ft × 15 ft = 75 ft2
Area of triangle VDA = ½ × 18 ft × 13 ft = 117 ft2
Therefore total surface area
= 180 ft2 + 75 ft2 + 117 ft2 + 75 ft2 + 117 ft2
= 564 ft2
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