Volume of Pyramid
The volume of pyramid is space occupied by it (or) it is defined as the number of unit cubes that can be fit into it. A pyramid is a polyhedron as its faces are made up of polygons. There are different types of pyramids such as a triangular pyramid, square pyramid, rectangular pyramid, pentagonal pyramid, etc that are named after their base. i.e., if the base of a pyramid is a square, it is called a square pyramid. All the side faces of a pyramid are triangles where one side of each triangle merges with a side of the base. Let us explore more about the volume of pyramid along with its formula, its proof, and a few solved examples.
|1.||What Is Volume of Pyramid?|
|2.||Formula of Volume of Pyramid|
|3.||Volume Formulas of Different Types of Pyramids|
|4.||FAQs on Volume of Pyramid|
What Is Volume of Pyramid?
The volume of a pyramid refers to the space enclosed between its faces. The volume of any pyramid is always one-third of the volume of a prism where the bases of the prism and pyramid are congruent and the heights of the pyramid and prism are also the same. i.e., three identical pyramids of any type can be arranged to form a prism of the same type such that the heights of the pyramid and the prism are the same and their bases are congruent, i.e., three rectangular pyramids can be arranged to form a rectangular prism. We can understand this by the following activity. Take a rectangular pyramid full of sand and take an empty rectangular prism whose base and height are as same as that of the pyramid. Pour the sand from the pyramid into the prism, we can see that the prism is exactly one-third full.
In the same way, we can see that in a cube, there are three square pyramids arranged invisibly.
Formula of Volume of Pyramid
Let us consider a pyramid and prism each of which has a base area 'B' and height 'h'. We know that the volume of a prism is obtained by multiplying its base by its height. i.e., the volume of the prism is Bh. In the earlier section, we have seen that the volume of pyramid is one-third of the volume of the corresponding prism (i.e., their bases and heights are congruent). Thus,
Volume of pyramid = (1/3) (Bh), where
- B = Area of the base of the pyramid
- h = Height of the pyramid (which is also called "altitude")
Note: The triangle formed by the slant height (s), the altitude (h), and half the side length of the base (x/2) is a right-angled triangle and hence we can apply the Pythagoras theorem for this. Thus, (x/2)2 + h2 = s2. We can use this while solving the problems of finding the volume of the pyramid given its slant height.
Volume Formulas of Different Types of Pyramids
From the earlier section, we have learned that the volume of a pyramid is (1/3) × (area of the base) × (height of the pyramid). Thus, to calculate the volume of a pyramid, we can use the areas of polygons formulas (as we know that the base of a pyramid is a polygon) to calculate the area of the base, and then by simply applying the above formula, we can calculate the area of the base. Here, you can see the volume formulas of different types of pyramids such as the triangular pyramid, square pyramid, rectangular pyramid, pentagonal pyramid, and hexagonal pyramid and how they are derived.
Solved Examples on Volume of Pyramid
Example 1: Cheops pyramid in Egypt has a base measuring about 755 ft. × 755 ft. and its height is around 480 ft. Calculate its volume.
Cheops Pyramid is a square pyramid. Its base area (area of square) is,
B = 755 × 755 = 570,025 square feet.
The height of the pyramid is, h = 480 ft.
Using the volume of pyramid formula,
Volume of pyramid, V = (1/3) (Bh)
V = (1/3) × 570025 × 480
V = 91,204,000 cubic feet.
Answer: The volume of the Cheops pyramid is 91,204,000 cubic feet.
Example 2: A pyramid has a regular hexagon of side length 6 cm and height 9 cm. Find its volume.
The side length of the base (regular hexagon) is, a = 6.
The base area (area of regular hexagon) is,
B = (3√3/2) × a2
B = (3√3/2) × 62 ≈ 93.53 cm2.
The height of the pyramid is, h = 9 cm.
The volume of the hexagonal pyramid is,
V = (1/3) (Bh)
V = (1/3) × 93.53 × 9
V = 280.59 cm3
Answer: The volume of the pyramid is 280.59 cm3.
Example 3: Tim built a rectangular tent (that is of the shape of a rectangular pyramid) for his night camp. The base of the tent is a rectangle of side 6 units × 10 units and the height is 3 units. What is the volume of the tent?
The base area (area of rectangle) of the tent is, B = 6 × 10 = 60 square units.
The height of the tent is, h = 3 units.
The volume of the tent using the volume of pyramid formula is,
V = (1/3) (Bh)
V = (1/3) × 60 × 3
V = 60 cubic units.
Answer: The volume of the tent = 60 cubic units.
FAQs on Volume of Pyramid
What Is Meany By Volume of Pyramid?
The volume of a pyramid is the space that the pyramid occupies. The volume of a pyramid whose base area is 'B' and whose height is 'h' is (1/3) (Bh).
What Is the Volume of Pyramid With a Square Base?
If 'B' is the base area and 'h' is the height of a pyramid, then its volume is V = (1/3) (Bh). Consider a square pyramid whose base is a square of length 'x'. Then the base area is B = x2 and hence the volume of the pyramid with a square base is (1/3)(x2h).
What Is the Volume of Pyramid With a Triangular Base?
To find the volume of a pyramid with a triangular base, first, we need to find its base area 'B' which can be found by applying a suitable area of triangle formula. If 'h' is the height of the pyramid, its volume is found using the formula V =(1/3) (Bh).
What Is the Volume of Pyramid With a Rectangular Base?
A pyramid whose base is a rectangle is a rectangular pyramid. Its base area 'B' is found by applying the area of the rectangle formula. i.e., if 'l' and 'w' are the dimensions of the base (rectangle), then its area is B = lw. If 'h' is the height of the pyramid, then its area is V =(1/3) (Bh) = (1/3) lwh.
What Is the Formula To Find the Volume of Pyramid?
The volume of a pyramid is found using the formula V = (1/3) Bh, where 'B' is the base area and 'h' is the height of the pyramid. As we know the base of a pyramid is any polygon, we can apply the area of polygons formulas to find 'B'.
How To Find Volume of Pyramid With Slant Height?
If 'x' is the base length, 's' is the slant height, and 'h' is the height of a regular pyramid, then they satisfy the equation (the Pythagoras theorem) (x/2)2 + h2 = s2. If we are given with 'x' and 's', then we can find 'h' first using this equation and then apply the formula V = (1/3) Bh to find the volume of the pyramid where 'B' is the base area of the pyramid.