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, proof, and a few solved examples. Show
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. Volume of Pyramid FormulaLet 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
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 PyramidsFrom 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 volume of pyramid. 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.
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FAQs on Volume of PyramidThe volume of a pyramid is the space that a pyramid occupies. The volume of a pyramid whose base area is 'B' and whose height is 'h' is (1/3) (Bh) cubic units. 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) cubic units. 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) cubic units. 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 volume is V =(1/3) (Bh) = (1/3) lwh cubic units. 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. Why is There a 1/3 in the Formula for the Volume of Pyramid?A cube of unit length can be divided into three congruent pyramids. So, the volume of pyramid is 1/3 of the volume of a cube. Hence, we have a 1/3 in the volume of pyramid.
A pyramid is a 3-dimensional diagram whose polygonal base is connected to the apex by triangular faces in geometry. The triangular faces of a pyramid are known as lateral faces, and the perpendicular distance from the apex (vertex) to the base of a pyramid is known as the height. Pyramids are named after the shape of their bases. For instance, a rectangular pyramid has a rectangular base, a triangular pyramid has a triangular base, a pentagonal pyramid has a pentagonal base, etc. How to Find the Volume of a Pyramid?In this article, we discuss how to find the volume of pyramids with different types of bases and solve word problems involving a pyramid’s volume. The volume of a pyramid is defined as the number of cubic units occupied by the pyramid. As stated before, the name of a pyramid is derived from the shape of its base. Therefore, the volume of a pyramid also depends on the shape of the base. To find the pyramid’s volume, you only need the dimensions of the base and the height. Volume of a pyramid formulaThe general volume of a pyramid formula is given as: Volume of a pyramid = 1/3 x base area x height. V= 1/3 Ab h Where Ab = area of the polygonal base and h = height of the pyramid. Note: The volume of a pyramid varies slightly depending on the polygonal base. Example 1 Calculate the volume of a rectangular pyramid whose base is 8 cm by 6 cm and the height is 10 cm. Solution For a rectangular pyramid, the base is a rectangle. Area of a rectangle = l x w = 8 x 6 = 48 cm2. And by the volume of a pyramid formula, we have, Volume of a pyramid = 1/3Abh = 1/3 x 48 cm2 x 10 cm = 160 cm3. Example 2 The volume of a pyramid is 80 mm3. If the pyramid’s base is a rectangle that is 8 mm long and 6 mm wide, find the pyramid’s height. Solution Volume of a pyramid = 1/3Abh ⇒ 80 = 1/3 x (8 x 6) x h ⇒ 80 = 15.9h By dividing both sides by 15.9, we get, h = 5 Thus, the height of the pyramid is 5 mm. Volume of a square pyramidTo obtain the formula for the volume of a square pyramid, we substitute the base area (Ab) with the area of a square (Area of a square = a2) Therefore, the volume of a square pyramid is given as: Volume of a square pyramid = 1/3 x a2 x h V = 1/3 a2 h Where a = side length of the base (a square) and h = height of the pyramid. Example 3 A square pyramid has a base length of 13 cm and a height of 20 cm. Find the volume of the pyramid. Solution Given: Length of the base, a = 13 cm height = 20 cm Volume of a square pyramid = 1/3 a2 h By substitution, we have, Volume = 1/3 x 13 x 13 x 20 = 1126.7 cm3 Example 4 The volume of a square pyramid is 625 cubic feet. If the height of the pyramid is 10 feet, what are the dimensions of the pyramid’s base? Solution Given: Volume = 625 cubic feet. height = 10 feet By the volume of a square formula, ⇒ 625 = 1/3 a2 h ⇒ 625 = 1/3 x a2 x 10 ⇒ 625 = 3.3a2 ⇒ a2 =187.5 ⇒ a = = √187.5 a =13.7 feet So, the dimensions of the base will be 13.7 feet by 13.7 feet. Example 5 The base length of a square pyramid is twice the height of the pyramid. Find the dimensions of the pyramid if it has a volume of 48 cubic yards. Solution Let the height of the pyramid = x the length = 3x volume = 48 cubic yards But, the volume of a square pyramid = 1/3 a2 h Substitute. ⇒ 48 = 1/3 (3x)2 (x) ⇒ 48 = 1/3 (9x3) ⇒ 48 = 3x3 Divide both sides by 3 to get, ⇒ x3 =16 ⇒ x = 3√16 x = 2.52 Therefore, the height of the pyramid = x ⇒2.53 yards, and each side of the base is 7.56 yards Volume of a trapezoidal pyramidA trapezoidal pyramid is a pyramid whose base is a trapezium or a trapezoid. Since we know, area of a trapezoid = h1 (b1 + b2)/2 Where h = height of the trapezoid b1 and b2 are the lengths of the two parallel sides of a trapezoid. Given the general formula for the volume of a pyramid, we can derive the formula for the volume of a trapezoidal pyramid as: Volume of a trapezoidal pyramid = 1/6 [h1 (b1 + b2)] H Note: When using this formula, always remember that h is the height of the trapezoidal base and H is the height of the pyramid. Example 6 The base of a pyramid is a trapezoid with parallel sides of length 5 m and 8 m and a height of 6 m. If the pyramid has a height of 15 m, find the volume of the pyramid. Solution Given; h = 6 m, H = 15 m, b1 =5 m and b2 = 8 m Volume of a trapezoidal pyramid = 1/6 [h1 (b1 + b2)] h = 1/6 x 6 x 15 (5 + 8) = 15 x 13 =195 m3. Volume of a triangular pyramidAs we know, the area of a triangle; Area of a triangle = 1/2 b h Volume of a triangular pyramid = 1/3 (1/2 b h) H Where b and h are the base length and height of the triangle. H is the height of the pyramid. Example 7 Find the area of a triangular pyramid whose base area is 144 in2 and the height is 18 in. Solution Given: Base area = 144 in2 H = 18 in. Volume of a triangular pyramid = 1/3 (1/2 b h) H = 1/3 x 144 x 18 = 864 in3 |
What is the relationship between Volume and pyramid when their bases and height are the same?
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