Volume Of An Oblique Pyramid
A pyramid is a 3D shape whose base is a polygon and whose side faces (that are triangles) meet at a point which is called the apex (or) vertex. The space occupied by a pyramid in a three-dimensional space is called the volume of a pyramid and it depends on its base area and elevation. Book of a pyramid is measured in terms of cubic units, such every bit $cm^{3}$, $m^{iii}$, $mm^{3}$, $in^{iii}$, $ft^{3}$, etc.
Let's learn how to find the volume of a pyramid and its formula.
Pyramid – A 3D Solid Shape
A pyramid is a three-dimensional shape. A pyramid has a polygonal base and flat triangular faces, which bring together at a mutual point chosen the apex. A pyramid is formed past connecting the bases to an apex. Each edge of the base of operations is connected to the apex and forms the triangular face, called the lateral face up. If a pyramid has an $north$-sided base of operations, then it has $n+i$ faces, $n+1$ vertices, and $2n$ edges.
Types of Pyramids
The pyramids are classified depending on the following factors
- Pyramids based on the blazon of polygon, of the base of operations
- Pyramids based on the alignment of the identical bases
- Pyramids based on the shape of the bases
A pyramid based on the type of the polygon of the base of operations tin can be of the following ii types.
- Regular Pyramid: If the base of the pyramid is in the shape of a regular polygon, the pyramid is a regular pyramid.
- Irregular Pyramid: If the base of the pyramid is in the shape of an irregular polygon, the pyramid is an irregular pyramid.
The location of the apex or the top of a pyramid decides whether a pyramid is a correct pyramid or an oblique pyramid.
- Right Pyramid: A pyramid is named a right pyramid when the location of the apex is exactly over the middle of the base of operations of the pyramid. In other words, when a perpendicular line from the apex intersects the middle of the base, it is a right pyramid.
- Oblique Pyramid: When the location of the apex is non exactly over the middle but slightly away, then that pyramid is called an oblique pyramid. When it does not intersect the eye of the base, it is an oblique pyramid.
There are unlike types of pyramids based on the shape of their base.
- Triangular Pyramid: If the base of a pyramid is in the shape of a triangle, it is said to be a triangular pyramid. A triangular pyramid has $six$ edges, $4$ vertices, and $four$ faces. This kind of pyramid can also be called a tetrahedron.
- Square Pyramid: A square pyramid is formed when the base of the pyramid is in the shape of a square. A square pyramid consists of one foursquare base and three triangular faces. In other words, it has $8$ edges, $5$ vertices, and $5$ faces.
- Rectangular Pyramid: A rectangular pyramid is formed when the base of the pyramid is in the shape of a rectangle. A rectangular pyramid consists of one rectangular base and three triangular faces. In other words, it has $8$ edges, $5$ vertices, and $5$ faces.
- Pentagonal Pyramid: A pentagonal pyramid is 1 that has its base shaped like a pentagon, with the rest of the faces as triangles. This pyramid has $half-dozen$ vertices, $10$ edges, and $six$ faces.
What is the Volume of a Pyramid?
The book of a pyramid is the space enclosed between its triangular faces. It is measured in cubic units such as $cm^{three}$, $grand^{3}$, $in^{3}$, etc.
A pyramid is a iii-dimensional shape where its base (a polygon) is joined to the vertex (noon) with the assist of triangular faces. The perpendicular distance from the apex to the middle of the polygon base is referred to as the superlative of the pyramid.
A pyramid's proper noun is derived from its base. For case, a pyramid with a square base is referred to every bit a square pyramid. The book of the pyramid is calculated past finding ane-third of the production of the base of operations area times its top.
$\text{Volume of a Pyramid} = \frac {1}{3} \times \text{Expanse of Base} \times \text{Top of Pyramid}$.
The general formula for finding the area of a $n$-sided polygon is given by $\text{Surface area} = \frac {ns^{2}}{four \tan \frac {180}{northward}}$
where $n$ is the number of edges (or sides) in a polygon
$s$ is the length of the edge(or side) of a polygon
Therefore, the formula for volume of a $n$-sided pyramid of edge length $southward$ and height $h$ is given by $\text{Volume} = \frac {1}{3} \times \frac {ns^{2}h}{iv \tan \frac {180}{n}} = \frac {ns^{2}h}{12 \tan \frac {180}{due north}}$.
Volume Formula for Different Pyramids
| Pyramid Type | Number of sides in Base of operations $n$ | Volume Formula |
| Triangular Pyramid | $three$ | $\frac {southward^{2}h}{4 \tan 60^{\circ}}$ |
| Square Pyramid | $four$ | $\frac {s^{ii}h}{3\tan 45^{\circ}}$ |
| Pentagonal Pyramid | $five$ | $\frac {5s^{2}h}{12 \tan 36^{\circ}}$ |
| Hexagonal Pyramid | $6$ | $\frac {southward^{2}h}{2 \tan 30^{\circ}}$ |
| Heptagonal Pyramid | $vii$ | $\frac {7s^{two}h}{12 \tan 25.71^{\circ}}$ |
| Octagonal Pyramid | $8$ | $\frac {2s^{ii}h}{three \tan 22.5^{\circ}}$ |
| Nonagonal Pyramid | $9$ | $\frac {3ns^{2}h}{4 \tan 20^{\circ}}$ |
| Decagonal Pyramid | $ten$ | $\frac {5s^{two}h}{6 \tan 18^{\circ}}$ |
Note:
- The above formulae are applicable simply when bases are regular polygons.
- In instance the bases are irregular polygons, so commencement find i-third of the expanse of the base of operations and multiply it by the meridian of the pyramid to detect the book.
Examples
Ex 1: What is the volume of a pyramid with a top of $15 m$ and a square base with a side length of $7 chiliad$?
The side length of the base of operations = $7 k$
Area of base of operations = $\text{B} = 7^{2} = 49 m^{2}$.
The height of a pyramid $\text{H} = 15 m$.
Volume of a pyramid = $\frac {i}{three} \text{BH} = \frac {ane}{three} \times 49 \times 15 = 245 m^{three}$.
Ex two: Find the book of the following pyramid.
It's a square pyramid.
Length of a side of a base = $10 m$.
Area of the base of operations $\text{B} = 10^{2} = 100 m^{2}$.
The elevation of the pyramid is $AM = H$.
To notice the height $H$, first, find the length of the diagonal $CE$ of the square base.
Using Pythagorean theorem $\text{CE} = \sqrt{\text{CD}^{two} + \text{DE}^{2}} = \sqrt{x^{2} + x^{ii}} = 10 \sqrt{two} g$
Therefore, $\text{ME} = five \sqrt{2} yard$.
Now, in correct-angled $\triangle \text{AME}$ right-angled at $\text{M}$, $\text{AM} = \sqrt{\text{AE}^{2} – \text{ME}^{2}} = \sqrt{13^{2} – \left(five \sqrt{2} \right)^{two}} = \sqrt{169 – 50} = \sqrt{119} m$.
Volume of pyramid = $\frac {i}{3}\text{B}{H} = \frac {1}{3} \times 100 \times \sqrt{119} = \frac {100 \sqrt{119}}{iii} yard^{3}$.
Ex 3: An builder wants to make a square pyramid and fill it with $12,000$ cubic feet of sand. If the pyramid'due south base is $xxx$ feet on each side, how tall does he need to make it?
Book of pyramid $\text{V} = 12,000 ft^{three}$.
Side length of square base = $30 ft$.
Area of the base $\text{B} = xxx^{2} = 900 ft^{2}$.
Volume of a pyramid $\text{5} = \frac {1}{3} \text{B} \text{H} => \text{H} = \frac {3 \text{V}}{\text{B}} = \frac {3 \times 12,000}{900} = 40 ft$.
Conclusion
The book of a pyramid is the number of cubic units, occupied past the pyramid completely and is calculated by finding the product of ane-3rd of the base area and the height. Since the formula for finding the area of dissimilar polygons are different, and then the formulas for calculating the book of different types of pyramids are different.
Practice Problems
Q 1. Find the volume of the given pyramid.
Q two. The volume of a $vi$-pes-alpine square pyramid is $eight$ cubic feet. How long are the sides of the base of operations?
Q iii. Detect the volume of the given pyramid.
Q four. Find the book of the given pyramid.
Recommended Reading
- Volume of a Prism(Formula, Derivation & Examples)
- Book of a Sphere – Formula, Derivation & Examples
- Volume of a Cone(Formula, Derivation & Examples)
- Book of a Cylinder(Formulas, Derivation & Examples)
- Volume of Cuboid – Formulas, Derivation & Examples
- Volume of a Cube – Derivation, Formula & Examples
- Surface Expanse of a Pyramid(Definition, Formula & Examples)
- Surface Area of a Prism(Definition, Formulas & Examples)
- Surface Surface area of a Sphere(Definition, Formulas & Examples)
- Area of a Cone(Definition, Formulas & Examples)
- Expanse of a Cylinder(Definition, Formulas & Examples)
- Surface Area of a Cone(Definition, Formulas & Examples)
- Surface Area of A Cube (Definition, Formula & Examples)
- Surface Area of Cuboid (Definition, Formula & Examples)
- Expanse of Rectangle – Definition, Formula & Examples
- Expanse of Foursquare – Definition, Formula & Examples
- Expanse of a Triangle – Formulas, Methods & Examples
- Surface area of a Circumvolve – Formula, Derivation & Examples
- Area of Rhomb – Formulas, Methods & Examples
- Area of A Kite – Formulas, Methods & Examples
- Perimeter of a Polygon(With Formula & Examples)
- Perimeter of Trapezium – Definition, Formula & Examples
- Perimeter of Kite – Definition, Formula & Examples
- Perimeter of Rhomb – Definition, Formula & Examples
- Circumference (Perimeter) of a Circle – Definition, Formula & Examples
- Perimeter of Square – Definition, Formula & Examples
- Perimeter of Rectangle – Definition, Formula & Examples
- Perimeter of Triangle – Definition, Formula & Examples
- What Are 2D Shapes – Names, Definitions & Backdrop
FAQs
What is the volume of a pyramid?
The amount of space occupied by a pyramid is referred to as the book of a pyramid. The volume of the pyramid depends on the base area of the pyramid and the pinnacle of the pyramid. The unit of measurement of volume of the pyramid is expressed in $m^{iii}$, $cm^{3}$, $in^{3}$, $ft^{3}$, etc.
What is the formula for the volume of a pyramid?
The formula for the volume of a pyramid is obtained by finding the product of one-third of the base surface area and height of the pyramid. The volume of a pyramid is given as $\text{V} = \frac {1}{3} \text{B} \times \text{H}$ where, $\text{5}$ is the volume of the pyramid, $\text{B}$ is the area of the base of operations of the pyramid, and $\text{H}$ is the height of the pyramid.
How does the volume of the pyramid modify if the type of pyramid changes?
The book of the pyramid depends on the base surface area of the pyramid. As the type of pyramid changes, the base of the pyramid changes thereby changing the base expanse of the pyramid. This change in the base of operations surface area of the pyramid changes the volume of the pyramid.
You May Besides Like
What are Algebraic Identities(With Definition, Types & Derivations)
Tabular array of Contents What are Algebraic Identities?Types of Algebraic IdentitiesTwo-Variable IdentitiesThree-Variable IdentitiesProofs
What is the Significant of Equation – Definition, Types & Examples
Table of Contents What is the Meaning of Equation?Writing an EquationParts of
Division of Algebraic Expressions(With Methods & Examples)
Table of Contents What is the Division of Algebraic Expressions?How to Divide
Volume Of An Oblique Pyramid,
Source: https://codinghero.ai/volume-of-a-pyramid/
Posted by: halldausay.blogspot.com
0 Response to "Volume Of An Oblique Pyramid"
Post a Comment