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Volume of 3D Shapes
In Geometry, 3D shapes are referred to as three-dimensional shapes or solids. These occupy space and have 3 dimensions. Usually, 3D shapes are obtained from the rotation of the 2D shapes. The faces of the solid shapes are 2D shapes only. Some examples of 3D shapes are a cube, cuboid, cone, cylinder, sphere, prism, and so on. The amount of any substance that these shapes can hold is called its volume.
|1.||Introduction to Volume of 3D Shapes|
|2.||Volume of 3D Shapes Formulas|
|3.||Solved Examples on Volume of 3D Shapes|
|4.||Practice Questions on Volume of 3D Shapes|
|5.||Faqs on Volume of 3D Shapes|
Introduction to Volume of 3D Shapes
The best way to visualize volume is to think of it in terms of the space enclosed/occupied by any 3-dimensional object or solid shape.
Do this simple exercise:
- Take a rectangular sheet of paper of length l cm and width h cm.
- Join the opposite sides of the sheet of paper without folding/creasing the sheet.
- See! You have made a 3-D object (cylinder) that encloses space within, from a 2-D sheet (rectangle).
Volume of a 3-d shape is defined as the total space enclosed/occupied by any 3-dimensional object or solid shape. It also can be defined as the number of unit cubes that can be fit into the shape. The SI unit of volume is cubic meters. Other units are fluid ounce, gallon, quart, pint, tsp, fluid dram, cubic yards, barrel, etc.
Volume of 3D Shapes Formulas
The volume of different 3D shapes can be calculated using different formulas for each shape. Some important ones along with their real-life examples are listed in the section below:
The Volume of a Cylinder
To compare the quantity of liquid contained in the cylinder-shaped drink cans of various sizes we will have to calculate the volume of the canned bottle. A cylinder is a tube-like structure with circular faces of the same radius at either end joined by the planar circular surface. Think of it as the area of a circle multiplied by a new dimension, the height. If you consider, r as the radius of the circular base (and top) and h as the height of the cylinder, the Volume of cylinder = πr2h
The Volume of a Cone
Suppose you and your friend are enjoying chilled summer drinks in different conical-shaped glasses. How will you figure out the amount of drink to be filled in each glass?
A cone is a three-dimensional shape that has a flat surface at one end and a curved surface pointing outward towards a point at the other end (which is called apex). The volume of a cone formula is given as, volume of a cone = (1/3) πr2h cubic units where,
- ‘r’ is the base radius of the cone
- ‘l’ is the slant height of a cone
- ‘h’ is the height of the cone
As we can see from the above cone formula, the capacity of a cone is one-third of the capacity of the cylinder. That means if we take 1/3rd of the volume of the cylinder, we get the formula for cone volume. Volume of a cone = (1/3) πr2h
The Volume of a Cube
The volume of a cube can be easily found out by just knowing the length of the edge of the cube. If the length of the cube is s, then the formula to calculate the volume of a cube is: Volume of the cube = s3 where 's is the length of the side of the cube.
The Volume of a Cuboid
What will happen if you stack a bundle of many sheets of paper? How does it look? It makes up a cuboid.
Let the area of a rectangular sheet of paper be A, the height up to which they are stacked be h and the volume of the cuboid be V. Then, the volume of the cuboid is given by multiplying the base area and height. Thus, the Volume of Cuboid = Base Area × Height = Volume of the cuboid = lbh
The Volume of a Sphere
The volume here depends on the diameter or radius of the sphere since if we take the cross-section of the sphere, it is a circle. The surface area of sphere is the area or region of its outer surface. To calculate the sphere volume, whose radius is ‘r’ we have the below formula: Volume of a sphere = 4/3 πr3
The Volume of a Hemisphere
Wondering what will be the total volume of the ice-cream scoop on your waffles? Since the scoop is hemispherical in shape, we will use the volume of a hemisphere formula to calculate this.
The Volume of Prism
A prism is a 3D object with flat faces, where faces are parallel to each other. It has the same cross-section along its length. Mathematically, the volume of a prism is the product of the area of the base and the height.
Therefore, the volume of a Prism = Base Area × Height
The Volume of a Pyramid
Pyramids can be classified into different types depending on their bases. They include Triangular Pyramid, Quadrilateral Pyramid, Pentagonal Pyramid, etc. 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. Thus, the 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")
- A photoreceptor located in the eye’s retina that assists in better vision is shaped like a cone.
- A cone has only one apex or vertex point.
- A prism is a 3D object with flat faces and the same cross-section along its length.
There are 3 different types of units of volumes and are discussed in the table below:
1. Capacity volume:
|1 cubic inch (in3)||16 mL|
|1 cubic foot (ft3)||1728 cu in|
|1 cubic yard (yd3)||27 cu ft|
|1 acre-foot||1613.333 cu yd|
2. Liquid volume:
|1 minim||1 drop|
|1 US fluid dram||60 minim|
|1 teaspoon||80 minim|
|1 tablespoon||3 tsp|
|1 US fluid ounce||2 Tbsp|
|1 US shot||3 Tbsp|
|1 US gill||4 fl oz|
|1 US cup||2 gi|
|1 (liquid) US pint||2 cp|
|1 (liquid) US quart||2 pt|
|1 (liquid) US gallon||4 qt|
|1 (liquid) barrel||31.5 gal|
|1 oil barrel||42 gal|
|1 hogshead||63 gal|
3. Dry volume:
|(dry) pint (pt)||0.6 L|
|1 (dry) quart (qt)||2 pt|
|1 (dry) gallon (gal)||4 qt|
|1 peck (pk)||2 gal|
|1 bushel (bu)||4 pk|
|1 (dry) barrel (bbl)||3.281 bu|
Mariam has a real sweet tooth. She has a jar filled with sweets. Each piece contains sugar syrup which is 30% of the volume of a piece. If each piece is shaped like a sphere having a diameter of 1 in, how much syrup would be needed for 45 sweets?
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Solved Examples on Volume of 3D Shapes
Example 1: A mason wants to build a 3D sphere using cement. He wants to know the amount of cement required to construct the sphere of radius 10 in. Can you help him to find the volume of the sphere? Round your answer to the nearest tenths.
Given, the radius of the sphere (r) = 10 in
We know the formula for the volume of a sphere: V= 4/3 πr3
The volume of the cement sphere (V)=4/3 πr3
Substituting the value of the radius in the above formula, we get: V = 4/3 πr3 = 4/3 π (10)3 = 4188.8 in3
Therefore, the volume of the cement sphere is 4188.8 in3.
Example 2: Jane wants to drink milk from a glass that is in the shape of a cylinder. The height of the glass is 15 in and the radius of the base is 3 in. What is the quantity of milk that Jasmin will drink from the glass? Round your answer to the nearest integer.
Given, height of the glass = 15 in and the radius of the glass base = 3 in.
To find the volume of the glass, we need to use the formula for the volume of a cylinder, which is V = πr2h
The volume of the glass, V = πr2h = π(3)2(15) = 135π = 424.11in3
Therefore, she needs roughly 425 in3 of milk to fill her glass.
Example 3: Little Joe loves playing with building blocks. He has built a structure with 15 cubic blocks. If the edge of each cube is 3 in, what would be the volume of his structure?
Let's calculate the volume of one cube.
The volume of cube = Edge × Edge × Edge = 3 in × 3 in × 3 in =27 in3
There are 15 cubes in his structure. So, the volume of the structure is, Volume of structure =15 × Volume of one cube = 15 × 27 in3 = 405 in3
Therefore, the volume of the structure is 405 in3.
FAQs on Volume of 3D Shapes
How Do You Find the Volume of a 3D Shape?
The volume of a 3-d shape is the amount of space it occupies. It is usually measured in cubic units. One can easily find the volume of any three-dimensional shape using simple volume formulas.
What Is the Volume of a 3D Shape?
The volume of any three-dimensional shape refers to the amount of space occupied by it. The volume is usually represented in terms of cubic units.
How To Find the Volume of An Irregular 3D Shape?
We can find the volume of an irregular 3-d shape, just the way we find the area of an irregular two-dimensional shape, that is by breaking it down into regular shapes.
- Step 1: Split the solid up into smaller parts until you reach the shapes that you can work with easily.
- Step 2: Identify all those shapes.
- Step 3: Find their volume.
- Step 4: Add their volumes.
- Step 5: Write that sum with an appropriate unit.
What Units are Used for the Volume of 3D Shapes?
The volume of any object in a three-dimensional space is measured in cubic units such as cubic centimeters, cubic inch, cubic foot, cubic meter, etc.
How To Measure the Volume of 3D Shapes?
The volume of different 3D shapes can be calculated using different formulas for each shape. In case, any shape is made of cubic unit blocks, one can count the cubes to find the shape's volume.
What Is the General Formula to Find the Volume of 3D Shapes?
The volume of three-dimensional geometrical shapes like cube, cuboid, cylinder, prism, and cone, etc. can be easily calculated by using simple arithmetic formulas., such as, the volume of cube is s3, where s is the edge of the cube, and the volume of cuboid is lwh where l is the length, w is the width and h is the height of the cuboid.
Is the Volume of 3D Shape the Same As Its Capacity?
We know that the volume and capacity are properties of three-dimensional objects. On one hand, volume depicts the measure of space that a three-dimensional object occupies. On the other hand, capacity describes how much amount of something a container can hold.