Volume is defined as how much three-dimensional space a solid, gas, or liquid occupies. The volume of most solids can be determined using various mathematical formulas involving the dimensions of the shapes. Cubic units are used to measure the volume of solids. Arithmetic formulas are commonly used to find the volume of simple shapes such as circular, regular, or straight-edged shapes. Integral calculus is used for more complicated shapes.

- Surface Area & Volume – Grade 9 lesson plan for students to find the surface area and volume of objects.

A cube, or regular hexahedron, is a 3-dimensional solid shape with six square sides. Three of the 6 sides meet at each of the 4 vertices. The cube contains congruent, polyhedral faces and angles. All edges of a cube are equal. To find the volume of a cube, measure the length of any side and multiply this length by itself 3 times.

Formula to find the volume of a cube:

V = [Length of Any Side] (Cubed)

- Volume of a Cube – How to find the volume of a cube and units of volume.
- Cube Lesson – Objective and lesson for finding the volume of a cube.

Rectangular prisms, or cuboids, are figures that contain six rectangular faces and all right angles. Unlike a 2-dimensional rectangle, a rectangular prism is similar to a box, as is a 3-dimensional shape with a length, width, and height. Finding the volume of a rectangular prism involves measuring the length, width, and height of the prism and multiplying the three dimensions together.

Formula to find the volume of a rectangular prism:

V = Length x Width x Height

- Rectangular Solids – Lesson on measuring the volume, surface area and length of 3 dimensional objects.
- Exploring Volume – Lesson plan using boxes and inch cubes as manipulatives to explore volume of a rectangular prism.

A prism has two parallel ends, or bases, that are identical in shape and size. Any polygon can form the bases, which determine the name of the prism. For example, a prism with two triangles for bases is a triangular prism. These two bases are connected to each other by rectangular sides. The number of rectangular sides is determined by the number of sides of the polygon at the bases. A prism has the same cross section along its length. To find the volume of a prism, first find the area of the polygon that forms the bases. Next, measure the perpendicular height from one base to the other. Multiplying these two values will produce the volume.

Formula to find the volume of a prism:

V = Area of Base x Height

- Surface Area of Prisms – How to find the surface area and volume of prisms.
- Volume of Prisms – Find worked solutions to volume problems and videos.
- Right Prisms – Information for finding the surface area and volume of right prisms.
- Prisms – Information on volume of prisms, illustrations and examples with solutions.

A pyramid is a polyhedron that is formed by attaching a polygonal base with an apex, or vertex. The apex and base edges form triangular sides. The number of triangular sides is determined by the number of sides of the polygon at the base. Like prisms, pyramids are classified by their base, which can be any polygon. For example, a pyramid with a square base is a square pyramid. To determine the volume of a pyramid, first find the area of the polygon that forms the base. Then, measure the perpendicular height from the base to the apex. Multiplying these two values together and then by 1/3 will produce the volume.

Formula to find the volume of a pyramid:

V = 1/3 x Area of Base x Height

- Volume of Pyramid Video – Math video demonstration of how to find the volume of a pyramid.
- Pyramid Volume – Step by step guide to finding the volume of pyramid and cone shape objects.
- Pyramid Applet – Set the width of base, length of base, and height of pyramid for a rectangular pyramid and watch the program build the pyramid and automatically calculate the surface area and volume.

A cylinder is a solid object that features two identical circular bases with a curved side connecting them. A cylinder can be thought of as like a prism with circular bases. Cylinders have a circular cross section and the bases are perpendicular to the curved surface. The volume of a cylinder can be determined by first measuring the radius of the the circular base and squaring this value. Next, measure the perpendicular height from one base to the other. Multiplying these two values together and then by pi will produce the volume.

Formula to find the volume of a cylinder:

V = Pi x Radius (Squared) x Height

- Volume of a Cylinder – Illustration and formula for find the volume of a cylinder.
- Cylinder Volume Video – Math demonstration video for finding the cylinder volume.

A cone is a shape that draws upwards smoothly from a circular base to form an apex, or vertex. A cone can be thought of as like a pyramid with a circular base. The vertex can be found parallel to the center of the base. To determine the volume of a cone, first measure the radius of the the circular base and square this value. Next, measure the perpendicular height from the base to the apex. Multiplying these two values together and then by 1/3 and by pi will produce the volume.

Formula to find the volume of a cone:

V = 1/3 x Pi x Radius (Squared) x Height

- Cone Lab – Lab and lesson plan for finding the area and volume of pyramids and cones.
- Encyclopedia – General information for finding the volume of a cone and cylinder.
- Cone Volume Formula – Breakdown of finding the volume of a cone for grades 9-12.

A sphere is perfectly round 3-dimensional shape that is a completely symmetrical from the center. All points on the surface of the sphere are the exact same distance from the center point. The radius of a sphere is the length from any point on the surface to the center. A sphere has no edges or vertexes, and is not a polyhedron. To find the volume of a sphere, first measure the length of the radius and multiply this value by itself 3 times. Multiply the resulting value by 4/3 and by pi to determine the volume.

Formula to find the volume of a sphere:

V = 4/3 x Pi x Radius (Cubed)

- Finding the Volume – What you should know about finding the volume of a sphere.
- Volume Sphere Video – Math demonstration video on how to calculate the volume of a sphere.
- Spheres – How to find the volume of spheres with multiple examples.

An ellipsoid is a solid 3-dimensional figure in which any section through it is a circle or an ellipse. Three radii can be measured in an ellipsoid, one parallel to each axis in a 3-dimensional plane. If the three radii are equivalent, then the object is a sphere. If the radii along the x and y axes are equal, and longer than than the radius along the z-axis, then the shape is an oblate spheroid, which looks like a disk. If the radii along the x and y axes are equal, and shorter than than the radius along the z-axis, then the shape is a prolate spheroid, which looks like a football. If all three radii have different measurements, then the shape is a scalene ellipsoid. To determine the volume of an ellipsoid, first measure each of the three radii and multiply the values together. Multiply the resulting value by 4/3 and by pi to produce the volume.

Formula to find the volume of a ellipsoid:

V = 4/3 x Pi x Radius 1 x Radius 2 x Radius 3

- Area & Volume – How to find the area and volume of ellipsoids with illustrations.
- Disk Method – Using the disk method to find the volume of ellipsoids.
- Volume Integrals – Find the volume of n-dimensional ellipsoids and review the table of units.
- Ellipsoid Algorithm – Information on the application for ellipsoid algorithm.

Calculus can be used to find the volume of more complex and irregular 3-dimensional shapes. There are two main methods to figuring out the volume of shapes in calculus. The first method is called ‘Method of Rings’ and the second is ‘Method of Cylinders’. The main concept behind these methods is that 2-dimensional shapes are formed under a curve. Revolving this shape around an axis forms a 3-dimensional shape. The volume of this shape can be found using integrals.

- Volumes of Solids of Revolution – Tutorial for ‘Metod of Rings’. Following page has a tutorial for “Method of Cylinders’.
- Revolving Figures – 3-D animations of various shapes and formulas for finding volume.
- Volume Lesson Plans – Using calculus to find the volume of solids with known cross sections.
- Maximizing Volume – Multiple examples and guides to finding the volume of shapes using calculus.
- Volume Integration – Introduces the two main methods of finding volume known as the disk method and shell method.
- Pyramid Examples – Multiple examples on finding the volume of revolution.
- Calculus Tutorial – Using calculus to find the volume of a sphere and other shapes.
- Calculating Volumes – Simple examples and description of a variety of geometric solids.

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