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Illustration used to show finding the volume of a pentagonal prism.

Volume Of Pentagonal Prism

Illustration used to show finding the volume of a pentagonal prism.

An illustration of a cross section of a concrete pier for a railway bridge with dimensions labeled. Illustration could be used to calculate volume and area of figures.

Cross Section of Concrete Pier for Railway bridge

An illustration of a cross section of a concrete pier for a railway bridge with dimensions labeled.…

A pile of books. One is open and must be about Abbeys and Castles.

Pile of Books

A pile of books. One is open and must be about Abbeys and Castles.

Comparison of the size of a pint, quart, peck, and bushel.

Pint, Quart, Peck, Bushel

Comparison of the size of a pint, quart, peck, and bushel.

A volume of a truncated right triangular prism is equal to the product of its base by one third the sum of its lateral edges.

Truncated Right Triangular Prism for Volume

A volume of a truncated right triangular prism is equal to the product of its base by one third the…

A volume of a truncated right triangular prism is equal to the product of its base by one third the sum of its lateral edges.

Truncated Right Triangular Prism for Volume

A volume of a truncated right triangular prism is equal to the product of its base by one third the…

A truncated triangular prism is equivalent to the sum of three pyramids, whose common base is the base of the prism and whose vertices are the three vertices of the inclined section.

Truncated Triangular Prism for Volume

A truncated triangular prism is equivalent to the sum of three pyramids, whose common base is the base…

A truncated triangular prism is equivalent to the sum of three pyramids, whose common base is the base of the prism and whose vertices are the three vertices of the inclined section.

Truncated Triangular Prism for Volume

A truncated triangular prism is equivalent to the sum of three pyramids, whose common base is the base…

Illustration of a prism used to demonstrate that the volume of any prism is equal to the product of its base by its altitude.

Prism Showing Volume

Illustration of a prism used to demonstrate that the volume of any prism is equal to the product of…

Illustration of a plane passing through a prismatoid.

Plane Passing Through Prismatoid

Illustration of a plane passing through a prismatoid.

Illustration to show how volume of a prismatoid is found. "The volume of a prismatoid is equal to the product of one sixth of its altitude into the sum of its bases and four times its mid-section." V=1/6H(B+b+4M)

Prismatoid Illustration for Volume

Illustration to show how volume of a prismatoid is found. "The volume of a prismatoid is equal to the…

Diagram used to prove the theorem: "The volume of a prismatoid is equal to the product of one-sixth the altitude into the sum of the two bases and four times the mid-section."

Volume of Prismatoid

Diagram used to prove the theorem: "The volume of a prismatoid is equal to the product of one-sixth…

Illustration of 2 right octagonal prisms with congruent bases, but different heights. The height of the smaller prism is one half that of the larger.

2 Octagonal Prisms

Illustration of 2 right octagonal prisms with congruent bases, but different heights. The height of…

Illustration of 2 right rectangular prisms. The bases are congruent, but the height of the smaller prism is one half that of the larger. Hidden edges are shown.

2 Rectangular Prisms

Illustration of 2 right rectangular prisms. The bases are congruent, but the height of the smaller prism…

Illustration of 2 right rectangular prisms. The bases are congruent, but the height of the smaller prism is one half that of the larger.

2 Rectangular Prisms

Illustration of 2 right rectangular prisms. The bases are congruent, but the height of the smaller prism…

Illustration of 2 Similar right octagonal prisms. The height and length of the edges of the smaller prism are one half that of the larger.

2 Similar Octagonal Prisms

Illustration of 2 Similar right octagonal prisms. The height and length of the edges of the smaller…

A cluster of 4 right hexagonal prisms with congruent bases, but varying heights.

4 Hexagonal Prisms

A cluster of 4 right hexagonal prisms with congruent bases, but varying heights.

Illustration of 2 rectangular prism used to show volume.

Rectangular Prisms Showing Volume

Illustration of 2 rectangular prism used to show volume.

Illustration of 2 similar right decagonal prisms. Both have regular decagons for bases and rectangular faces. The height of the prism and length of the side of the decagon on the smaller decagonal prism are one half that of the larger.

Similar Decagonal Prisms

Illustration of 2 similar right decagonal prisms. Both have regular decagons for bases and rectangular…

Illustration of 2 similar right heptagonal/septagonal prisms. Both have regular heptagons/septagons for bases and rectangular faces. The height of the prism and length of the side of the heptagon on the smaller heptagonal prism are one half that of the larger.

Similar Heptagonal/Septagonal Prisms

Illustration of 2 similar right heptagonal/septagonal prisms. Both have regular heptagons/septagons…

Illustration of 2 similar right hexagonal prisms. The height of the prism and length of the side of the hexagon on the smaller hexagonal prism are one half that of the larger.

Similar Hexagonal Prisms

Illustration of 2 similar right hexagonal prisms. The height of the prism and length of the side of…

Illustration of 2 similar right hexagonal prisms. The height of the prism and length of the side of the hexagon on the smaller hexagonal prism are one half that of the larger.

Similar Hexagonal Prisms

Illustration of 2 similar right hexagonal prisms. The height of the prism and length of the side of…

Illustration used to compare the volumes of a pyramid and a prism by emptying sand from the pyramid into the prism.

Comparative Volumes Of A Pyramid And Prism

Illustration used to compare the volumes of a pyramid and a prism by emptying sand from the pyramid…

The volume of the frustum of any pyramid is equal to the sum of the volumes of three pyramids whose common altitude is the altitude of the frustum, and whose bases are the lower base, the upper base, and the mean proportional between the bases of the frustum.

Frustum Pyramid for Volume

The volume of the frustum of any pyramid is equal to the sum of the volumes of three pyramids whose…

An illustration of a pyramid with the top cut off by a plane parallel to the base. The remaining part is called a frustum of a pyramid or a cone.

Pyramid Frustum

An illustration of a pyramid with the top cut off by a plane parallel to the base. The remaining part…

An illustration of a pyramid with the top cut off by a plane parallel to the base. The remaining part is called a frustum. This frustum has hexagonal bases, one with 10 inch side and the other with a 6 inch side. Height is 18 inches.

Pyramid Frustum With Hexagonal Bases and 6 inch and 10 inch Sides

An illustration of a pyramid with the top cut off by a plane parallel to the base. The remaining part…

An illustration of a pyramid with the top cut off by a plane parallel to the base. The remaining part is called a frustum. This frustum has right triangular bases, one with 20 inch side and the other with a 30 inch side. Height is 27 inches.

Pyramid Frustum With Triangular Bases and Height of 27 inches

An illustration of a pyramid with the top cut off by a plane parallel to the base. The remaining part…

An illustration of a pyramid with the top cut off by a plane parallel to the base. The remaining part is called a frustum. This frustum has triangular bases with 14 inch sides. The other sides are 16 and 22 inches. The altitude is 24 inches.

Pyramid Frustum With Triangular Bases and Height of 27 inches

An illustration of a pyramid with the top cut off by a plane parallel to the base. The remaining part…

An illustration of a pyramid with a hexagonal base.

Pyramid With Hexagonal Base

An illustration of a pyramid with a hexagonal base.

Illustration of pentagonal pyramid used to show that the volume is equal to one third of the product of its base by its altitude.

Pentagonal Pyramid For Volume

Illustration of pentagonal pyramid used to show that the volume is equal to one third of the product…

An illustration of a regular pyramid with a square base and labels on lateral edge, slant height, altitude, and all corners.

Regular Pyramid With Square Base

An illustration of a regular pyramid with a square base and labels on lateral edge, slant height, altitude,…

An illustration of a pyramid with a triangular base.

Pyramid With Triangular Base

An illustration of a pyramid with a triangular base.

Illustration of triangular pyramid used to show that the volume is the limit of the sum of the volumes of a series of inscribed, or circumscribed prisms of equal altitude, if the number of prisms is indefinitely increased.

Triangular Pyramid For Volume

Illustration of triangular pyramid used to show that the volume is the limit of the sum of the volumes…

Illustration of triangular pyramid used to show that the volume is equal to one third of the product of its base by its altitude.

Triangular Pyramid For Volume

Illustration of triangular pyramid used to show that the volume is equal to one third of the product…

Illustration of 2 right pentagonal pyramids with hidden edges shown. The pentagonal bases are congruent, but the height of the smaller pyramid is one half that of the larger.

2 Right Pentagonal Pyramids

Illustration of 2 right pentagonal pyramids with hidden edges shown. The pentagonal bases are congruent,…

Illustration of 2 right rectangular pyramids with hidden edges shown. The rectangular bases are congruent, but the height of the smaller pyramid is one half that of the larger.

2 Right Rectangular Pyramids

Illustration of 2 right rectangular pyramids with hidden edges shown. The rectangular bases are congruent,…

Illustration of 2 similar right pentagonal pyramids with hidden edges shown. The height of the pyramid and length of the side of the pentagon (base) on the smaller pentagonal pyramid are one half that of the larger.

Similar Pentagonal Pyramids

Illustration of 2 similar right pentagonal pyramids with hidden edges shown. The height of the pyramid…

Illustration showing that the volume of 2 triangular pyramids, having the same trihedral angle of the one equal to a trihedral angle of the other, are to each other as the products of the three edges of these trihedral angles.

Triangular Pyramids for Volume

Illustration showing that the volume of 2 triangular pyramids, having the same trihedral angle of the…

An illustration of a rectangular prism with dimensions of 4 ft. by 1 ft. by 1 ft..

4 By 1 By 1 Rectangular Prism

An illustration of a rectangular prism with dimensions of 4 ft. by 1 ft. by 1 ft..

An illustration of a rectangular prism with dimensions of 4 ft. by 2 ft. by 1 ft..

4 By 2 By 1 Rectangular Prism

An illustration of a rectangular prism with dimensions of 4 ft. by 2 ft. by 1 ft..

An illustration of a rectangular prism with dimensions of 4 ft. by 2 ft. by 2 ft..

4 By 2 By 2 Rectangular Prism

An illustration of a rectangular prism with dimensions of 4 ft. by 2 ft. by 2 ft..

Illustration of a 5 in. by 3 in. by 4 in. rectangular solid with each cube in the solid representing one cubic inch. This can be used when explaining volume.

Volume Of A Rectangular Solid

Illustration of a 5 in. by 3 in. by 4 in. rectangular solid with each cube in the solid representing…

An illustration of a rectangular solid/prism.

Rectangular Solid

An illustration of a rectangular solid/prism.

An 8-inch sphere cut by parallel planes, one 2 inches from center and the other 6 inches from center. This illustration can be used to find the area of the zone and the volume of the segments.

Sphere With 8-inch Diameter Cut by Planes

An 8-inch sphere cut by parallel planes, one 2 inches from center and the other 6 inches from center.…

An illustration of a sphere cut into polygons as bases with their vertices at the center of sphere.

Sphere Cut Into Pyramids.

An illustration of a sphere cut into polygons as bases with their vertices at the center of sphere.

An illustration of a sphere inside of a cylinder with radius/diameter labeled. Illustration for showing how volume is found.

Sphere Inside of Cylinder.

An illustration of a sphere inside of a cylinder with radius/diameter labeled. Illustration for showing…

An illustration of a sphere showing diameter, arcs, and circles. A sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within, called the center.

Sphere with arcs and circles labeled.

An illustration of a sphere showing diameter, arcs, and circles. A sphere is a solid bounded by a curved…

An illustration of a sphere showing diameter, arcs, and circles. A sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within, called the center. This shows the zones of a sphere.

Sphere With Arcs, Circles, and Zones Labeled.

An illustration of a sphere showing diameter, arcs, and circles. A sphere is a solid bounded by a curved…

Illustration showing that "the solids formed by the dissected part of the sphere are pyramids."

Sphere Formed By Pyramids

Illustration showing that "the solids formed by the dissected part of the sphere are pyramids."

An illustration of a sphere showing a radius of 1 foot. Illustration could be used for calculating volume.

Sphere With a Radius of 1 foot

An illustration of a sphere showing a radius of 1 foot. Illustration could be used for calculating volume.

An illustration of a zone of a sphere. A zone occurs when a sphere is cut by parallel planes that are equal distances apart. This illustration is the segment of one base.

Zones or Segments of Spheres

An illustration of a zone of a sphere. A zone occurs when a sphere is cut by parallel planes that are…

An illustration of a zone of a sphere. A zone occurs when a sphere is cut by parallel planes that are equal distances apart. This illustration is the segment of two bases.

Zones or Segments of Spheres

An illustration of a zone of a sphere. A zone occurs when a sphere is cut by parallel planes that are…

An illustration of a flanged spherical segment with a diameter of 10 inches. Illustration could be used to find the area.

Flanged Spherical Segment

An illustration of a flanged spherical segment with a diameter of 10 inches. Illustration could be used…

Diagram used to prove the theorem: "The volume of a spherical segment is equal to the sum of two cylinders and a sphere, the altitudes of the cylinders being one half the altitude of the segment, and their bases the upper and lower bases of the segment, respectively, and the diameter of the sphere being the altitude of the segment."

Volume of Spherical Segment

Diagram used to prove the theorem: "The volume of a spherical segment is equal to the sum of two cylinders…

Illustration used to show finding the volume of a square prism.

Volume Of Square Prism

Illustration used to show finding the volume of a square prism.

Cylindrical water tank with a height of 10 ft., thickness of 3 inches, and diameter of 3 feet. The diagram can be used to find volume.

Cylindrical Water Tank

Cylindrical water tank with a height of 10 ft., thickness of 3 inches, and diameter of 3 feet. The diagram…

Illustration used to show finding the volume of a triangular prism.

Volume Of Triangular Prism

Illustration used to show finding the volume of a triangular prism.

Diagram used to prove the theorem: "The volume of a triangular prism is equal to the product of its base and its altitude."

Volume of Triangular Prism

Diagram used to prove the theorem: "The volume of a triangular prism is equal to the product of its…

Diagram used to prove the theorem: "The volume of a triangular pyramid is equal to one third of a triangular prism of the same base and altitude."

Volume of Triangular Pyramid

Diagram used to prove the theorem: "The volume of a triangular pyramid is equal to one third of a triangular…

Cross section of Pullman car water tank. The diagram can be used to find volume.

Cross Section of Water Tank

Cross section of Pullman car water tank. The diagram can be used to find volume.