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Illustration of a conical bifrustum created by three parallel planes of circles with the middle plane largest. The top and bottom circles are congruent. It is constructed from two congruent frustums across a plane of symmetry.

Conical Bifrustum

Illustration of a conical bifrustum created by three parallel planes of circles with the middle plane…

Illustration of a decagonal bifrustum created by three parallel planes of decagons with the middle plane largest. The top and bottom decagons are congruent. It can be constructed from two congruent frustums across a plane of symmetry, or as a bipyramid with truncated vertices.

Decagonal Bifrustum

Illustration of a decagonal bifrustum created by three parallel planes of decagons with the middle plane…

Illustration of a hexagonal bifrustum created by three parallel planes of hexagons with the middle plane largest. The top and bottom hexagons are congruent. It can be constructed from two congruent frustums across a plane of symmetry, or as a bipyramid with truncated vertices.

Hexagonal Bifrustum

Illustration of a hexagonal bifrustum created by three parallel planes of hexagons with the middle plane…

Illustration of a octagonal bifrustum created by three parallel planes of octagons with the middle plane largest. The top and bottom octagons are congruent. It can be constructed from two congruent frustums across a plane of symmetry, or as a bipyramid with truncated vertices.

Octagonal Bifrustum

Illustration of a octagonal bifrustum created by three parallel planes of octagons with the middle plane…

Illustration of a square bifrustum created by three parallel planes of squares with the middle plane largest. The top and bottom squares are congruent. It can be the combination of two congruent frustums across a plane of symmetry, or as a bipyramid with truncated vertices.

Square Bifrustum

Illustration of a square bifrustum created by three parallel planes of squares with the middle plane…

An illustration of a composite figure made up of a quadrilateral frustum and half of a cylinder. Frustum edges range between 6 feet and 11 feet 5 inches and cylinder has a height of 12 feet.

Composite Figure of Quadrilateral Frustum With Half of a Cylinder Attached

An illustration of a composite figure made up of a quadrilateral frustum and half of a cylinder. Frustum…

Illustration of a right circular cone that has been cut by a plane parallel to the base. The lower part is known as the frustum.

Cone Cut By Plane

Illustration of a right circular cone that has been cut by a plane parallel to the base. The lower part…

Illustration of a right circular cone that has been cut by a plane parallel to the base. The top section has been removed and placed in front of the lower section. The lower part is known as the frustum.

Cone Cut By Plane

Illustration of a right circular cone that has been cut by a plane parallel to the base. The top section…

An illustration of a circular cone 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.

Circular Cone Frustum

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

Illustration of a frustum of a cone.

Frustum of a Cone

Illustration of a frustum of a cone.

Frustum of a cone.

Frustum of Cone

Frustum of a cone.

Frustum of a cone.

Frustum of Cone

Frustum of a cone.

Illustration of a frustum of a right circular cone with hidden edges shown. When a cone is cut by a plane parallel to the base, the lower part is known as the frustum.

Frustum of A Cone

Illustration of a frustum of a right circular cone with hidden edges shown. When a cone is cut by a…

Illustration of a frustum of a right circular cone. When a cone is cut by a plane parallel to the base, the lower part is known as the frustum.

Frustum of A Cone

Illustration of a frustum of a right circular cone. When a cone is cut by a plane parallel to the base,…

"The lateral area of a frustum of a cone of revolution is equal to half the sum of the circumferences of its bases multiplied by the slant height."

Frustum of Cone to Find Lateral Area

"The lateral area of a frustum of a cone of revolution is equal to half the sum of the circumferences…

"The volume of a frustum of a circular cone is equivalent to the sum of the volumes of three cones 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 of Cone to Find Volume

"The volume of a frustum of a circular cone is equivalent to the sum of the volumes of three cones whose…

Illustration of a pyramid and with a regular polygon inscribed in and circumscribed about a cone. "If a pyramid whose base is a regular polygon is inscribed in or circumscribed about a circular cone, and if the number of sides of the base of the pyramid is indefinitely increased, the volume of the cone is the limit of the volume of the pyramid, and the lateral area of the cone is the limit of the lateral area of the pyramid."

Cone With Regular Polygon Inscribed and Circumscribed About

Illustration of a pyramid and with a regular polygon inscribed in and circumscribed about a cone. "If…

An illustration of a cube with the faces shaded.

Cube With Faces Shaded

An illustration of a cube with the faces shaded.

An illustration of a cube divided into 6 equal pyramids to illustrate how volume can be found.

Cube for Illustrating Volume

An illustration of a cube divided into 6 equal pyramids to illustrate how volume can be found.

Illustration of a decagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Decagonal Pyramid

Illustration of a decagonal pyramid that has been cut by a plane parallel to the base. The top section…

Illustration of a hollow decagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section has been turned upside down. It is known as the frustum of the pyramid.

Frustum Of A Decagonal Pyramid

Illustration of a hollow decagonal pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a decagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid.

Frustum Of A Decagonal Pyramid

Illustration of a decagonal pyramid that has been cut by a plane parallel to the base. The top section…

Illustration of a hollow septagonal/heptagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section has been turned upside down. It is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Decagonal Pyramid

Illustration of a hollow septagonal/heptagonal pyramid that has been cut by a plane parallel to the…

If a pyramid be cut by a plan parallel to the base, so as to form two parts, the lower part is called the frustum of the pyramid.

Frustum

If a pyramid be cut by a plan parallel to the base, so as to form two parts, the lower part is called…

If a cone be cut in a similar manner, the lower part is called the frustum of the cone.

Frustum

If a cone be cut in a similar manner, the lower part is called the frustum of the cone.

"The part of any solid between two planes, which may be either parallel or inclined to each other: as, the frustum of a cone ... In the figure the dotted line, c, indicates the part of the cone cut off to form the frustum, f." -Whitney, 1911

Frustum of a Cone

"The part of any solid between two planes, which may be either parallel or inclined to each other: as,…

Diagram used to prove the theorem: "The volume of the frustum of a pyramid (cone) is equal to the sum of three pyramids (cones) whose common altitude is the altitude of the frustum and whose bases are respectively the upper base, the lower base, and a mean proportional between them."

Volume of Frustum of a Pyramid

Diagram used to prove the theorem: "The volume of the frustum of a pyramid (cone) is equal to the sum…

The frustum of a pyramid is the portion of a pyramid included between the base and a section parallel to the base.

Pyramid Frustum

The frustum of a pyramid is the portion of a pyramid included between the base and a section parallel…

The frustum of a triangular pyramid is equivalent to the sum 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 two bases of the frustum.

Triangular Pyramid Frustum

The frustum of a triangular pyramid is equivalent to the sum of three pyramids whose common altitude…

Illustration of a right hexagonal pyramid that has been cut by a plane parallel to the base. The lower part is known as the frustum.

Hexagonal Pyramid Cut By Plane

Illustration of a right hexagonal pyramid that has been cut by a plane parallel to the base. The lower…

Illustration of a frustum of a hexagonal pyramid (including cross-section).

Frustum of Hexagonal Pyramid

Illustration of a frustum of a hexagonal pyramid (including cross-section).

Illustration of a right hexagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Hexagonal Pyramid

Illustration of a right hexagonal pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a right hexagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid.

Frustum Of A Hexagonal Pyramid

Illustration of a right hexagonal pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a right hexagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid.

Frustum Of A Hexagonal Pyramid

Illustration of a right hexagonal pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a skewed hexagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Skewed Hexagonal Pyramid

Illustration of a skewed hexagonal pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a hollow nonagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section has been turned upside down. It is known as the frustum of the pyramid.

Frustum Of A Nonagonal Pyramid

Illustration of a hollow nonagonal pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a right octagonal pyramid, with hidden edges shown, that has been cut by a plane parallel to the base. The lower part is known as the frustum.

Octagonal Pyramid Cut By Plane

Illustration of a right octagonal pyramid, with hidden edges shown, that has been cut by a plane parallel…

Illustration of a right octagonal pyramid that has been cut by a plane parallel to the base. The lower part is known as the frustum.

Octagonal Pyramid Cut By Plane

Illustration of a right octagonal pyramid that has been cut by a plane parallel to the base. The lower…

Illustration of a hollow octagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section has been turned upside down. It is known as the frustum of the pyramid.

Frustum Of A Octagonal Pyramid

Illustration of a hollow octagonal pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a frustum of a pentagonal pyramid (including cross-section).

Frustum of Pentagonal Pyramid

Illustration of a frustum of a pentagonal pyramid (including cross-section).

Illustration of a right pentagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Pentagonal Pyramid

Illustration of a right pentagonal pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a hollow pentagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section has been turned upside down. It is known as the frustum of the pyramid.

Frustum Of A Pentagonal Pyramid

Illustration of a hollow pentagonal pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a right pentagonal pyramid, with a height much larger than the edge lengths of the base, that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Pentagonal Pyramid

Illustration of a right pentagonal pyramid, with a height much larger than the edge lengths of the base,…

Illustration of a right pentagonal pyramid, with a height smaller than the edge lengths of the base, that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Pentagonal Pyramid

Illustration of a right pentagonal pyramid, with a height smaller than the edge lengths of the base,…

Illustration of a hollow pentagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section has been turned upside down. It is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Pentagonal Pyramid

Illustration of a hollow pentagonal pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a skewed pentagonal pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Skewed Pentagonal Pyramid

Illustration of a skewed pentagonal pyramid that has been cut by a plane parallel to the base. The top…

The frustum of a pyramid is the portion of a pyramid included between the base and a section parallel to the base.

Frustum Pyramid

The frustum of a pyramid is the portion of a pyramid included between the base and a section parallel…

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…

The frustum of a pentagonal pyramid is the portion of a pyramid included between the base and a section parallel to the base.

Frustum of Pentagonal Pyramid

The frustum of a pentagonal pyramid is the portion of a pyramid included between the base and a section…

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…

Illustration of pentagonal and triangular pyramids cut by a plane parallel to the bases.

Pyramids With Pentagonal and Triangular Bases

Illustration of pentagonal and triangular pyramids cut by a plane parallel to the bases.

Illustration of a right rectangular pyramid that has been cut by a plane parallel to the base. The lower part is known as the frustum.

Rectangular Pyramid Cut By Plane

Illustration of a right rectangular pyramid that has been cut by a plane parallel to the base. The lower…

Illustration of a right rectangular pyramid that has been cut by a plane parallel to the base. The top section has been removed and placed next to the lower section. The lower part is known as the frustum.

Rectangular Pyramid Cut By Plane

Illustration of a right rectangular pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a right rectangular pyramid that has been cut by a plane parallel to the base. The top section has been removed and placed next to the lower section. The lower part is known as the frustum.

Rectangular Pyramid Cut By Plane

Illustration of a right rectangular pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a right rectangular pyramid that has been cut by a plane parallel to the base. The lower part is known as the frustum.

Rectangular Pyramid Cut By Plane

Illustration of a right rectangular pyramid that has been cut by a plane parallel to the base. The lower…

Illustration of a right rectangular pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid. The hidden edges are shown in this illustration.

Frustum Of A Rectangular Pyramid

Illustration of a right rectangular pyramid that has been cut by a plane parallel to the base. The top…

Illustration of a right rectangular pyramid that has been cut by a plane parallel to the base. The top section has been removed and the remaining section is known as the frustum of the pyramid.

Frustum Of A Rectangular Pyramid

Illustration of a right rectangular pyramid that has been cut by a plane parallel to the base. The top…