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Solids for which Volume = 1/3Mean Area of Base times Altitude
Frustum Frustum of a pyramid (cone) is a portion of pyramid (cone) included between the base and the section parallel to the base not passing through the vertex.Formula for Volume of a FrustumThe volume of a frustum is equal to one- third the product of the altitude and the sum of the upper base, the lower base, and the mean proportional between the bases. In symbols
Derivation of formula for volume of a frustum of pyramid/cone
Where: h = perpendicular distance between A1 and A2 (h is called the altitude of the frustum)
A1 = area of the lower base
A2 = area of the upper base
Note that A1 and A2 are parallel to each other.
Derivation: By similar solids
Derivation: Substitute y to Equation (1),
Frustum of a Regular PyramidFrustum of a regular pyramid is a portion of right regular
pyramid included between the base and a section parallel to the base.
Properties of a Frustum of Regular Pyramid - The slant height of a fr stum of a regular pyramid is the altitude of the face. - The lateral edges of a frustum of a regular pyramid are equal, and the faces are equal isosceles trapezoids. - The bases of a frustum of a regular pyramid are similar regular polygons. If these polygons become equal, the frustum will become prism.
Elements of a Frustum of Regular Pyramid a = upper base edge b = lower base edge e = lateral edge h = altitude L = slant height A1 = area of lower base A2 = area of upper base
n = number of lower base edges
Formulas for Frustum of a Regular PyramidVolume
Formulas for Frustum of a Regular PyramidLateral Area, AL The lateral area of frustum of regular pyramid is equal to one-half the sum of the perimeters of the
bases multiplied by the slant height.
The relationship between slant height L, lower base edge b, upper base edge a, and lateral edge e, of the frustum of regular pyramid is given by
Frustum of a Right Circular ConeFrustum of a right circular cone is that portion of right circular cone included between the base and a section parallel to the base not passing through the vertex.
Properties of Frustum of Right Circular Cone- The altitude of a frustum of a right circular cone is the
perpendicular distance between the two bases. It is denoted by h. - All elements of a frustum of a right circular cone are equal. It is denoted by L.Formulas for Frustum of Right Circular
Cone Area of lower base, A1
Area of upper base, A2
Lateral Area, AL
1. The frustum of a square pyramid has a lower base 24 cm x 24 cm and an upper base of 15cm x 15 cm. If the altitude is 10 cm. What is the volume and lateral area.
2. The bases of the frustum of a regular triangle pyramid are equilateral triangle whose edges are 3m fro the upper base and 9m for the lower base. The bases are 6m apart. What is the volume of the frustum in cu.m
3. A regular square pyramid has a horizontal base 40cm x 40cm and an altitude of one meter. A plane is passed parallel to the based and 20cm from it. Find the lateral area of the frustum below the plane
4. The volume of a frustum of a right circular cone is 370 cubic cm. its altitude 10cm and the sum of the radii of its bases is 12cm. Find the circumferences of its bases.