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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 Frustum**The 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.

**Derivati
on:
**

**Equation
1**

**Derivation: By
similar solids **

**Derivation:
**Substitute y to Equation
(1),

**Frustum of a
Regular
Pyramid**Frustum 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 Pyramid**Volume

**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
Cone**Frustum 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.

**Exam
ples**