# Frustum of Pyramid

Sides of the lower base: $${a_1},{a_2}, \ldots ,{a_n}$$
Sides of the top base: $${b_1},{b_2}, \ldots ,{b_n}$$
Scale factor: $$k$$
Height: $$h$$
Slant height: $$m$$
Perimeters of the bases: $${P_1}$$, $${P_2}$$
Areas of the bases: $${S_1}$$, $${S_2}$$
Total surface area: $$S$$
Lateral surface area: $${S_L}$$
Volume: $$V$$
1. A frustum of a pyramid is a polyhedron that lies between the base of the pyramid and a plane through it parallel to the base.
2. The polygons lying in the bases of a frustum of a pyramid are similar to each other:
$${\large\frac{{{b_1}}}{{{a_1}}}\normalsize} = {\large\frac{{{b_2}}}{{{a_2}}}\normalsize} = {\large\frac{{{b_3}}}{{{a_3}}}\normalsize} = \ldots$$ $$= {\large\frac{{{b_n}}}{{{a_n}}}\normalsize} = {\large\frac{b}{a}\normalsize} = k,$$
where $$k$$ is a scale factor.
3. Ratio of the base areas
$${\large\frac{{{S_2}}}{{{S_1}}}\normalsize} = {k^2}$$
4. Lateral surface area of a frustum of a regular pyramid
$${S_L} = m{\large\frac{{{P_1} + {P_2}}}{2}\normalsize},$$
where $$m$$ is the slant height, $${P_1}$$, $${P_2}$$ are the perimeters of the top and bottom bases.
5. Total surface area
$$S = {S_L} + {S_1} + {S_2}$$
6. Volume of a frustum of a pyramid
$$V =$$ $${\large\frac{h}{3}\normalsize} \left( {{S_1} + \sqrt {{S_1}{S_2}} + {S_2}} \right)$$