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Formula for calculating the volume of a truncated pyramid, the lateral area and total area of a truncated pyramid
Let's learn what a truncated pyramid is, how to calculate the volume of a truncated pyramid, the lateral area and total area of a truncated pyramid in the article below.
Table of Contents
- Definition of frustum
- Formula for calculating the area of a truncated pyramid
- Formula for calculating the volume of a truncated pyramid
Definition of frustum
A truncated pyramid is a part of a polyhedron, located between the base and the cross-section cut by a plane parallel to the base of the pyramid.
- The figure consisting of polygons A1A2...An,B1B2...BnA1A2...An,B1B2...Bn and trapezoids A1A2B2B1,A2A3B3B2,...,AnA1B1BnA1A2B2B1,A2A3B3B2,...,AnA1B1Bn is called a truncated pyramid, denoted as A1A2...An.B1B2...Bn.A1A2...An.B1B2...Bn.
- Simply put, a truncated pyramid is formed from the pyramid S.A1A2...AnS.A1A2...An after cutting off the pyramid S.B1B2...Bn.S.B1B2...Bn.
+ The polygons A1A2...An,B1B2...BnA1A2...An,B1B2...Bn are called the two bases,
+ Trapezoids A1A2B2B1,A2A3B3B2,...,AnA1B1BnA1A2B2B1,A2A3B3B2,...,AnA1B1Bn are called lateral faces.
+ The line segments A1B1,A2B2,...,AnBnA1B1,A2B2,...,AnBn are called the side edges, the edges of the base surface are called the base edges.
+ The distance between the two bases is called the height of the truncated pyramid.
Properties of truncated pyramid:
- The two bases are two polygons (triangles, quadrilaterals, pentagons, etc.) with corresponding parallel sides and equal ratios of corresponding sides.
- The side faces are all trapezoids.
- The lines containing the lateral edges will converge at a point (the top of the pyramid).
- A truncated pyramid is a pyramid whose bases are regular polygons (with equal sides). Therefore, the lateral faces of a truncated pyramid are all equal isosceles trapezoids.
Formula for calculating the area of a truncated pyramid
Surface area of truncated pyramid
The lateral area of a truncated pyramid is the area of the surrounding faces, the part surrounding the truncated pyramid, excluding the area of the two bases.
How to calculate the lateral area of a truncated pyramid: Calculate the area of each side face (trapezoids) of the truncated pyramid according to the formula for calculating the area of a normal trapezoid , then calculate the total area.
Formula for calculating the area of a regular truncated pyramid:
In there:
- Sxq: surrounding area
- n: number of lateral faces of the truncated pyramid (equal to the number of edges of the base polygon)
- a, b: side lengths of the upper and lower bases respectively
- h: height of the side quadrilaterals.
Example 1:
Calculate the lateral area of a regular quadrilateral truncated pyramid with base edges of 10cm and 15cm and side height of 12cm.
Prize:
The side face of a regular quadrilateral truncated pyramid is an isosceles trapezoid, so the area of one side face is:
A regular quadrilateral truncated pyramid has 4 equal sides, so its lateral area is:
150 x 4 = 600 (cm2)
Example 2: Calculate the lateral area of a regular quadrilateral truncated pyramid with base edges of 6cm and 8cm and side height of 5cm. Calculate the lateral area of a regular quadrilateral truncated pyramid with base edges of 6cm and 8cm.
Prize:
The side face of a regular quadrilateral truncated pyramid is an isosceles trapezoid, so the area of one side face is equal to
A regular quadrilateral truncated pyramid has four equal sides so its lateral area is equal to
35 x 4 = 140 (cm2)
Total surface area of a truncated pyramid
The total area of a truncated pyramid is equal to the sum of the lateral area and the area of the two bases.
Formula: Stp = Sxq + Big bottom + Small bottom
In there:
- Stp: Total area
- Sxq: Surrounding area
- Large base: Large base area
- Small base: Small base area
For example:
Calculate the total area of a regular truncated pyramid according to the dimensions given in the figure.
Prize:
We have:
AD = 12 ⇒ OJ = 6
Draw II1 ⊥ OJ we have: I1J = 3
Applying the Pythagorean theorem to right triangle II1J we have:
IJ2 = II12 + I1J2 = 92 + 32 = 90
Hence:
The area of one side of a trapezoid is:
The surrounding area is equal to:
The area of the upper base is: S = 6 x 6 = 36 (units)
The area of the bottom base is: S = 12 x 12 = 144 (units)
The total area of a truncated pyramid is equal to:
Formula for calculating the volume of a truncated pyramid
Recipe:
In there:
- V: volume of truncated pyramid
- S, S' are the areas of the large and small bases of the pyramid, respectively.
- h: height of the pyramid, that is the distance between the two large and small bases
Is a truncated pyramid a square (regular quadrilateral):
In there:
- V: Volume
- h: Height of the pyramid
- a, b are the lengths of the edges of the large base and the small base respectively.
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