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Explain in Detail : Frustum
In geometry, a frustum (from the Latin for “morsel”; plural: frusta or frustums) is the portion of a solid (normally a pyramid or a cone) that lies between one or two parallel planes cutting it. The base faces are polygonal, the side faces are trapezoidal. A right frustum is a right pyramid or a right cone truncated perpendicularly to its axis. Volume of a Frustum – Explanation Formula .
If a frustum has all its edges of the same length (equilateral figure), then it is a uniform prism.
In computer graphics, the viewing frustum is the three-dimensional region which is visible on the screen. It is formed by a clipped pyramid; in particular, frustum culling is a method of hidden surface determination.
In the aerospace industry, a frustum is the fairing between two stages of a multistage rocket (such as the Saturn V), which is shaped like a truncated cone.
Volume of a Frustum – Explanation Formula Property and FAQs.
A frustum is a truncated pyramid.
Similar Property of Triangles to Find Derivation of Volume of Frustum
The similar property of triangles states that if two triangles have the same shape (i.e., they are similar), then the ratio of their corresponding sides is always the same. This property can be used to derive the formula for the volume of a frustum, which is the shape that results when you cut off the top of a cone or pyramid.
To derive the formula for the volume of a frustum, let’s consider a cone with radius r and height h. If we cut off the top of the cone to create a frustum, the height of the frustum will be h’ and the radii of the top and bottom of the frustum will be r1 and r2, respectively.
[asy] pair A,B,C,D,E,F,G,H; A=(0,0); B=(2,0); C=(1,23^.5); D=(1,0); E=(.5,0); F=(1,1); G=(1,.53^.5); H=(.5,.53^.5); draw(A–B–C–cycle); draw(D–E–F–cycle); draw(D–G–H–cycle); draw(D–C,dashed); label(“$r$”,(1,1.23^.5),N); label(“$h$”,(1,.5),W); label(“$r_1$”,(.5,.23^.5),SW); label(“$r_2$”,(.5,.73^.5),NE); label(“$h’$”,(1,.85),E); [/asy]In case of confusion in reading the above text :
[/asy] is a tag used in the Asymptote programming language to indicate the end of an Asymptote block of code. Asymptote is a powerful programming language for creating technical drawings and scientific figures.The label and pair commands are Asymptote commands used to add labels and pairs of coordinates to a drawing. A label is used to add text to a drawing, and a pair is used to specify a pair of coordinates in the drawing.
Here is an example of how to draw a frustum using Asymptote:
[asy] size(200);pair A,B,C,D,E,F; A=(0,0); B=(2,0); C=(1,23^.5); D=(1,0); E=(.5,0); F=(1,1); draw(A–B–C–cycle); draw(D–E–F–cycle); draw(D–C,dashed); label(“$r_1$”,(.5,.23^.5),SW); label(“$r_2$”,(.5,.7*3^.5),NE); label(“$h’$”,(1,.85),E); [/asy]
This code creates a frustum with base radius r1, top radius r2, and height h’. The base of the frustum is a circle with radius r1, and the top is a circle with radius r2. The lateral surface of the frustum is a sloping surface connecting the bases. The dashed line represents the cut that was made to create the frustum.
We can draw a cross-section of the frustum and the original cone as shown above. We can see that the cross-section of the frustum is similar to the cross-section of the original cone. This means that the ratio of the corresponding sides of the two triangles is constant.
We can write the ratio of the corresponding sides of the two triangles as:
(r1/r2) = (h’/h)
We can rearrange this equation to solve for h’:
h’ = (r1/r2) * h
The volume of the frustum is the volume of the original cone minus the volume of the smaller cone at the top of the frustum. The volume of the original cone is:
Volume = (1/3) * π * r^2 * h
The volume of the smaller cone at the top of the frustum is:
Volume = (1/3) * π * r1^2 * h’
Substituting the value of h’ into the equation for the volume of the smaller cone, we get:
Volume = (1/3) * π * r1^2 * (r1/r2) * h
The volume of the frustum is the difference between these two volumes:
Volume = (1/3) * π * r^2 * h – (1/3) * π * r1^2 * (r1/r2) * h
Combining like terms, we get:
Volume = (1/3) * π * h * (r^2 – r1^2 * (r1/r2))
This simplifies to:
Volume = (1/3) * π * h
How to Find Total Surface Area and Curved Surface Area in a Volume Truncated Cone
To find the total surface area and curved surface area of a volume-truncated cone, you can use the following formulas:
Total surface area = π * (r1 + r2) * √((r1 – r2)^2 + h^2) + π * r1^2 + π * r2^2
Curved surface area = π * (r1 + r2) * √((r1 – r2)^2 + h^2)
where:
- r1 is the radius of the base of the cone
- r2 is the radius of the top of the cone (also known as the “smaller radius”)
- h is the height of the cone
The total surface area of a volume-truncated cone is the sum of the curved surface area and the areas of the bases (the top and bottom circles). The curved surface area is the area of the lateral surface of the cone (the sloping part).
Volume of a Frustum – Explanation Formula Property and FAQs.
A frustum is the shape that results when you cut off the top of a cone or pyramid. The volume of a frustum can be calculated using the formula:
Volume = (1/3) * π * h * (r1^2 + r2^2 + r1*r2)
where:
- h is the height of the frustum
- r1 is the radius of the base of the frustum
- r2 is the radius of the top of the frustum
Here are some properties of the volume of a frustum:
- The volume of a frustum is always less than the volume of the cone or pyramid from which it was derived.
- The volume of a frustum is directly proportional to the height of the frustum and the radii of the base and top.
Here are some frequently asked questions about the volume of a frustum:
Q: Can a frustum have a negative volume?
A: No, the volume of a frustum cannot be negative because it is the space enclosed by the frustum.
Q: Can the volume of a frustum be zero?
A: Yes, the volume of a frustum can be zero if the height or one of the radii is zero.
Q: Can the volume of a frustum be greater than the volume of a cone or pyramid?
A: No, the volume of a frustum cannot be greater than the volume of a cone or pyramid because the frustum is derived by cutting off the top of the cone or pyramid.
