How to Identify and Apply Plane Surfaces in Geometry
Have you ever noticed which type of surfaces we deal with every day? We deal with both types of surfaces in our everyday life, i.e. plane surfaces as well as curved surfaces. There is nothing possible without the surface in one's life. In the below article, students will acquire knowledge about what surfaces in maths are, types of surfaces, and single curved surfaces. This is one of the most basic topics of geometry, thus, should be on the tips of every student to excel in further mathematics. Now, let's begin with our learning.
What is Surface in Maths?
A surface in maths is defined as the continuous space available for the collection of points that may or may not be two-dimensional. Surface examples are cones, ellipsoids, flat surfaces of 3-dimensional solids, etc.
Types of Surfaces
There are several types of surfaces whose list is given below:
Minimal Surface
Ruled Surface
Quadrics
Non-orientable Surface
Pseudospherical Surface
Algebraic Surface
Miscellaneous Surface
What is a Plane Surface?
A two-dimensional flat surface extended indefinitely is called a plane surface. In simple terms, a plane surface is a flat surface that can be extended in all possible directions. Plane surface examples are flat surfaces of cubes and cuboids, floors with zero thickness, etc.
Showing Plane surfaces that are parallel
Properties of Planes
If there are two distinct planes, they are either parallel or intersect in a line.
A line is either parallel to a plane, intersects at a single point, or exists in the plane.
If there are two distinct lines, which are perpendicular to the same plane, then they must be parallel to each other.
If two different planes are perpendicular to the same line, they must be parallel.
What is a Curved Surface?
A curved surface is defined as a surface which is not flat. These surfaces are two-dimensional as well as three-dimensional. For example, water bottles, ice cream cones etc. Curved Surfaces can further be divided into two categories, namely:
Single Curved Surface
Double Curved Surface
What is Single Curved Surface
A single curved surface represents the curved surface in which only one is curved out of two given curves. This makes the surface developable, the one whose Gaussian Curvature is always zero.
Single Curved Surface
Solved Surface Examples
Q 1. Which three-dimensional geometrical figure has only one curved surface with no plane surface?
Ans: The geometrical figure possessing only one curved surface is a sphere. It has no flat surface due to its round or circular shape.
Showing a three-dimensional figure, sphere
Q 2. Which shape represents each flat surface of a cylinder?
Ans: Cuboid is the shape that represents the flat surfaces of a cylinder. Also, the two circular surfaces represent the flat surface of a cylinder. It has three flat surfaces with two circular ends depicted by the circles and one rectangular surface.
Showing the open sketch of a three-dimensional figure, cylinder
Practice Problems
Q 1. State three plane surface examples that belong to one's environment and real life.
Ans: Top of a table, Wall and Floor.
Q 2. A plane represents the _______ surface with no thickness.
Ans: flat.
Q 3. Which surface is called the developable surface?
Ans: Single curved surface is sometimes called a developable surface.
Summary
Summing up here with the concept of surfaces. This writing covers numerous topics, including surfaces, types of surfaces, plane surface examples, single curved surfaces and solved surface examples. Practice problems are assigned based on these topics to check students’ understanding. The language is kept simple and interesting to grab children's kind attention. In addition, images make the article more engaging for the students. Wishing you enjoyed reading the article. Feel comfortable to ask your problems by writing in the comments.
FAQs on Plane Surface: Key Concepts, Properties & Examples
1. What is a plane surface in Maths?
In mathematics, a plane surface is defined as a perfectly flat, two-dimensional (2D) surface that has length and width but no thickness. A key characteristic is that it extends infinitely in all directions. A straight line joining any two points on it lies completely within the surface.
2. What are some real-life examples of a plane surface?
While a perfect mathematical plane cannot exist physically, many everyday objects are excellent approximations of a plane surface. These include:
- The top of a table
- A blackboard or whiteboard
- The floor of a room
- A calm, still surface of water in a pool
- A flat sheet of paper
3. What is the main difference between a plane surface and a curved surface?
The main difference lies in their geometry. A plane surface is completely flat and has zero curvature. In contrast, a curved surface is not flat and has at least one bend or curve. For example, the face of a cube is a plane surface, while the surface of a sphere or a cylinder is a curved surface.
4. Why is a plane surface considered two-dimensional (2D)?
A plane surface is considered two-dimensional because any point on it can be uniquely located using just two coordinates (like an x-axis and y-axis). It possesses length and width but lacks the third dimension of depth or height. All shapes drawn on it, like squares and circles, are also 2D.
5. What are the key properties of a plane?
According to geometry, a plane has several distinct properties:
- A straight line connecting any two points on a plane lies entirely within that plane.
- Three points that are not on the same line (non-collinear) will always define a unique plane.
- Two separate planes in space can either be parallel (meaning they never intersect) or they will intersect along a single straight line.
6. How are plane surfaces important for understanding basic geometric shapes?
Plane surfaces are fundamental to geometry because they are the 'canvas' on which plane figures (2D shapes) are defined. Shapes such as triangles, squares, rectangles, and circles can only exist on a plane where all their points lie on the same flat surface. The study of their properties, like area and perimeter, is a core part of plane geometry.
7. What happens when two distinct planes meet or do not meet?
There are only two possibilities for two distinct planes in three-dimensional space:
- They intersect: If two planes meet, their intersection is always a single straight line.
- They are parallel: If two planes never meet, no matter how far they are extended, they are called parallel planes.
8. Can a shape have both plane and curved surfaces?
Yes, many common three-dimensional shapes have a combination of both plane and curved surfaces. A perfect example is a cylinder, which has two flat, circular plane surfaces (the top and bottom) and one continuous curved surface around its side. Another example is a cone, which has one flat, circular plane surface at its base and one curved surface leading to its apex.
