Geometry
Category : 3rd Class
Geometry
Learning Objectives
Introduction
There are shapes all around us. Shapes can be plane and solid. Lines, rays, line segments are basic concept of geometrical shapes. Geometry is a part of Mathematics which deals with points lines, rays, and line segments. The word 'Geometry' comes from the Greek word 'Geo’ means earth and 'Metria’ means measure. In this chapter, we will learn about the basic geometrical shapes.
Classifying Shapes
We recognize geometrical shapes by their appearances without paying attention to their component parts such as sides and angles.
Plane Shapes and Solid Shapes
Shapes can be either plane or solid. Let us compare the plane shapes and solid shapes.
Plane Shape |
Solid Shape |
(a) they are two dimension (2D) shape |
(a) they are three dimension (3D) shape |
(b) They are only length and breadth but no thickness |
(b) They have length breadth but no thickness |
(c) They have only one face |
(c) They have many face |
(d) They do not occupy volume. |
(d) They occupy volume. |
(e) For example: |
e) For example: |
Solid Geometrical Shapes
Cube, cuboid, cylinder, sphere and cone are examples of solid geometrical shapes.
Cube;
A cube has 6 faces of the same shape and size. It has 12 equal edges and 8 vertices (corners),
Cuboid:
A cuboid has 6 faces in which the opposite faces are of the same shape and size. It has 12 edges where opposite edges are of the same size. It has 8 vertices,
Cylinder:
A cylinder has 3 faces, 2 planes and 1 curved surface. It has 2 round edges and has no vertex or corner.
Sphere:
A sphere has 1 face or 1 curved surface. It has no edge and no vertex.
Cone:
A cone has 2 faces, 1 plane and 1 curved surface and 1 vertex.
Basic Geometric Elements
Some basic geometric elements are given below. Let us study about these geometric elements.
Point
A point has no dimensions. It has only position. It is denoted by a dot (.). For example, point A is marked in the given figure. A point is generally represented by a capital letter.
Line Segment
A line segment is a straight path between 2 points. It has a fixed length. In the given figure, a line segment or BC is CD given which is denoted as \[\overline{BC}\] or \[\overline{CB}\],
Line
A line is a straight path that goes on forever in both directions. It does not have fixed length. In the figure given below, a line PQ or QP is given which is denoted as \[\overleftrightarrow{PQ}\] or \[\overleftrightarrow{QP}\]
Closed Figure and Open Figure
Closed Figure: A closed figure is a figure which can be traced using the same starting and stopping points, and without crossing or retracing any section of the figure. Square, triangle, circle, etc, are example of closed figures,
Open Figure: An open figure is figure which can be traced using the different starting and stopping points. Given figures are examples of open figures which have an opening.
A simple closed figure formed of three or more line segments is called a polygon. In the given figure, triangle, quadrilateral, pentagon and hexagon, all are examples of polygon.
Regular and Irregular Polygon
A regular polygon has all sides equal and all angles equal, otherwise, it is an irregular polygon.
Simple and Complex Polygon
A simple polygon has only one boundary and it does not cross over itself. A complex polygon intersects itself.
Vertex and Angle
A point where two or more straight lines meet is called its vertex. The plural of word 'vertex' is vertices. A vertex is also known as corner.
Inclination between two rays having the common end point is called an angle. In the figure given above, OA and OB are two rays which have a common end point O. The inclination between the rays OA and OB is an angle AOB and it is denoted as \[\angle AOB\] Point 0 is called vertex and ray OA and OB are called arms of the angle AOB. Angles are usually measured in units called degrees (°).
Triangle
A geometrical shape or a simple closed figure bounded by three line segments is called a triangle. The line segments are called sides of the triangle.
Parts of a Triangle
Rectangle
A rectangle is a special type of quadrilateral in which:
(a) opposite sides are equal.
(b) both diagonals are equal, and
(c) each angle is a right angle\[\left( 90{}^\circ \right)\].
In the rectangle ABCD given above, AB = CD and AD = BC
Diagonals AC = BD and each angle is of\[90{}^\circ \].
Square
A square is a special type of quadrilateral in which:
(a) all four sides are equal
(b) both diagonals are equal, and
(c) each angle is a right angle\[\left( 90{}^\circ \right)\].
In the square ABCD given above, AB = BC = CD = AD and diagonal, AC = BD. Each angle is of\[90{}^\circ \]. A square is a special type of rectangle, but not every rectangle is a square.
Circle
A circle is a simple closed shape with curved boundary each of whose points is at the same distance from a fixed point inside it. A circle has no sides or corners, like the square, rectangle and triangle.
We need a fixed point to be able to draw a perfect circle around it. This fixed point is called the centre of the circle. The fixed distance from centre 0, as given in the figure above, to any point on the boundary is called the radius of the circle. The plural of the word 'radius' is radii. There can be any number of radii. All the radii of a circle are equal in length.
Example-1
1. Which one of the following is a simple closed curve?
(a) (d)
(c) (d)
(e) None of these
Answer (c) is correct.
Explanation: A simple closed curve is a figure which can be traced using the same starting and stopping points, and without crossing or retracing any section of the figure. Figure given in option (c) is a simple closed curve.
Rest of the options is incorrect because of the correctness of option (c).
2. How many line segments are required to make the alphabet W?
(a) 6 (b) 4
(c) 3 (d) 5
(e) None of these
Answer (b) is correct.
Explanation: Letter W has 4 line segments as shown below:
Rest of the options is incorrect because of the correctness of option (b).
3. Which one of the following is not a ray in the given figure?
(a)\[\overrightarrow{OA}\] (b) \[\overrightarrow{PQ}\]
(c)\[\overrightarrow{OB}\] (d) \[\overrightarrow{MA}\]
(e) None of these
Answer (b) is correct.
Explanation: A ray has one starting point. PQ, is a line with no starting point.
Rest of the options is incorrect because of the correctness of option (b).
4. A regular hexagon does not have:
(a) six equal sides (b) six equal angles
(c) six vertices (d) six pairs of parallel sides
(e) None of these
Answer (d) is correct.
Explanation: A regular hexagon has six equal sides, six equal angles and has six vertices. A regular hexagon has three pairs of parallel sides.
Rest of the options is incorrect because of the correctness of option (d).
Common Asked Question
1. Which one of the following represents a rectangular solid?
(a) (b)
(c) (d)
(e) None of these
Answer (d) is correct.
Explanation: A cuboid is a rectangular solid. Image given in option (d) shows a rectangular solid or a cuboid.
Rest of the options is incorrect because of the correctness of option (d).
2. Which of the following shapes is a circle?
(a) Shape E (b) Shape C
(c) Shape A (d) Shape H
(e) None of these
Answer (b) is correct.
Explanation: Only the shapes represented by B, C and G are the shapes of a circle.
Rest of the options is incorrect because of the correctness of option (b).
3. Which of the following shapes is not an isosceles triangle?
(a) (b)
(c) (d)
(e) None of these
Answer (c) is correct.
Explanation: An isosceles triangle has two sides of equal length.
Triangle given in option (c) has all unequal sides.
Rest of the options is incorrect because of the correctness of option (c).
4. How many lines are there in the given figure?
(a) 10 (b) 5
(c) 8 (d) 13
(e) None of these
Answer (e) is correct.
Explanation: A line has no starting point and has no end point. There are only 4 lines in the given figure.
Rest of the options is incorrect because of the correctness of option (e).
5. Which one of the following solids has more than one corner?
(a) (b)
(c) (d)
Answer (d) is correct.
Explanation: A cube has exactly 12 vertices or corners.
Rest of the options is incorrect because of the correctness of option (d).
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