GEOMETRY AND SPATIAL
Geometry is the study of different types of shapes, figures and sizes in Maths or in real life. In geometry, we learn
about different angles, transformations and similarities in the figures.
The basics of geometry depend on majorly point, line, angles and plane. All the geometrical shapes are based on
these basic geometrical concepts.
Definition of point
A geometric figure that has no length, width and height, it has only position is called a point.
Types of Point in Geometry
There are three types of points. Namely – collinear point, noncollinear point, concurrent point.
Definition of Collinear Point in Geometry
If more than one point is located on a certain straight line, they are called collinear points.
Definition of Noncollinear Point in Geometry
If three or more points cannot be joined by a straight line, those points are called noncollinear points.
Definition of Concurrent Point in Geometry
If two or more straight lines meet at a point, that point is called concurrent point.
The straight lines in the figure meet at a point, so the point is a concurrent point.
Angle
Once you have learned about points, line, line segments, and plane, the next thing to know is what happens when two
lines meet at a point. That is when an angle is formed.
Angles are one of the basic concepts of geometry. We cannot think of defining any shape, be it triangles,
quadrilaterals, or polygons, without their angles. An angle thus forms a part of each geometric shape.
What is an Angle
Mathematically, an angle is defined as a figure that forms when two rays meet at a common point. It is represented by
the symbol ∠. An angle is usually measured in degrees, denoted with ‘ ° ‘. The term ‘angle’ comes from the Latin
word ‘angulus’, meaning ‘corner’.
A degree is a measure of rotation. A full rotation around a point gives us a full circle which measures 360°, a half
rotation gives us a half-circle, which is 180°, and a quarter rotation gives us a right-angle measuring 90°.
1. Arms: The two straight or curved sides that join to form the angle are called the arms. Here OX and OY are the
arms of the angle ∠XOY.
2. Vertex: The common endpoint where the two rays meet to form an angle. Here the point ‘O’ is the vertex.
Types of Angles
All angles are commonly classified based on their magnitude or degree of rotation, into six main types:
1. Acute Angle: An angle that measures less than 90° is called an acute angle. In other words, it lies between 0° to
90°.
2. Right Angle: An angle that measures exactly 90° is called a right angle. It is formed when two sides of an angle
are perpendicular to each other. The sign of a right angle can also be shown using a quarter circle along with the
standard sign as shown in the figure.
3. Obtuse Angle: An angle that measures greater than 90° and less than 180° is called an obtuse angle.
4. Straight Angle: An angle that measures exactly 180° is called a straight angle. It is similar to a straight line and
hence the name straight angle.
5. Reflex Angle: An angle that measures greater than 180° and less than 360° is called a reflex angle.
6. Complete Angle: An angle whose measure is equal to 360° is called a complete angle. It is formed by one
complete rotation of one of its arms.
Examples in Real Life
We see angles almost everywhere around us in our daily life. Some of the examples are given below:
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Angles formed by the hands of a clock.
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Alphabets A, K, M, N, V, W, X, Y, and Z contain acute angles while E, F, H, L, and T have the right angles.
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Objects of daily use such as cloth hangers, scissors, the edge of tables, blades of a fan, cycle spokes, and wheels all
have angles in them.
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Study accessories such as a ruler, an opened compass, and set squares all contain angles.
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Angles are used in the navigation of airplanes and ships.
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Engineers and architects use angles to form the design of roads, buildings, and sports complexes.
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Artists use their knowledge of angles to make sketches and paintings.
Rectangular Prisms or Cuboids
A rectangular prism is also called a rectangular solid or a cuboid. In a rectangular prism, the length, width and height
may be of different lengths.
The volume of the above rectangular prism would be the product of the length, width and height that is
Volume of rectangular prism = lwh
Total area of top and bottom surfaces is lw + lw = 2lw
Total area of front and back surfaces is lh + lh = 2lh
Total area of the two side surfaces is wh + wh = 2wh
Surface area of rectangular prism = 2lw + 2lh + 2wh = 2(lw + lh + wh)
Prisms
A prism is a solid that has two congruent parallel bases that are polygons. The polygons form the bases of the prism
and the length of the edge joining the two bases is called the height.
The above diagrams show two prisms: one with a triangle-shaped base called a triangular prism and another with a
pentagon-shaped base called a pentagonal prism.
A rectangular solid is a prism with a rectangle-shaped base and can be called a rectangular prism.
The volume of a prism is given by the product of the area of its base and its height.
Volume of prism = area of base × height
The surface area of a prism is equal to 2 times area of base plus perimeter of base times height.
Surface area of prism = 2 × area of base + perimeter of base × height
Cylinders
A cylinder is a solid with two congruent circles joined by a curved surface.
In the above figure, the radius of the circular base is r and the height is h. The volume of the cylinder is the area of the
base × height.
Volume of cylinder = πr2h
The net of a solid cylinder consists of 2 circles and one rectangle. The curved surface opens up to form a rectangle.
Surface area = 2 × area of circle + area of rectangle
Surface area of cylinder = 2πr2 + 2πrh = 2πr (r + h)
Spheres
A sphere is a solid with all its points the same distance from the center.
Cones
A circular cone has a circular base, which is connected by a curved surface to its vertex. A cone is called a right
circular cone, if the line from the vertex of the cone to the center of its base is perpendicular to the base.
The net of a solid cone consists of a small circle and a sector of a larger circle. The arc of the sector has the same
length as the circumference of the smaller circle.
Surface area of cone = Area of sector + area of circle
= πrs + πr2 = πr(r + s)
Pyramids
A pyramid is a solid with a polygon base and connected by triangular faces to its vertex. A pyramid is a regular
pyramid if its base is a regular polygon and the triangular faces are all congruent isosceles triangles.
Nets Of A Solid
An area of study closely related to solid geometry is nets of a solid. Imagine making cuts along some edges of a solid
and opening it up to form a plane figure. The plane figure is called the net of the solid.
The following figures show the two possible nets for the cube.
Classifying Triangles
A triangle is any closed figure made by three line segments intersecting at their endpoints. Every triangle
has three vertices (the points where the segments meet), three sides (the segments), and three interior
angles (formed at each vertex). All of the following shapes are triangles.
The sum of the interior angles in a triangle is 180∘. This is called the Triangle Sum Theorem and is
discussed further in the "Triangle Sum Theorem" concept.
Angles can be classified by their size as acute, obtuse or right. In any triangle, two of the angles will
always be acute. The third angle can be acute, obtuse, or right. We classify each triangle by this angle.
Example 1
Which of the figures below are not triangles?
I. Fill in the blanks:
(i) The triangle with equal sides is …………………….
(ii) A ……………………. triangle has all sides of different length.
(iii) Each angle of equilateral triangle is …………………….
(iv) An ……………………. triangle has two equal sides.
(v) In obtuse angled triangle one angle is ……………………. 90°.
(vi) Each angle of an acute triangle is ……………………. than 90°.
II. Classify the following triangle:
(i) Sides of triangle are 4 cm, 4 cm and 7 cm
(ii) Angles of triangle is 90°, 60° and 30°
(iii) Angles of triangle is 110°, 40° and 30°
(iv) Sides of triangle are 5 cm, 13 cm and 12 cm
(v) Angles of triangle is 60°, 60° and 60°