Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Polygons
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Polygons
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Subject: Technical Drawings
Class: Senior Secondary 1
Term: 2nd Term
Week: 4
Theme: Geometrical Constructions
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Explain the term polygons. Identify and name types of polygons Construct polygons using different methods.
This section provides a detailed breakdown of essential concepts related to polygons, their properties, and methods of construction. 2.1 Definition of a Polygon A polygon is a closed two-dimensional plane figure made up of three or more straight line segments connected end-to-end to form a closed shape. The line segments are called sides, and the points where the sides meet are called vertices (singular: vertex). 2.2 Types of Polygons Regular Polygon: A polygon with all its sides equal in length and all its interior angles equal in measure. Examples include an equilateral triangle, square, regular pentagon, and regular hexagon.
Irregular Polygon: A polygon whose sides are not all equal in length, or whose interior angles are not all equal in measure, or both.
Convex Polygon: A polygon where all interior angles are less than 180 degrees, and all vertices point outwards. A line segment connecting any two points inside the polygon lies entirely inside the polygon.
Concave Polygon: A polygon where at least one interior angle is greater than 180 degrees (a reflex angle), and at least one vertex points inwards. A line segment connecting two points inside the polygon may pass outside the polygon. 2.3 Properties of Regular Polygons For a regular polygon with 'n' sides: Number of sides = n Number of vertices = n Number of exterior angles = n Number of interior angles = n All sides are equal in length. All interior angles are equal. All exterior angles are equal. Sum of interior angles = (n - 2) 180° Measure of each interior angle = [(n - 2) 180°] / n Sum of exterior angles = 360° Measure of each exterior angle = 360° / n An exterior angle and its adjacent interior angle sum to 180°. 2.4 Naming Polygons Polygons are named based on the number of their sides: | Number of Sides (n) | Name of Polygon | | :------------------ | :--------------- | | 3 | Triangle | | 4 | Quadrilateral | | 5 | Pentagon | | 6 | Hexagon | | 7 | Heptagon | | 8 | Octagon | | 9 | Nonagon | | 10 | Decagon | | 11 | Hendecagon | | 12 | Dodecagon | | n | n-gon | 2.5 Construction Methods for Regular Polygons This section details common construction methods for regular polygons using a compass, ruler, and protractor (where necessary). Emphasize accuracy and neatness.
Method 1: General Method for Constructing any Regular Polygon (Given Circumscribing Circle) This method is applicable for constructing any regular polygon if the circumscribing circle is known.
Steps:
1. Draw a circle with the given radius (O is the center). This will be the circumscribing circle.
2. Draw a vertical diameter AB.
3. Divide the diameter AB into as many equal parts as the number of sides of the polygon (e.g., 5 parts for a pentagon, 6 parts for a hexagon). Label the divisions along AB starting from A as 1, 2, 3, ...
4. With A as center and AB as radius, draw an arc.
5. With B as center and BA as radius, draw another arc to intersect the first arc at point P.
6. Draw a straight line from P through the second division point on the diameter AB (for a pentagon, through point '2'; for a hexagon, through point '2'). This line intersects the circle at point C.
7. The line segment AC is the side length of the required polygon.
8. With A as center and AC as radius, mark off successive arcs around the circle, intersecting it at points D, E, F, etc.
9. Join the points A, C, D, E, F, ... in order to form the regular polygon. Self-correction/Refinement for General Method: The second division point rule (step 6) is a common approximation for general polygons. A more accurate general method for any regular polygon involves calculating the central angle (360°/n) and using a protractor, or a more sophisticated geometric subdivision. For SS1 Technical Drawings, the approximation is often taught first.
However, to maintain and AC as radius, mark off successive arcs around the circle, intersecting it at points D, E, F, etc.
9. Join the points A, C, D, E, F, ... in order to form the regular polygon. Self-correction/Refinement for General Method: The second division point rule (step 6) is a common approximation for general polygons. A more accurate general method for any regular polygon involves calculating the central angle (360°/n) and using a protractor, or a more sophisticated geometric subdivision. For SS1 Technical Drawings, the approximation is often taught first.
However, to maintain accuracy, it's crucial to mention the central angle method.
Method 1a: General Method using Central Angle (More Accurate)
Steps:
1. Draw a circle with the given radius (O is the center).
2. Calculate the central angle: $\theta = 360^\circ / n$, where 'n' is the number of sides.
3. Draw a radius OA.
4. Using a protractor, measure and mark the central angle $\theta$ from OA, drawing a new radius OB. The segment AB is the side length.
5. Repeat step 4, measuring $\theta$ from OB to OC, and so on, until all 'n' segments are marked around the circle.
6. Join the points A, B, C, D, ... in order around the circle to form the regular polygon.
Method 2: Specific Constructions for Common Regular Polygons 2.5.1 Equilateral Triangle (3 sides)
Method A (Given side length):
1. Draw the base line segment AB of the given length.
2. With A as center and AB as radius, draw an arc above AB.
3. With B as center and BA as radius, draw another arc to intersect the first arc at point C.
4. Join A to C and B to C to complete the equilateral triangle AB
C. Method B (Given circumscribing circle):
1. Draw the circumscribing circle with center O.
2. Draw a vertical diameter AD.
3. With D as center and radius DO, draw an arc intersecting the circle at points B and C.
4. Join A to B, B to C, and C to A to form the equilateral triangle ABC. 2.5.2 Square (4 sides)
Method A (Given side length):
1. Draw the base line segment AB of the given length.
2. At A, construct a perpendicular line segment AX (using compass and ruler for 90°).
3. With A as center and AB as radius, mark off point D on AX.
4. With D as center and AB as radius, draw an arc.
5. With B as center and AB as radius, draw another arc to intersect the previous arc at point C.
6. Join B to C and C to D to complete the square ABC
D. Method B (Given diagonal length):
1. Draw the diagonal AC of the given length.
2. Bisect AC at O (find the midpoint using perpendicular bisector method).
3. With O as center and OA as radius, draw a circle.
4. Draw a perpendicular bisector to AC, passing through O. This line intersects the circle at B and D.
5. Join A to B, B to C, C to D, and D to A to form the square ABCD. 2.5.3 Regular Pentagon (5 sides)
Method A (Given circumscribing circle):
1. Draw a circle with center O and the given radius.
2. Draw a vertical diameter AB.
3. Bisect radius OB at point C.
4. With C as center and radius CA, draw an arc to intersect the horizontal diameter extended (or a horizontal line through O) at point D.
5. The length AD is the side length of the regular pentagon.
6. With A as center and radius AD, mark off point E on the circle.
7. Now, with A and E as starting points, and AD as radius, mark off the remaining vertices around the circle.
8. Join the five points to form the pentagon.
Method B (Given side length AB):
1. Draw the given side length AB.
2. Construct a line from A perpendicular to AB. Mark point P on this line such that AP = AB.
3. Bisect AB at M.
4. With M as center and MP as radius, draw an arc intersecting and radius AD, mark off point E on the circle.
7. Now, with A and E as starting points, and AD as radius, mark off the remaining vertices around the circle.
8. Join the five points to form the pentagon.
Method B (Given side length AB):
1. Draw the given side length AB.
2. Construct a line from A perpendicular to AB. Mark point P on this line such that AP = AB.
3. Bisect AB at M.
4. With M as center and MP as radius, draw an arc intersecting AB extended at Q.
5. With A as center and AQ as radius, draw an arc.
6. With B as center and AB as radius, draw an arc intersecting the arc from step 5 at C.
7. With A as center and AB as radius, draw an arc.
8. With B as center and AQ as radius, draw an arc intersecting the arc from step 7 at D.
9. With C as center and AB as radius, draw an arc intersecting the arc from step 5 at E.
1
0. Join the points A, B, C, E, D to form the pentagon. (This method can be complex for students; Method A is generally preferred for its relative simplicity). 2.5.4 Regular Hexagon (6 sides) Method A (Given circumscribing circle - most common and easiest):
1. Draw a circle with center O and the given radius.
2. Draw a horizontal diameter AB.
3. With A as center and radius AO (which is also the side length of the hexagon), draw an arc intersecting the circle at points C and D.
4. With B as center and radius BO, draw an arc intersecting the circle at points E and F.
5. The six points A, C, F, B, E, D are the vertices of the hexagon.
6. Join the six points in order to form the regular hexagon.
Method B (Given side length):
1. Draw the given side length AB.
2. With A as center and AB as radius, draw an arc.
3. With B as center and BA as radius, draw another arc to intersect the first arc at point O (this is the center of the hexagon).
4. With O as center and OA (or OB) as radius, draw a circle.
5. Using AB as the radius, mark off points around the circle starting from A and B, until all six vertices are found.
6. Join the points to form the regular hexagon. 2.5.5 Regular Octagon (8 sides)
Method A (Given circumscribing circle):
1. Draw a circle with center O and the given radius.
2. Draw two perpendicular diameters (one horizontal AB, one vertical CD).
3. Bisect the four right angles formed by the diameters (e.g., bisect angle AOD, AOB, BOC, COD). This can be done by drawing arcs from A, D, B, C with the same radius to find intersection points, then drawing lines from O through these intersections.
4. These eight lines from the center will intersect the circle at eight equally spaced points.
5. Join these eight points in order to form the regular octagon. * Method B (Given side length):
1. Draw the given side length AB.
2. At A and B, draw lines perpendicular to AB.
3. From A and B, draw lines at 45° to AB, intersecting the perpendiculars at C and D respectively.
4. With C as center and CB as radius, draw an arc.
5. Continue this process of marking side lengths and 45° angles to complete the octagon. (This method is more complex and typically less preferred for beginners compared to the circumscribing circle method). 3.1 Introduction (10 minutes)
Teacher Activity: Begins by displaying various real-life objects or images that contain polygonal shapes (e.g., a traffic sign, floor tiles, a football, a section of an Ankara fabric, building structures). Asks students to identify the shapes they observe in these objects. Guides a brief discussion on the importance of shapes in design and construction. Introduces the term "polygon" and states the lesson objectives clearly.
Student Activity: Observe and identify shapes in displayed objects. Participate in the discussion, sharing their observations. Listen attentively to the introduction and learning objectives. 3.2 Explanation of Key Concepts (25 minutes)
Teacher Activity: Defines polygons, regular vs. irregular, convex vs. concave polygons using clear language and visual aids (diagrams drawn on the board). Explains the properties of regular polygons (sides, angles, sum of angles) with examples. Presents a table of polygon names (triangle to dodecagon) and their corresponding number of sides. Engages students with questions to check understanding (e.g., "Is a kite a regular polygon? Why?"). Emphasizes the importance of understanding properties for construction.
Student Activity: Listen and take notes on definitions and properties. Copy the table of polygon names. Answer questions posed by the teacher, justifying their responses. Ask clarifying questions. 3.3 Demonstration of Construction Methods (30 minutes)
Teacher Activity: Demonstrates step-by-step the construction of at least three different regular polygons (e.g., equilateral triangle, square, regular hexagon) using different methods on the chalkboard, whiteboard, or with an overhead projector (OHP)/document camera if available. For example, demonstrate: Equilateral triangle (given side length). Square (given side length). Regular Hexagon (given circumscribing circle - Method A). Regular Pentagon (given circumscribing circle - Method A). Emphasizes the correct use of drawing instruments (ruler, compass, protractor). Provides clear, verbal instructions for each step while performing the construction. Encourages students to follow along roughly on scrap paper or watch closely.
Student Activity: Observe carefully the teacher's demonstration. Pay attention to the sequence of steps and the use of instruments. Ask questions about any unclear steps. Some students may attempt to follow along on scrap paper. 3.4 Guided Practice (40 minutes)
Teacher Activity: Distributes drawing paper and ensures students have their complete drawing sets. Assigns specific construction tasks to students (e.g., "Construct a regular pentagon with a side of 40mm using the general method," or "Construct a regular hexagon in a circle of 50mm radius"). Circulates among students, providing individual guidance, correcting errors, and answering questions. Monitors students' use of instruments and accuracy. Selects a few students to demonstrate their constructions or specific steps to the class if time permits.
Student Activity: Attempt the assigned construction tasks independently or in pairs. Apply the methods demonstrated by the teacher. Seek assistance from the teacher or peers when facing difficulties. Practice neatness and accuracy in their drawings. 3.5 Conclusion and Review (15 minutes)
Teacher Activity: Recap the key concepts learned: definition of polygons, types, properties, and various construction methods. Asks students to reiterate the steps for constructing a particular polygon. Assigns homework/independent practice questions. Collects guided practice drawings for assessment.
Student Activity: Participate in the recap session. Answer review questions. Note down homework assignments.
Understanding polygons is not just an academic exercise; it has significant practical relevance in various aspects of Nigerian life and industry.
Architecture and Construction: Many buildings and structures in Nigeria incorporate polygonal shapes. For example, roof trusses often form triangular shapes for strength. Modern architectural designs for public buildings, residential homes, and even traditional compounds sometimes feature octagonal or hexagonal elements in floor plans, window designs, or decorative patterns. The construction of a multi-sided kiosk or a specific tile pattern for a patio in a Nigerian home directly applies polygonal construction skills.
Arts and Crafts / Textile Design: Polygons are fundamental to various Nigerian artistic expressions. The intricate geometric patterns found in traditional textiles like Ankara, Adire, and Kente often consist of tessellations or arrangements of polygons (squares, triangles, hexagons). Similarly, patterns on carved calabashes, pottery, and wall murals frequently use polygons. Students studying these crafts can apply their knowledge of polygon construction to create or reproduce complex designs accurately.
Engineering and Design: In fields like civil engineering, polygonal shapes are used in designing road signs (e.g., octagonal stop signs, triangular yield signs), structural components, bridges, and even in urban planning for land subdivision. Mechanical engineering often uses polygonal components in gear designs or fastening elements. Understanding how to construct and calculate properties of polygons is critical for these applications, ensuring stability, aesthetics, and functionality. For instance, designing a park bench with a hexagonal base or a triangular bracing for a metal gate involves applying polygon principles.