Embed Size (px)
Citation preview
Unit 12.1 Measurement
Topic 5: Scales and dimensions - regular polygons
PREPARED BY VSHANKAR 1
LEARNING OUTCOME
After you have completed the presentation, you will be able to:
•calculate the area of regular polygons by using various formulas for the area of a triangle.
PREPARED BY VSHANKAR 2
RECAP
•REVIEW OF PRIOR KNOWLEDGE •Polygon: is a closed shape with many straight sides.•Regular polygon: a polygon in which all sides and angles are equal/congruent.
PREPARED BY VSHANKAR 3
NEW TERMS
•Apothem: A line connecting the centre of a polygon with a midpoint of once of the sides which functions as a perpendicular bisector.
•Radius: a line connecting the centre of a polygon with a vertex.
PREPARED BY VSHANKAR 4
G
•Regular polygons are polygons with all sides and angles congruent or equal.
•To calculate the area of any regular polygon, the easiest way is to divide it into triangles, and use the formula for the area of a triangle. The area of that one triangle is calculated and is multiplied to the number of triangles that are present.
•There are two methods that we will explore to find the area of a regular polygon. The 1st one is fairly straight forward but the second one will need a lot of concentration.
PREPARED BY VSHANKAR 5
STEPS FOR FINDING THE AREA OF A REGULAR POLYGON
STEP 1: Break the polygon into ‘n’ number of equal triangles. Draw the radii and one apothem.
STEP 2: Remove one triangle from the polygon.
STEP 3: Find the area of the triangle using special rules. i.e. Heron’s formula, sine rule for area of a triangle etc.
STEP 4: Multiply the area of 1 triangle by the number of triangles identified in STEP 1.
METHOD 1
PREPARED BY VSHANKAR 6
EXAMPLE 1 : Find the area of a regular hexagon whose side is 3cm and apothem 4cm.
STEP 1: Break the polygon into ‘n’ number of equal triangles. Draw the radii and one apothem.
PREPARED BY VSHANKAR 7
EXAMPLE 1 : Find the area of a regular hexagon whose side is 3cm and apothem 4cm.
STEP 2: Remove one triangle from the polygon.
STEP 3: Find the area of the triangle using special rules. i.e. Heron’s formula, sine rule for area of a triangle etc.
𝐴 =1
2𝑏𝑎𝑠𝑒 × 𝑎𝑝𝑜𝑡ℎ𝑒𝑚 × 6 𝑖𝑑𝑒𝑛𝑡𝑖𝑐𝑎𝑙 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠
STEP 4: Multiply the area of 1 triangle by the number of triangles identified in STEP 1.
𝐴 =1
23 × 4 × 6
∴ 𝐴 = 36𝑐𝑚2
APOTHEM = 4cm
3cm
Apothem can also be referred to as the
height of the triangle.
PREPARED BY VSHANKAR 8
EXAMPLE 2 : Find the area of a regular hexagon whose side is 12.5cm and apothem 14.5cm.
STEP 1: Break the polygon into ‘n’ number of equal triangles. Draw the radii and one apothem.
STEP 2: Remove one triangle from the polygon.
STEP 3: Find the area of the triangle using special rules. i.e. Heron’s formula, sine rule for area of a triangle etc. 𝐴 =
1
2𝑏𝑎𝑠𝑒 × 𝑎𝑝𝑜𝑡ℎ𝑒𝑚 × 6 𝑖𝑑𝑒𝑛𝑡𝑖𝑐𝑎𝑙 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠
APOTHEM = 14.5cm
12.5cm
PREPARED BY VSHANKAR 9
EXAMPLE 2 : Find the area of a regular hexagon whose side is 12.5cm and apothem 14.5cm.
𝐴 =1
2𝑏𝑎𝑠𝑒 × 𝑎𝑝𝑜𝑡ℎ𝑒𝑚 × 6 𝑖𝑑𝑒𝑛𝑡𝑖𝑐𝑎𝑙 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠
𝐴 =1
212.5 × 14.5 × 6
STEP 4: Multiply the area of 1 triangle by the number of triangles identified in STEP
∴ 𝐴 = 543.75𝑐𝑚2
APOTHEM = 14.5cm
12.5cm
PREPARED BY VSHANKAR 10
STEPS FOR FINDING THE AREA OF A REGULAR POLYGON
STEP 1: Calculate the perimeter ‘p’ of the polygon. STEP 2: Find the apothem ‘a’ of the polygon by using the formula
a =𝑠
2𝑡𝑎𝑛180𝑛
Where a= apothem ; s = length of each side ; n = number of sides.
STEP 3: To find the area of the regular polygon use the formula:
𝐴𝑟𝑒𝑎 =𝑎 × 𝑝
2
METHOD 2
PREPARED BY VSHANKAR 11
EXAMPLE 3 : Find the area of a regular hexagon if the sides has a length of 10m as shown in the diagram.
STEP 1: Calculate the perimeter ‘p’ of the polygon.
Number of sides = 6
Perimeter of the shape is 10m x 6 sides = 60m
PREPARED BY VSHANKAR 12
EXAMPLE 3 : Find the area of a regular hexagon if the sides has a length of 10m as shown in the diagram.
STEP 2: Find the apothem ‘a’ of the polygon by using the formula
a =𝑠
2𝑡𝑎𝑛180𝑛
Where a= apothem ; s = length of each side ; n = number of sides. Since a = ? ; s = 10m ; n = 6
a =10
2𝑡𝑎𝑛180
6
= 8.66𝑐𝑚
PREPARED BY VSHANKAR 13
EXAMPLE 3 : Find the area of a regular hexagon if the sides has a length of 10m as shown in the diagram.
STEP 3: To find the area of the regular polygon use the formula:
𝐴𝑟𝑒𝑎 =𝑎 × 𝑝
2
𝐴𝑟𝑒𝑎 =8.66 × 60
2∴ 𝐴𝑟𝑒𝑎 = 259.8𝑚2
PREPARED BY VSHANKAR 14
EXAMPLE 4 : Find the area of a regular nonagon if the sides are of length 5m as shown in the diagram.
STEP 1: Calculate the perimeter ‘p’ of the polygon.
Number of sides = 9
Perimeter of the shape is 5m x 9 sides = 45m
STEP 2: Find the apothem ‘a’ of the polygon by using the formula
a =𝑠
2𝑡𝑎𝑛180𝑛
Where a= apothem ; s = length of each side ; n = number of sides.
Since a = ? ; s = 5m ; n = 9
a =5
2𝑡𝑎𝑛180
9
= 6.87𝑚PREPARED BY VSHANKAR 15
EXAMPLE 4 : Find the area of a regular nonagon if the sides are of length 5m as shown in the diagram.
STEP 3: To find the area of the regular polygon use the formula:
𝐴𝑟𝑒𝑎 =𝑎 × 𝑝
2
𝐴𝑟𝑒𝑎 =45 × 6.87
2
∴ 𝐴𝑟𝑒𝑎 = 154.58𝑚2
PREPARED BY VSHANKAR 16
ACTIVITY Find the area of each regular polygon using the given side length. Round your answer to two decimal places. The final answers are given so please work towards those answers.
A. B.
12m10m
𝐴𝑅𝐸𝐴= 15.59𝑚2
𝐴𝑅𝐸𝐴= 86.02m2
CLICK HERE TO START TIMER
PREPARED BY VSHANKAR 17
You may now work on your assignment for
Module 3. The due date for the assignment is
stated on the school calendar.
PREPARED BY VSHANKAR 18
