This is the Mathematics Grade 7 Quarter 3 Module 8 self-learning module, “Solving Problems Involving Side and Angle of a Polygon.” It was developed by DepEd SOCCSKSARGEN (Region XII) for home-based learning, and it is meant for Grade 7 learners, their parents, and their teachers. By the end of the module, learners should be able to find perimeters, angle sums, and angle measures of polygons and use them to solve word problems.
On this page 7 sections
What learners will learn
The module has one lesson, “Sides and angles of a polygons.” After going through it, learners are expected to:
- Solve problems involving sides and angles of polygons.
Key ideas: sides and perimeter
The module starts by recalling that a polygon is a closed figure made up of three or more line segments joined at their endpoints. When all sides and all angles are congruent, it is a regular polygon (or regular n-gon).
The perimeter is the sum of the lengths of all the sides. The module’s examples:
- Pentagon with sides of 5 cm: 5 + 5 + 5 + 5 + 5 = 25 cm
- Equilateral triangle with sides of 7 cm: 7 + 7 + 7 = 21 cm
- Square with sides of 26 mm: 26 + 26 + 26 + 26 = 104 mm
The word problems also use perimeter. For example, learners work out how much red piping Belen needs to sew around a table cloth that measures 1.75 m by 2.85 m, and the largest square, rectangular, or pentagon frame Eddie can make from 1 meter of wooden siding.
Key ideas: angles of a polygon
In the discovery activity, learners draw convex polygons with four to eight sides. They divide each one into triangles by drawing diagonals from one vertex. Since each triangle’s angles add up to 180°, the sum of the polygon’s interior angles is the number of triangles times 180°.
| Polygon | Sides | Triangles formed | Sum of interior angles |
|---|---|---|---|
| Quadrilateral | 4 | 2 | 360° |
| Pentagon | 5 | 3 | 540° |
| Hexagon | 6 | 4 | 720° |
| Heptagon | 7 | 5 | 900° |
| Octagon | 8 | 6 | 1,080° |
This pattern leads to the formulas the module asks learners to remember:
- Sum of the interior angles of an n-gon: S = (n – 2) × 180°, where n is the number of sides.
- Sum of the exterior angles of any polygon: 360°.
- Each interior angle of a regular polygon: (n – 2) × 180° ÷ n.
- Each exterior angle of a regular polygon: 360° ÷ n.
Worked examples from the module
- Sum of the angles of a pentadecagon (15 sides): S = (15 – 2) × 180° = 13 × 180° = 2,340°.
- Number of sides of a regular polygon whose exterior angle is 45°: 45° = 360° ÷ n, so n = 360° ÷ 45° = 8. The polygon has 8 sides.
You can also work backward from an angle sum. If the interior angles add up to 1,260°, then 1,260° ÷ 180° = 7 triangles, and 7 + 2 = 9 sides.
Parts of the module
- What I Know: a 10-item multiple-choice pretest.
- What’s In: a review of polygons and perimeter.
- What’s New: the triangle-dividing activity and table.
- What is It: the angle formulas and worked examples.
- What’s More: two perimeter word problems (Activity 1 and Activity 2).
- What I Have Learned: a matching-type exercise on angle sums and numbers of sides.
- What I Can Do: Activity 3, finding the side of a regular hexagon inscribed in a circle with a 10 cm radius.
- Assessment: a 10-item multiple-choice post-test.
- Additional Activities: finding the sums of the interior and exterior angles of a hexagon and an octagon.
- Answer Key: answers for the tests and activities.
How to use this module
- Learners: Start with the pretest to check what you already know. Then work through the parts in order and finish with the post-test. Write all your answers on a separate sheet of paper, and do not put marks on the module. Check your work with the answer key honestly, after you have answered.
- Parents and guardians: Help your child follow the parts in order. Encourage them to draw the polygons when solving a problem, as the module asks.
- Teachers: The module includes notes for facilitators. The pretest and the post-test give a quick before-and-after check of the competency. The real-life tasks in What’s More work well for checking whether learners can apply perimeter.
If anything in the module is unclear, the module tells learners to ask their teacher or facilitator.
Frequently asked questions
What grade and quarter is this module for?
It is for Grade 7 Mathematics, Quarter 3, Module 8. It is a first-edition (2020) self-learning module from DepEd Region XII.
What is the formula for the sum of the interior angles of a polygon?
The sum is S = (n – 2) × 180°, where n is the number of sides. For example, a 15-sided pentadecagon has interior angles that add up to 2,340°.
How do you find the number of sides of a regular polygon from its exterior angle?
Each exterior angle of a regular polygon measures 360° ÷ n, so divide 360° by the exterior angle. An exterior angle of 45° gives 360° ÷ 45° = 8 sides.
Does the module have an answer key?
Yes. It includes answers for the pretest, the activities, What I Have Learned, the assessment, and the additional activities.
How many lessons are in this module?
It has one lesson, on solving problems involving the sides and angles of a polygon.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics Quarter 3 – Module 8: Solving Problems Involving Side and Angle of a Polygon
[illustration: students studying outdoors and indoors, with a teacher, a student beside a globe, and a child studying by lamplight]
CO_Q3_Mathematics 7_ Module 8
Copyright and Development Team
Mathematics – Grade 7 Alternative Delivery Mode Quarter 3 – Module 8: Solving Problems Involving Side and Angle of a Polygons First Edition, 2020
Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties.
Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.
Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio
Printed in the Philippines by_______________________
Department of Education – SOCCSKSARGEN – Region XII
Office Address: Regional Center, Brgy. Carpenter Hill, City of Koronadal Telefax: (083) 2288825/ (083)2281893 E-mail Address: region12@deped.gov.ph
| Development Team of the Module | |
|---|---|
| Writer: | Arnel C. Lapitan |
| Editor: | Venus Jane E. Rosete |
| Reviewer: | Reynaldo C. Tagala |
| Layout Artist: | Maylene F. Grigana James B. De Los Reyes |
| Management Team: | Allan G. Farnazo, CESO IV, Regional Director |
| Gilbert B. Barrera, Chief, CLMD | |
| Arturo D. Tingson Jr., REPS, LRMS | |
| Peter Van C. Ang-ug, REPS, ADM | |
| Donna S. Panes, CID Chief | |
| Elizabeth G. Torres, EPS – LRMS | |
| Judith B. Alba, Division ADM Coordinator |
Mathematics Quarter 3 – Module 8: Solving Problems Involving Side and Angle of a Polygons
Introductory Message
This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.
Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.
In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.
Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.
If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.
Thank you.
What I Need to Know
This module was designed and written with you in mind. It is here to help you learn solving problems involving side and angle of polygons. The scope of this module may be used in many different learning situations. The language used recognizes your vocabulary level. The lessons are arranged to follow the standard sequence of the course. But the order in which you study can be rearranged to correspond with the textbook you are now using.
The lesson presented in this module is:
- Sides and angles of a polygons
After going through this module, you are expected to:
- Solve problems involving sides and angle of a polygons.
What I Know
Multiple choice. Read each item carefully. Choose the letter of the best answer and write it on a separate sheet of paper.
What is the sum of the measures of the vertex angles of octagon?
a. 1,080°
b. 900°
c. 1,260°
d. 720°What is the sum of the measures of the vertex angles of a pentagon?
a. 540°
b. 360°
c. 720°
d. 900°The number of sides of the polygons with 1,260° as the sum of the measures of the vertex angles is _____.
a. 6
b. 7
c. 8
d. 9What is the number of sides of the polygons when the sum of the measures of the vertex angles is 1,620°?
a. 11
b. 12
c. 13
d. 14What is the perimeter of an isosceles triangle with the base measuring 12.3 cm. and one of the equal sides equal to 15cm?
a. 27.3 cm
b. 27.3 cm²
c. 42.3 cm
d. 42.3 cm²What is the sum of all interior angle of a polygon having 29 sides?
a. 4,860°
b. 5,040°
c. 5,220°
d. 5,400°If the sum of the measure of the interior angle of a polygon is 3,240, what is the number of sides of the polygon?
a. 18
b. 19
c. 20
d. 21What is the sum of the interior angles of a decagon?
a. 90°
b. 180°
c. 360°
d. 1,440°The sum of all interior angles of a polygon is 3,060°. How many sides does the polygon have?
a. 12
b. 14
c. 17
d. 19What is the measure of each interior angle of a regular polygon with 9 sides?
a. 90°
b. 140°
c. 160°
d. 240°
Lesson 1 Solving Problems Involving Side and Angle of a Polygon
In this lesson, you will learn how to solve problems involving side and angle of a polygon.
What’s In
Recall that a polygon is a closed figure made up of three or more line segments joined at their endpoints. When all sides and all angles of a polygons are congruent, the polygons is a regular polygons or regular n-gon.
Illustrative examples:
Find the perimeter of each polygon.
A. Pentagon with a sides 5cm B. Equilateral triangle with sides 7cm C. Square with sides 26mm
Solution:
A. Pentagon with sides 5cm P = s₁ + s₂ + s₃ + s₄ + s₅ = 5 cm + 5 cm + 5 cm + 5 cm + 5 cm = 25 cm
B. Equilateral triangle with sides 7cm P = s₁ + s₂ + s₃ = 7 cm + 7 cm + 7 cm = 21 cm
C. Square with sides 26mm P = s₁ + s₂ + s₃ + s₄ = 26 mm + 26 mm + 26 mm + 26 mm = 104 mm
What’s New
To help you understand problems involving sides and angles of a polygon, recall that the sum of the measures of the interior angles of a triangle is 180⁰. Now, try to determine inductively the sum of the measures of the interior angles of any polygon by doing the following activity.
Activity 1.
- Draw convex polygons with four, five, six, seven and eight sides.
- Divide each polygon into triangles by drawing diagonals from one of its vertices.
- Find the sum of the measures of the interior angles of each polygon by multiplying the number of triangles formed by 180⁰.
- Record all the resulting values in a separate sheet of paper using a table similar to table 1.1. The first one has been done for you.
Table 1.1 Sum of the Interior Angles of a Polygon
| Polygon | No. of Sides | No. of Triangles Formed in the Figure | Sum of the Angle Measure |
|---|---|---|---|
| Quadrilateral | 4 | [illustration: quadrilateral with vertices F, E, R, A, one diagonal drawn] 2 | 2 x 180⁰ = 360⁰ |
| Pentagon | 5 | [illustration: pentagon with vertices R, O, B, A, L, diagonals drawn from one vertex] _____ | _____ x 180⁰ = _____ |
| Hexagon | 6 | [illustration: hexagon with vertices P, O, C, E, R, Y, diagonals drawn from one vertex] _____ | _____ x 180⁰ = _____ |
| Heptagon | 7 | [illustration: heptagon with vertices D, C, B, A, Z, Y, X, diagonals drawn from one vertex] _____ | _____ x 180⁰ = _____ |
| Polygon | No. of Sides | No. of Triangles Formed in the Figure | Sum of the Angle Measure |
|---|
[transcriber’s note: table continued from the previous page]
| Polygon | No. of Sides | No. of Triangles Formed in the Figure | Sum of the Angle Measure |
|---|---|---|---|
| Octagon | 8 | [illustration: octagon with vertices E, U, L, I, D, S, T, N, diagonals drawn from one vertex] _____ | _____ x 180⁰ = _____ |
Remember!
The sum of the measures of the interior angles of an n-gon is , S = (n – 2 )180⁰, where n is the number of sides.
Questions:
What did you discover? What is the sum of the measures of the interior angles of a polygon with 10 sides? of an n-gon?
What is It
The sum of the measures of the interior angles of an n-gon is, S = (n – 2 )180⁰, where n is the number of sides. The sum of the measures of the exterior angles of any polygon is equal to 360⁰. The measure of each angle of a regular polygon with n sides is (n−2)180⁰ / n.
Example 1. Find the sum of the measures of the vertex angles for pentadecagon.
Solution: Pentadecagon is a 15-sided polygon
S = ( n – 2)180⁰
= ( 15 – 2 ) 180⁰
= ( 13 ) 180⁰
= 2,340⁰
Therefore, the sum of the measures of the angles of a pentadecagon is 2,340⁰.
Example 2. Find the number of sides of a regular polygon when the measure of an exterior angle is 45⁰.
Solution:
Recall that each exterior angle of a regular polygon measures 360⁰/n. Since each exterior angle measures 45⁰, hence,
45⁰ = 360⁰/n
45⁰n = 360⁰
n = 360⁰/45⁰
n = 8
Therefore, the polygon has 8 sides.
What’s More
Activity 1. Draw the figure as a guide. Eddie estimated that he would need at least 1 meter of wooden sidings to make a picture frame. What are the dimensions of the largest frame he could make if the frame is
a. a square? b. rectangular shape? c. pentagon?
Activity 2
For their project in TLE, Belen decided to sew red piping along the edges of the table cloth measures 1.75m by 2.85m, at least how many meters of piping should she prepare?
What I Have Learned
On a separate sheet of paper, match the correct answer in column B. Write only the letter of your answer in the space provided.
| Column A | Column B |
|---|---|
| _____ 1. Find the sum of the measures of the vertex angles for octagon. | a. 900⁰ |
| _____ 2. Find the sum of the measures of the vertex angles for hexagon | b. 9 |
| _____ 3. Find the sum of the measures of the vertex angles for heptagon | c. 1,080⁰ |
| _____ 4. Find the number of sides of the regular polygon when the sum of the measures of the vertex angle is 1,260⁰ | d. 13 |
| _____ 5. Find the number of sides of the regular polygon when the sum of the measures of the vertex angle is 1,980⁰ | e. 720⁰ |
| f. 1,260⁰ |
[transcriber’s note: the printed Column A and Column B items are not ruled into rows; they are paired here by printed order, one per row]
Good job! Now you’re up for the next challenge of this lesson.
What I Can Do
Activity 3
Use your learning on solving problems involving sides and angles of polygons.
A six sided regular polygon is inscribed in a circle of radius 10cm, find the length of one side of hexagon.
Excellent work! You did a good job in applying what you have learned!
Assessment
Multiple choice. Read each item carefully. Choose the letter of the best answer and write it on a separate sheet of paper.
Find the sum of the interior angles of a heptagon.
- A. 900⁰
- B. 135⁰
- C. 1,800⁰
- D. 40⁰
Find the interior angle of a regular octagon.
- A. 160⁰
- B. 135⁰
- C. 1,800⁰
- D. 40⁰
Find the measure of each interior angle, each exterior angle and the sum of the interior angle of a regular dodecagon.
- A. 150⁰ , 40⁰, 1800⁰
- B. 150⁰ , 60⁰, 1800⁰
- C. 150⁰ , 30⁰, 1800⁰
- D. 150⁰ , 20⁰, 1800⁰
Jacob has a large sandbox, the sandbox is 12meters long and 8meters wide. He wants to surround the sandbox with new pieces of wood. How many meters of wood will he need?
- A. 40 meters
- B. 50 meters
- C. 60 meters
- D. 70 meters
The perimeter of Sophia’s Garden is 42 meters. The garden is 6 meters wide. What is the length of the garden?
- A. 30 meters
- B. 25 meters
- C. 15 meters
- D. 10 meters
What is the sum of all interior angle of a regular hexagon?
- A. 540⁰
- B. 720⁰
- C. 1,260⁰
- D. 1,440⁰
If the sum of the measure of the interior angle of a polygon is 2,160⁰, what is the number of sides of the polygon?
- A. 11
- B. 12
- C. 13
- D. 14
What is the measure of each interior angle of a regular octagon?
- A. 135⁰
- B. 180⁰
- C. 360⁰
- D. 540⁰
A regular polygon has an exterior angle that measures 40⁰. How many sides does the polygon have?
- A. 7
- B. 8
- C. 9
- D. 10
What is the measure of each interior angle of a polygon with 12 sides?
- A. 120⁰
- B. 140⁰
- C. 150⁰
- D. 160⁰
Additional Activities
In each of the following polygons, find the sum of the interior angles and the sum of exterior angles.
- Hexagon
- Octagon
Answer Key
[transcriber’s note: this page is printed upside down; the text below is transcribed in the order it reads when the page is turned right side up]
What I Know
- A
- A
- D
- A
- C
- A
- C
- D
- D
- B
What I have learned
- c
- e
- a
- b
- d
What’s New
| No. of triangles formed in the figure | Sum of the angle measure |
|---|---|
| 2 | 360⁰ |
| 3 | 540⁰ |
| 4 | 720⁰ |
| 5 | 900⁰ |
| 6 | 1080⁰ |
What’s More
Activity 1
- S = 25cm
- W =254cm, [?]
L =25cm - S = 20cm
Activity 2
- 9.2 meter
Assessment
- A
- B
- C
- A
- C
- B
- D
- A
- C
- C
What Can I Do
- 10cm
Additional Activity
- Sum of interior angle is 720⁰
Sum of exterior angle is 360⁰ - Sum of interior angle is 1080⁰
Sum of exterior angle is 360⁰
References
Department of Education-National Program Support for Basic Education Elementary Algebra for First Year.
Department of Education-Instructional Materials Council Secretariat (DepEd – IMCS), Mathematics – Grade 7 Teachers Guide First Edition, 2013 ISBN: 978-971-9990-61-1
Charles P Mckeague, et.al. (2013). Basic College Mathematics Anvil Publishing, Inc. Mandaluyong City 1550 Philippines
For inquiries or feedback, please write or call:
Department of Education – Bureau of Learning Resources (DepEd-BLR)
Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600
Telefax: (632) 8634-1072; 8634-1054; 8631-4985
Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph
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