12. polygons

# 12. polygons ## Polygons: Trapeziums, pentagons, regular hexagon, regular octagon, parallelograms ### Introduction Welcome to this lesson on polygons! In mathematics, polygons are fundamental geometric shapes that you encounter everywhere, from the tiles on your floor to the designs of buildings. In this lesson, we will explore what polygons are and then focus on some specific types: trapeziums, parallelograms, pentagons, regular hexagons, and regular octagons. We will learn about their properties, how to identify them, and how to calculate their angles. ### Key Concepts #### What is a Polygon? A polygon is a closed two-dimensional figure made up of straight line segments. These segments are called sides, and where two sides meet is called a vertex (plural: vertices). Important characteristics of polygons include: * They must be closed (all sides connect). * They must have straight sides (no curves). * They must be two-dimensional. * The sides must not cross each other. #### Types of Polygons Polygons are named according to the number of sides they have: * 3 sides: Triangle * 4 sides: Quadrilateral * 5 sides: Pentagon * 6 sides: Hexagon * 7 sides: Heptagon * 8 sides: Octagon * 9 sides: Nonagon * 10 sides: Decagon #### Trapeziums A trapezium (also known as a trapezoid in some regions) is a quadrilateral with **at least one pair of parallel sides**. The parallel sides are called the bases, and the non-parallel sides are called the legs. If the non-parallel sides are equal in length, it is called an isosceles trapezium. * **Property:** Has at least one pair of parallel sides. #### Parallelograms A parallelogram is a special type of quadrilateral where **both pairs of opposite sides are parallel**. Due to this, parallelograms have several distinct properties: * **Property 1:** Opposite sides are equal in length. * **Property 2:** Opposite angles are equal. * **Property 3:** Consecutive angles (angles next to each other) are supplementary (they add up to \( 180^ \) ). * **Property 4:** The diagonals bisect each other (they cut each other into two equal parts). #### Pentagons A pentagon is a polygon with **five sides and five angles**. * **Regular Pentagon:** A pentagon where all five sides are equal in length and all five angles are equal in measure. * **Sum of Interior Angles:** For any pentagon, the sum of its interior angles is \[ (n-2) \times 180^ = (5-2) \times 180^ = 3 \times 180^ = 540^ \] * **Each Interior Angle of a Regular Pentagon:** \[ \frac{540^ }{5} = 108^ \] #### Regular Hexagons A regular hexagon is a polygon with **six equal sides and six equal angles**. * **Sum of Interior Angles:** For any hexagon, the sum of its interior angles is \[ (n-2) \times 180^ = (6-2) \times 180^ = 4 \times 180^ = 720^ \] * **Each Interior Angle of a Regular Hexagon:** \[ \frac{720^ }{6} = 120^ \] * **Each Exterior Angle of a Regular Hexagon:** The sum of exterior angles of any convex polygon is \( 360^ \). So, for a regular hexagon, each exterior angle is \[ \frac{360^ }{6} = 60^ \] #### Regular Octagons A regular octagon is a polygon with **eight equal sides and eight equal angles**. * **Sum of Interior Angles:** For any octagon, the sum of its interior angles is \[ (n-2) \times 180^ = (8-2) \times 180^ = 6 \times 180^ = 1080^ \] * **Each Interior Angle of a Regular Octagon:** \[ \frac{1080^ }{8} = 135^ \] * **Each Exterior Angle of a Regular Octagon:** \[ \frac{360^ }{8} = 45^ \] #### Interior and Exterior Angles of Polygons For any polygon with \( n \) sides: * **Sum of Interior Angles:** \[ S_i = (n-2) \times 180^ \] * **Each Interior Angle of a Regular Polygon:** \[ I = \frac{(n-2) \times 180^ }{n} \] * **Sum of Exterior Angles:** The sum of the exterior angles of any convex polygon is always \( 360^ \). \[ S_e = 360^ \] * **Each Exterior Angle of a Regular Polygon:** \[ E = \frac{360^ }{n} \] * **Relationship between Interior and Exterior Angle:** For any vertex, the interior angle and its corresponding exterior angle add up to \( 180^ \). So, \( I + E = 180^ \). ### Vocabulary * **Polygon:** A closed two-dimensional figure made up of straight line segments. * **Side:** A line segment forming part of a polygon. * **Vertex (plural: Vertices):** The point where two sides of a polygon meet. * **Quadrilateral:** A polygon with four sides. * **Trapezium:** A quadrilateral with at least one pair of parallel sides. * **Parallel Sides:** Sides that are always the same distance apart and never meet. * **Parallelogram:** A quadrilateral with two pairs of parallel sides. * **Pentagon:** A polygon with five sides. * **Hexagon:** A polygon with six sides. * **Octagon:** A polygon with eight sides. * **Regular Polygon:** A polygon where all sides are equal in length and all angles are equal in measure. * **Interior Angle:** An angle inside a polygon formed by two adjacent sides. * **Exterior Angle:** An angle formed by one side of a polygon and the extension of an adjacent side. It is outside the polygon. * **Diagonal:** A line segment connecting two non-adjacent vertices of a polygon. ### Examples **Example 1: Calculating Interior Angle Sum** What is the sum of the interior angles of a 7-sided polygon (heptagon)? * **Solution:** Using the formula \( S_i = (n-2) \times 180^ \) where \( n=7 \): \( S_i = (7-2) \times 180^ \) \( S_i = 5 \times 180^ \) \( S_i = 900^ \) The sum of the interior angles of a heptagon is \( 900^ \). **Example 2: Finding Each Interior Angle of a Regular Octagon** Find the measure of each interior angle of a regular octagon. * **Solution:** For a regular octagon, \( n=8 \). First, find the sum of interior angles: \( S_i = (8-2) \times 180^ = 6 \times 180^ = 1080^ \). Then, divide by the number of sides: \( I = \frac{1080^ }{8} = 135^ \). Alternatively, using the exterior angle formula: \( E = \frac{360^ }{8} = 45^ \). Then, \( I = 180^ - E = 180^ - 45^ = 135^ \). Each interior angle of a regular octagon is \( 135^ \). **Example 3: Identifying Polygon from Exterior Angle** If the exterior angle of a regular polygon is \( 72^ \), how many sides does it have? What is its name? * **Solution:** Using the formula \( E = \frac{360^ }{n} \): \( 72^ = \frac{360^ }{n} \) \( n = \frac{360^ }{72^ } \) \( n = 5 \) The polygon has 5 sides, so it is a **regular pentagon**. **Example 4: Properties of a Parallelogram** In parallelogram ABCD, angle A is \( 70^ \). Find the measures of angles B, C, and D. * **Solution:** * Since opposite angles are equal, angle C = angle A = \( 70^ \). * Since consecutive angles are supplementary, angle B + angle A = \( 180^ \). So, angle B = \( 180^ - 70^ = 110^ \). * Since opposite angles are equal, angle D = angle B = \( 110^ \). Therefore, angle B = \( 110^ \), angle C = \( 70^ \), and angle D = \( 110^ \). ### Exercises 1. Calculate the sum of the interior angles of a regular 9-sided polygon. 2. What is the measure of each interior angle of a regular 10-sided polygon? 3. A regular polygon has an exterior angle of \( 30^ \). How many sides does it have? What is the name of this polygon? 4. Sketch a trapezium and mark its parallel sides. Can a trapezium have all its sides equal? Explain. 5. In parallelogram PQRS, side PQ is 8 cm, and side QR is 5 cm. Angle P is \( 65^ \). Find: a. The length of side RS. b. The length of side PS. c. The measure of angle R. d. The measure of angle Q. ### Assignment 1. The sum of the interior angles of a regular polygon is \( 1440^ \). How many sides does it have? What is the measure of each exterior angle? 2. Draw a regular pentagon. Carefully measure and record one of its interior angles. Does your measurement match the calculated value? Explain any differences. 3. Explain the key differences between a trapezium and a parallelogram using diagrams. 4. A road sign for a stop sign is in the shape of a regular octagon. If the sign is to be made with equal side lengths, and you need to know the angle at each corner for cutting, what would that angle be? 5. If one interior angle of a regular polygon is \( 162^ \), how many sides does it have? ### Summary In this lesson, we have learned that polygons are closed 2D shapes with straight sides. We explored specific types of quadrilaterals: trapeziums (at least one pair of parallel sides) and parallelograms (two pairs of parallel sides with opposite sides and angles equal). We also studied pentagons (5 sides), regular hexagons (6 equal sides/angles), and regular octagons (8 equal sides/angles). Key formulas for angles in an \( n \)-sided polygon are: * Sum of interior angles: \( (n-2) \times 180^ \) * Each interior angle of a regular polygon: \( \frac{(n-2) \times 180^ }{n} \) * Sum of exterior angles: \( 360^ \) * Each exterior angle of a regular polygon: \( \frac{360^ }{n} \) * Interior angle + Exterior angle = \( 180^ \) Understanding these shapes and their properties is crucial for further studies in geometry and for applications in the real world. ## Properties of: regular polygons and parallelograms ### Introduction Welcome to this lesson on polygons! In geometry, polygons are fundamental shapes that you encounter everywhere. Today, we will dive into two specific types of polygons: **regular polygons** and **parallelograms**. Understanding their unique properties will help you solve many geometrical problems and better appreciate the shapes around you. ### Key Concepts #### What is a Polygon? A polygon is a closed two-dimensional figure made up of three or more straight line segments. These line segments are called **sides**, and they meet at points called **vertices**. Polygons are named based on the number of their sides (e.g., triangle for 3 sides, quadrilateral for 4 sides, pentagon for 5 sides, etc.). #### Regular Polygons A **regular polygon** is a special type of polygon where all its sides are of equal length, and all its interior angles are equal. ##### Properties of Regular Polygons: * All sides are equal in length. * All interior angles are equal. * All exterior angles are equal. * A regular polygon can be inscribed in a circle and a circle can be inscribed in it. * It has rotational symmetry and line symmetry. ##### Angle Formulas for Regular Polygons (where \(n\) is the number of sides): * **Sum of Interior Angles:** The sum of the interior angles of any polygon (regular or irregular) with \(n\) sides is given by the formula: \[ S = (n-2) \times 180^\circ \] * **Each Interior Angle:** For a regular polygon, since all interior angles are equal, each interior angle \(I\) can be found by dividing the sum by the number of sides: \[ I = \frac{(n-2) \times 180^\circ}{n} \] * **Sum of Exterior Angles:** The sum of the exterior angles of any convex polygon (regular or irregular) is always \(360^\circ\). \[ \sum E = 360^\circ \] * **Each Exterior Angle:** For a regular polygon, each exterior angle \(E\) can be found by: \[ E = \frac{360^\circ}{n} \] * **Relationship between Interior and Exterior Angles:** An interior angle and its corresponding exterior angle always add up to \(180^\circ\): \[ I + E = 180^\circ \] #### Parallelograms A **parallelogram** is a quadrilateral (a four-sided polygon) with two pairs of parallel sides. ##### Properties of Parallelograms: * **Opposite sides are parallel.** This is the defining characteristic. * \( AB \parallel DC \) and \( AD \parallel BC \) * **Opposite sides are equal in length.** * \( AB = DC \) and \( AD = BC \) * **Opposite angles are equal.** * \( \angle A = \angle C \) and \( \angle B = \angle D \) * **Consecutive (adjacent) angles are supplementary.** This means they add up to \(180^\circ\). * \( \angle A + \angle B = 180^\circ \) * \( \angle B + \angle C = 180^\circ \) * \( \angle C + \angle D = 180^\circ \) * \( \angle D + \angle A = 180^\circ \) * **Diagonals bisect each other.** The point where the diagonals intersect divides each diagonal into two equal parts. * If diagonals \( AC \) and \( BD \) intersect at point \( M \), then \( AM = MC \) and \( BM = MD \). ### Vocabulary * **Polygon:** A closed two-dimensional figure made up of straight line segments. * **Regular Polygon:** A polygon with all sides equal and all interior angles equal. * **Irregular Polygon:** A polygon whose sides or angles (or both) are not all equal. * **Parallelogram:** A quadrilateral with two pairs of parallel sides. * **Side:** A line segment forming part of the boundary of a polygon. * **Vertex (plural: Vertices):** A point where two sides of a polygon meet. * **Interior Angle:** An angle inside a polygon formed by two adjacent sides. * **Exterior Angle:** An angle formed by one side of a polygon and the extension of an adjacent side. * **Diagonal:** A line segment connecting two non-adjacent vertices of a polygon. * **Congruent:** Having the same shape and size. * **Supplementary Angles:** Two angles whose sum is \(180^\circ\). * **Bisect:** To divide into two equal parts. ### Examples #### Example 1: Finding Angles in a Regular Polygon **Question:** Find the measure of each interior angle and each exterior angle of a regular octagon. **Solution:** 1. A regular octagon has \(n = 8\) sides. 2. **Each Exterior Angle:** \( E = \frac{360^\circ}{n} = \frac{360^\circ}{8} = 45^\circ \) 3. **Each Interior Angle:** \( I = 180^\circ - E = 180^\circ - 45^\circ = 135^\circ \) Alternatively, using the interior angle formula: \( I = \frac{(n-2) \times 180^\circ}{n} = \frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ \) #### Example 2: Using Properties of a Parallelogram **Question:** In parallelogram ABCD, \( AB = 10 \text{ cm} \), \( BC = 7 \text{ cm} \), and \( \angle A = 70^\circ \). Find the lengths of \( CD \) and \( AD \), and the measure of \( \angle B \) and \( \angle C \). **Solution:** 1. **Opposite sides are equal:** * \( CD = AB = 10 \text{ cm} \) * \( AD = BC = 7 \text{ cm} \) 2. **Consecutive angles are supplementary:** * \( \angle B + \angle A = 180^\circ \Rightarrow \angle B + 70^\circ = 180^\circ \Rightarrow \angle B = 110^\circ \) 3. **Opposite angles are equal:** * \( \angle C = \angle A = 70^\circ \) * (Also, \( \angle D = \angle B = 110^\circ \)) ### Exercises 1. A regular polygon has an exterior angle of \( 30^\circ \). How many sides does it have? What is the measure of each interior angle? Name the polygon. 2. The sum of the interior angles of a polygon is \( 1440^\circ \). How many sides does it have? 3. In parallelogram PQRS, \( PQ = 12 \text{ cm} \) and \( PS = 8 \text{ cm} \). If \( \angle P = 60^\circ \): a. Find the length of \( RS \) and \( QR \). b. Find the measure of \( \angle Q \), \( \angle R \), and \( \angle S \). 4. The diagonals of a parallelogram ABCD intersect at O. If \( AO = 5 \text{ cm} \) and \( DO = 6 \text{ cm} \), find the lengths of \( AC \) and \( BD \). ### Assignment 1. Each interior angle of a regular polygon is \( 150^\circ \). Calculate the number of sides of the polygon. 2. One angle of a parallelogram is \( 45^\circ \). Find the measure of the other three angles. 3. Draw a regular hexagon and label its interior and exterior angles. Calculate their measures. 4. A quadrilateral has vertices at A(1, 1), B(4, 1), C(5, 3), and D(2, 3). Show that it is a parallelogram by proving that its opposite sides are parallel (you can use slope). 5. In a parallelogram, the perimeter is \( 48 \text{ cm} \). If one side is \( 10 \text{ cm} \), find the lengths of the other three sides. ### Summary In this lesson, we explored the fascinating world of polygons, focusing specifically on **regular polygons** and **parallelograms**. We learned that regular polygons have equal sides and equal angles, and we mastered the formulas to calculate their interior and exterior angles using the number of sides, \(n\). We also delved into parallelograms, understanding that their opposite sides are parallel and equal, opposite angles are equal, consecutive angles are supplementary, and their diagonals bisect each other. These properties are crucial for solving various geometric problems and understanding the structure of many shapes in our environment. ## Area of a parallelogram and of a trapezium. ### Introduction Welcome to this lesson! Today, we will explore how to calculate the area of two important quadrilaterals: the parallelogram and the trapezium. Understanding these concepts is essential as they appear in many real-world applications, from construction to design. By the end of this lesson, you will be able to confidently find the area of both shapes. ### Key Concepts #### Area of a Parallelogram A parallelogram is a quadrilateral (a four-sided shape) with two pairs of parallel sides. This means opposite sides are parallel and equal in length. To find the area of a parallelogram, we can imagine transforming it into a rectangle. If you cut off a triangular piece from one end and move it to the other, it forms a rectangle. The area of a parallelogram is given by the product of its base and its perpendicular height. ##### Formula for the Area of a Parallelogram The area \(A\) of a parallelogram is: \[ A = \text{base} \times \text{height} \] Or simply: \[ A = b \times h \] Where: * \(b\) is the length of the base. * \(h\) is the perpendicular height (the shortest distance between the base and the opposite side). #### Area of a Trapezium (or Trapezoid) A trapezium (also known as a trapezoid in some regions) is a quadrilateral with *at least one* pair of parallel sides. These parallel sides are called the bases of the trapezium, and the non-parallel sides are called legs. To find the area of a trapezium, we can think of it as two triangles and a rectangle, or by duplicating it to form a parallelogram. ##### Formula for the Area of a Trapezium The area \(A\) of a trapezium is given by half the sum of its parallel sides multiplied by its perpendicular height. \[ A = \frac{1}{2} (\text{sum of parallel sides}) \times \text{height} \] Or using common variables: \[ A = \frac{1}{2} (a+b) h \] Where: * \(a\) and \(b\) are the lengths of the two parallel sides. * \(h\) is the perpendicular height (the shortest distance between the two parallel sides). ### Vocabulary * **Parallelogram:** A quadrilateral with two pairs of parallel and equal opposite sides. * **Trapezium (Trapezoid):** A quadrilateral with at least one pair of parallel sides. * **Base:** A side of a polygon, often used as a reference for calculating area. For a parallelogram, it's any side. For a trapezium, it refers to the parallel sides. * **Height:** The perpendicular distance from the base to the opposite side (for a parallelogram) or between the two parallel bases (for a trapezium). * **Parallel Sides:** Sides that are always the same distance apart and will never meet, no matter how far they are extended. * **Area:** The amount of space a two-dimensional shape occupies, measured in square units (e.g., \(cm^2\), \(m^2\)). ### Examples #### Example 1: Area of a Parallelogram Find the area of a parallelogram with a base of 12 cm and a perpendicular height of 7 cm. **Solution:** Given: \(b = 12\text{ cm}\), \(h = 7\text{ cm}\) Formula: \[ A = b \times h \] Substitute values: \[ A = 12\text{ cm} \times 7\text{ cm} \] Calculate: \[ A = 84\text{ cm}^2 \] The area of the parallelogram is \(84\text{ square centimetres}\). #### Example 2: Area of a Trapezium A trapezium has parallel sides of length 9 cm and 15 cm. The perpendicular distance between these parallel sides is 6 cm. Calculate its area. **Solution:** Given: \(a = 9\text{ cm}\), \(b = 15\text{ cm}\), \(h = 6\text{ cm}\) Formula: \[ A = \frac{1}{2} (a+b) h \] Substitute values: \[ A = \frac{1}{2} (9\text{ cm} + 15\text{ cm}) \times 6\text{ cm} \] Add parallel sides: \[ A = \frac{1}{2} (24\text{ cm}) \times 6\text{ cm} \] Multiply: \[ A = 12\text{ cm} \times 6\text{ cm} \] Calculate: \[ A = 72\text{ cm}^2 \] The area of the trapezium is \(72\text{ square centimetres}\). #### Example 3: Finding Height of a Parallelogram The area of a parallelogram is \(105\text{ m}^2\). If its base is 15 m, what is its perpendicular height? **Solution:** Given: \(A = 105\text{ m}^2\), \(b = 15\text{ m}\) Formula: \[ A = b \times h \] Substitute known values: \[ 105 = 15 \times h \] Divide both sides by 15: \[ h = \frac{105}{15} \] Calculate: \[ h = 7\text{ m} \] The perpendicular height of the parallelogram is \(7\text{ metres}\). ### Exercises Try these questions to test your understanding. 1. A parallelogram has a base of 10.5 m and a perpendicular height of 8 m. Calculate its area. 2. Find the area of a trapezium whose parallel sides are 11 cm and 17 cm, and the perpendicular height is 5 cm. 3. The area of a parallelogram is \(120\text{ cm}^2\). If its height is 10 cm, what is the length of its base? 4. A field is shaped like a trapezium with parallel sides measuring 200 m and 350 m. The perpendicular distance between these sides is 150 m. What is the area of the field in square metres? 5. A parallelogram has sides of 8 cm and 6 cm. Its perpendicular height corresponding to the 8 cm base is 5 cm. What is its area? ### Assignment Complete these questions at home to further practice your skills. 1. Calculate the area of a parallelogram with a base of 25 mm and a height of 18 mm. 2. A trapezium has parallel sides of 2.5 m and 4.0 m. The height of the trapezium is 1.5 m. Find its area. 3. If the area of a trapezium is \(80\text{ m}^2\) and its parallel sides are 12 m and 8 m, what is its perpendicular height? 4. A plot of land is in the shape of a parallelogram. Its area is \(450\text{ km}^2\) and its height is 18 km. What is the length of its base? 5. Draw a parallelogram and a trapezium, label their dimensions, and calculate their areas based on your chosen measurements. ### Summary In this lesson, we have learned how to calculate the area of parallelograms and trapeziums. Remember the key formulas: * **Area of a Parallelogram:** \( A = b \times h \), where \(b\) is the base and \(h\) is the perpendicular height. * **Area of a Trapezium:** \( A = \frac{1}{2} (a+b) h \), where \(a\) and \(b\) are the lengths of the parallel sides, and \(h\) is the perpendicular height. Always ensure you use the *perpendicular* height when applying these formulas. Practice makes perfect, so keep working through problems to master these concepts!