Polygons

Polygons A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others. How figures that are closed and not closed. Be sure they understand all polygons must be closed and made up of line segments!!

Polygons Which are polygons? 7. 2. 6. 1. 5. 4. 8. 3. 9. Talk about each figure. Ask participant to explain why it is or is not a polygon. # 1, 5, 6 & 7 are polygons. #2, 3, 4, 8 & 9 are NOT polygons. 4. 8. 3. 9.

Regular Polygons A regular polygon is a polygon whose sides are all the same length, and whose angles are all the same. Talk about the difference between polygons in general and the special case of regular polygons. Use quadrilaterals as an example. Which quadrilaterals are regular polygons?

Regular Polygons Why is a square considered a regular polygon? Name another shape that you see almost everyday that is a regular polygon. STOP SIGN!

Common Polygons Triangles Quadrilaterals Pentagon Hexagon Octagon Ask participants to think of traffic signs with these shapes.

Triangles A triangle having all three sides of equal length. The angles of an equilateral triangle all measure 60 degrees. STOP SIGN!

Kinds of Triangles Equilateral Isosceles Scalene All Sides Equal NO Sides Equal Equilateral – all sides equal – a regular triangle Isosceles – 2 sides equal Scalene – no sides equal Can you call an equilateral triangle an isosceles triangle? Can you call an isosceles triangle an equilateral triangle?

Kinds of Triangles Acute Obtuse Right One angle = 90o All angles less than 90o Has one angle greater than 90o but less than 180o A right triangle is a triangle with a right angle (i.e. 90°).The side opposite the right angle is always the triangle's longest side. It is called the hypotenuse of the triangle. The other two sides are called the legs. The lengths of the sides of a right triangle are related by the Pythagorean Theorem. An acute triangle is a triangle whose angles are all acute or less than 90° An obtuse triangle has one obtuse angle (greater than 90º). The longest side is always opposite the obtuse angle. One angle = 90o

Kinds of Triangles Acute Obtuse Right One angle = 90o All angles less than 90o Has one angle greater than 90o but less than 180o A right triangle is a triangle with a right angle (i.e. 90°).The side opposite the right angle is always the triangle's longest side. It is called the hypotenuse of the triangle. The other two sides are called the legs. The lengths of the sides of a right triangle are related by the Pythagorean Theorem. An acute triangle is a triangle whose angles are all acute or less than 90° An obtuse triangle has one obtuse angle (greater than 90º). The longest side is always opposite the obtuse angle. One angle = 90o

Quadrilaterals A quadrilateral is a four-sided polygon with four angles. There are many kinds of quadrilaterals.

Quadrilaterals The five most common types are the parallelogram, the rectangle, the square, the trapezoid, and the rhombus.

Parallelogram A parallelogram is a quadrilateral with opposite sides parallel. Ask participants to draw 4 different parallelograms. The definition of opposite sides parallel could also describe a square, a rectangle, a rhombus,

Rectangle A rectangle is a quadrilateral with four 90-degree angles. Ask participants to draw 2 different rectangles. This could also describe a square.

Square A square is a quadrilateral with four equal sides and four 90 degree angles. Make sure participants understand the definition is about 4 EQUAL angles as well as 4 EQUAL sides!

Trapezoid A trapezoid is a quadrilateral that has two sides parallel Some definitions say EXACTLY 2 sides parallel… and some definitions say 2 sides parallel. One could consider a square as a trapezoid as it has 2 sides parallel. Same for a rectangle, parallelogram and a rhombus!

Rhombus A rhombus is a parallelogram with four equal sides. Ask participants: Can a square be considered a rhombus? Can a Rhombus be called a square? Explain

PENTAGON A five-sided polygon. Explain to participants pentagons can be regular (all 5 sides equal) or irregular.

Hexagon A six-sided polygon. Explain to participants hexagons can be regular (all 6 sides equal) or irregular.

Octagon An eight-sided polygon. Explain to participants octagons can be regular (all 8 sides equal) or irregular.

Perimeter The perimeter of a polygon is the distance around the outside of the polygon. To find the perimeter of a polygon, take the sum of the length of each side.

Perimeter 5 9 11

Perimeter 8 11 3

Perimeter Find the perimeter of a square with each side measuring 2 inches. Find the perimeter of an equilateral triangle with each side measuring 4 centimeters. Find the perimeter of a regular pentagon with each side measuring 3 inches. 11

Area The area of a shape is a number that tells how many square units are needed to cover the shape. Area can be measured in different units, such as square feet, square meters, or square inches.

Area You can find an area by drawing a shape on graph paper, and counting the squares inside the shape.

Area But this is not very practical… Most polygons can be described as having a base and height. h h h b b b

Rectangle The area of a rectangle is equal to the product of the length of its base and the length of its height. The height is a segment that is perpendicular to the base. A = b • h A rectangle is a good, simple shape to begin with. Remind participants that the base and height are often called the "length" and the "width", and sometimes the height is referred to as the "altitude.” Ask participants to find the area of this rectangle, with a base measuring 4 feet and a height measuring 6 feet. Using the formula, multiply 4 feet times 6 feet, to get 24 square feet. h b

Square The square is a special rectangle, and you can find its area using the rectangle formula. The area of a square is equal to the length of one side squared. A = b2 A = h2 A = s2 Remember, since the base and height are always the same number for a square, we usually call them "sides." The formula is usually A = s2. It could also be base (b2) or height (h2). If the length of one side of this square is 4 centimeters, what is the area? Substitute the value "4 cm" into the formula, and we find the area to be 16 square centimeters (16 cm2). h b

Parallelogram To find the area of a parallelogram, we use the same formula that we used for the area of a rectangle, multiplying the length of the base times the length of the height. A = b • h Remind participants that a parallelogram can be cut and made to be a rectangle. Find the area of a parallelogram that has a base of 23 cm and a height of 7 cm. Substitute the values into the formula. The parallelogram has an area of 161 square centimeters (161 cm2). h b

Triangle Look at this rectangle: Now, what do you have? h b Spend some time showing how a triangle is no more than ONE HALF of a rectangle. This makes using the formula very easy! b

Triangle 2 triangles from one rectangle. Formula to find the area of a rectangle is A = b • h h Spend some time showing how a triangle is no more than ONE HALF of a rectangle. This makes using the formula very easy! b

Triangle What is formula for the area of ONE triangle? A = ½ b • h h b What is the area of a triangle with a base length of 23 feet and a height of 16 feet? Substitute the values into the formula, and we find the area to be 184 square feet (184 ft2). b

2-D & 3-D Non Euclidian geometry, is the study of 2-D and 3-D shapes. A three-dimensional solid consists of a collection of polygons, usually joined at their edges. Explain to participants that from the study of two dimensional (2D) shapes, they will quickly move into the realm of 3D geometry. This is where they develop the spatial sense to see the relationship between 2D and 3D geometry.

Polyhedra A polyhedra is simply a three-dimensional solid which consists of a collection of polygons, usually joined at their edges. A polygon is the name for a 3-D shape, made up of polygons.

Polygon vs Polyhedra The words are derived from the Greek word poly (many) and gon (side) or hedra (face) . A polyhedra is a 3-D shape with many faces. A polygon is a 2-D shape with many sides. The plural of polyhedron is "polyhedra" (or sometimes "polyhedrons").

Polygon Poly Gon Penta 5 sides Hexa 6 sides Hepta 7 sides Octa 8 sides Deca 10 sides Make sure participants understand how to associate the ROOT word with the number of sides of a polygon.

Polyhedra Poly Hedra Tetra 4 faces Hexa 6 faces Octa 8 faces Deca Dodeca 12 faces Make sure participants understand how to associate the ROOT word with the number of faces of a polyhedra.

Polygon Polygon sides angles vertices Triangle 5 Quadrilateral 6 Pentagon Hexagon Heptagon 7 Octagon 8 Decagon 10 Have participants complete the chart. It is important they understand the vocabulary.