GEOMETRY

IN M

Y

WORLD

LIZA F

RENCH

Block 3

Geometry H, T3 & T4

2012

Explanation

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Geometry is not a concept only used in math class, it simply helps us identify real world applications of things like shapes and lines. We can use the lessons we learn in class to discover new things about the world around us. For example, a stop sign is a octagon, a regular polygon. The legs of tables and chairs will often be parallel lines. This PowerPoint will show examples of Geometry I found in my life. I will explain how each picture is an example of Geometry and where I found it.

Decorative Plate – Circle (L. 23)

Definitions:

A center of a circle is the point inside a circle that is equidistant from every point on the circle.

A circle is a closed plane curve consisting of all points at a given distance from a point within it called the center.

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This is a picture of a decorative plate that hangs in my house.

The plate is an example of a circle. The center point (Point A) of the circle was used in the design by the artist to determine where to paint the birds.

This is a stain glass table lamp that sits next to the reading chair upstairs in my computer room.This lamp is a regular decagon. On the lamp, the congruency of the sides is shown with pink congruency markings, the diagonals, forming central angles are marked with blue, and the sides are yellow.

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Table Lamp – Decagon (L. 15)

Definitions:

A decagon is a ten-sided polygon.

A regular polygon is a polygon that is both equilateral (all the sides are the same length) and equiangular (all the angles are the same measure).

Stove Fan – Frustum of a Pyramid (L.103)

This picture is of the fan above my stove, which is used to bring better air circulation when cooking food that produces a lot of smoke.

The red lines represent the shape of the fan, which creates a frustum. The blue lines signify where the top of the pyramid would be, if it were complete.

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Definitions:

A frustum of pyramid is a part of a pyramid with two square parallel bases.

A pyramid is a polyhedron formed by a polygonal base and triangular lateral faces that meet at a common vertex.

Interesting Note*

Volume of a Frustum:

V = 1/3h(B1 + √B1(B2) + B2

Back Yard Fence – Perimeter (L. 19)

At my house, there is a square back yard. Each side of the yard is 40 ft. Around that yard, There is a wooden fence, which is shown in the picture here. However, there is a 4 foot pathway

The fence represents the perimeter of the yard, which in a square can be found with the formula “P = 4s” where s is the length of the sides, in the case of my yard, 40 ft.

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Working out the Formula: Step 1: P = 4s – 4 Step 2: P = 4(40) – 4Step 3: P = 160 – 4 Step 4: P = 156 ft

Definitions:

A perimeter is the sum of the side lengths of a closed plane figure.

Coat Hanger – Rhombus (L.52)

This is a coat hanger that I use daily in my room to hang clothes on.

When pulled apart, the coat hanger forms a rhombus. The diagonals of the rhombus bisect each other, forming congruent segments and right angles.

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Definitions:

A rhombus is a quadrilateral with four congruent sides.

The Properties of a Rhombus state that the diagonals of a rhombus are perpendicular and that each diagonal of a rhombus bisects opposite angles. Because opposite angles of a rhombus are equal, when they are bisected by a diagonal, four congruent angles result.

CD Case – Tangent (L. 43)This is a picture of a blank CD waiting to be used on my computer desk.

The edge of the CD case, Line Segment AB which is represented in yellow, lies tangent to Circle P, represented in red around the CD. The point of tangency in this picture is Point C.

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Definitions:

The tangent of a circle is a line in the same plane as the circle that intersects the circle at exactly one point.

The point of tangency is the single point that the tangent line intersects a circle with.

Nesting Boxes – Ratios (L. 41)

This is a picture of some nesting boxes that I received as a gift. The boxes are similar shapes because the sides are equivalent ratios. The square of the side lengths of the square top to the height is ¾.

For instance, a proportion of these numbers would be: 6/8 = 4.5/6 = 3/4 = 1.5/2.

Definitions:

A ratio is a comparison of two quantities by division.Similar polygons are polygons whose corresponding angles are congruent and whose corresponding sides are proportional.A proportion is a statement that two ratios are equal. 9

Window – Plane Intersections (L. 4)

This is a window on the second story of my house, looking out into my front yard.

In this picture, there are 8 planes. 3 are highlighted as plane M, plane R, and plane P. Plane M and Plane P intersect at line segment AB. Planes M, R, and P intersect at a point, Point A.

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Postulate 7: If two planes intersect, then their intersection is a line.

Definitions:

A plane is an undefined term in geometry; a flat surface that has no thickness and extends infinitely.

Mirror and Door – Chord (L.43)

This is a picture of a mirror in my house that is reflecting a closet door.

The seam of the closet door, represented by line segment AB, is a chord to the circle of the mirror, represented as Circle P because points A and B lie on the circle.

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Definitions:

A chord is a segment whose endpoints lie on a circle.

Porch Railing – Parallel and Perpendicular Lines (L. )

This is a picture of the wooden railing on my porch. The poles represent line segments AE, BF, CG, and DH. These line segments are parallel, as signified by the white markings. The white right angles show that line segment AD is perpendicular to line segments AE, BF, CG, and DH.

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Rain Gauge – Rectangular Prism (L. 59)

This is a (broken) rain gauge that I have placed outside to measure the amount of rainfall.

The rain gauge is in the shape of a rectangular prism. It has a two bases, one at the top and one at the bottom, and four equivalent sides.

To find the volume of this prism, you would use the formula V = Bh, where B is the area of one base and h is the volume.

To find the surface area, you would use the formula SA = ph + 2B, where p is the perimeter of the base, h is the height, and B is the area of one base.

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Working out Volume:

Step 1: V = Bh

Step 2: V = 1(1.5)(7)

Step 3: V = 10.5 in²

Working out Surface Area:

Step 1: SA = ph + 2B

Step 2: SA = 2(1 + 1.5)(7) + 2(1)(1.5)

Step 3: SA = 2(2.5)(7) + 2(3)

Step 4: SA = 41 in

Definitions:

A rectangular prism is a prism with six rectangular faces.

Volume is the number of non-overlapping unit cubes of a given size that will exactly fill the interior of a three-dimensional figure.

Surface Area is the total area of all faces and curved surfaces of a three-dimensional figure.

Macaroni Box – Nets (Investigation 5)

This is a deconstructed macaroni box, found in my recycling and the box in its completed form, found in the pantry.

The deconstructed box is a net of the original box. The single plane deconstruction could be folded to form the original box. The yellow sections of the net correspond to the yellow sections of the box, the red to the red, and the blue to the blue.

Definitions:

A net is a diagram of the faces of a three-dimensional figure, arranged so that thee diagram can be folded to form the three-dimensional figure. 14

Scented Candle – Cylinder (L. 62)This is a scented candle that my mother keeps on the kitchen counter.

The candle is in the shape of cylinder. The height of the candle is 15 cm, and the radius, or distance from the center point of the circle base, which is the wick, to the edge of the candle is 5 cm.

To find the volume of the cylinder, you use the formula V = πr²h.

To find the Surface Area, the formula SA = 2πrh + 2B is used. The variable h represents the height, B represents the area of the base, and r represents the radius.

Working out Volume:

Step 1: V = πr²h

Step 2: V = π5²(15)

Step 3: V = π(25)(15)

Step 4: V = π(375)

Step 5: V = 1178.1 cm²

Working out Surface Area:

Step 1: SA = 2πrh + 2B

Step 2: SA = 2π5(15) + 2π5²

Step 3: SA = 150π + 2π25

Step 4: SA = 150π + 50 π

Step 5: SA = 628.2 cm

Definition:

A cylinder is a three-dimensional figure with two parallel congruent circular bases and a curved lateral surface that connects the bases.

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Stairs – Kite (L. 19, L. 69)

This is a segment of my stairway, enabling the stairs to curve around the wall.

The shape of the step outlined in yellow is a kite. The dotted lines represent the hidden parts of the shape, as it was impossible to capture the entire shape from any angle. Two pairs of the kite are congruent and its diagonals form four right angles.

Definitions:

A kite is a quadrilateral with exactly two pairs of congruent consecutive sides. 16

Properties of Kites:

The diagonals of a kite are perpendicular.

Salt Shaker – Regular Polygons (L. 15 )

This is an aerial view of a salt shaker that I found on my kitchen table.

The base of the salt shaker forms a regular octagon. The eight sides of the octagon are marked with red lines and yellow congruency markings, and the angles are marked with blue.

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Definitions:

A regular polygon is a polygon (a closed plane figure formed by three or more segments) that is both equiangular and equilateral.

Garden Gate – Special Right Triangles (L. 53, L.56)

This is a picture of a gate that my father had built surrounding his garden to protect it from animals that would eat his plants.

The right triangles in the gate are 45-45-90 triangles, classified by the measurements of their angles. The legs of these triangles are congruent. The hypotenuse is the length of the legs multiplied by √2.

The right triangle to the right of the gate is a 30-60-90 right triangle, also classified by the measure of its angles. The length of the smallest side (x) is doubled (2x) to get the length of the hypotenuse. To find the length of the middle side, it is multiplied by radical 3, (x √3).

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Floor Tiles – Tessellation (Investigation 9)

This is a picture of a tiled floor hallway in my house.

I have left this photograph fairly untouched because the lines between the tiles are fairly distinct. I have, however, outlined the perimeter of the hallway lightly in black.

The squares in picture form a very simple regular tessellation.

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Definitions:

A tessellation is a repeating pattern of plane figures that completely covers a plane with no gaps or overlaps.

A regular tessellation is the simplest kind of tessellation, a repeating pattern of congruent regular polygons.

Geometry in Nature – Triangular Dog’s Head (L. 51)

This is the head of my greyhound, Opal. Her head is shaped like an acute isosceles triangle, Triangle QRS. <Q and <R have congruent measures, while the vertex angle, <S does not. This is because the angle across from the line segment that is not congruent to the other two line segments will not be congruent to the other two angles of the triangle. In Triangle QRS, Line Segment QS and Line Segment RS, the legs, are congruent, while Line Segment QR is not.

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Definition:

An acute triangle is a triangle with three acute angles (angles with measures less than 90°)

An isosceles triangle is a triangle with at least two congruent sides.

A vertex angle is the angle formed by the legs of the triangle.

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Theorem 51-1: Isosceles Triangle Theorem – If a triangle is isosceles, then its base angles are congruent.

Geometry in Nature - Starfish

This is a starfish I found on a trip to the beach last summer. I then took it home and created an ornament using a nail and a piece of string.

A starfish has five legs, each leg with two sides. This gives it a total of ten sides, making it a decagon¹.

21¹See slide # 4 for a definition of “decagon.”

Geometry in Nature – Spherical Plants

This plant is called an Allium, which, when bloomed, is a sphere of purple blossoms. My mother plants many Alliums in her garden at home, and this is where I found it.

From the plant, you can see the center from which the blossoms stems has been labeled as the center of the sphere (Point A), and two of the buds have been labeled as points (Point C and Point B). These buds represent points on the sphere’s exterior.

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Definitions:

A sphere is a set of points in space that are a fixed distance from a given point called the center of the sphere.

Bibliography• Saxon Geometry. Student ed. Austin: HMH Supplemental Publishers

Inc., 2009. 800-871. Print.

• "circle." Dictionary.com Unabridged. Random House, Inc. 11 Jun. 2012. <Dictionary.com http://dictionary.reference.com/browse/circle>.

• Simmons, Bruce. "Frustum of a Cone or Pyramid." Mathwords. N.p., March 24, 2011. Web. 11 Jun 2012. <http://www.mathsisfun.com/quadrilaterals.html>.

• "Quadrilaterals - Square, Rectangle, Rhombus, Trapezoid, Parallelogram." MathIsFun.com. N.p., 2012. Web. 11 Jun 2012. <http://www.mathsisfun.com/quadrilaterals.html>.

• . "Rectangular Prism - Geometry - Math Dictionary." icoachmathc.com. High Points Learning Inc, 199-2011. Web. 12 Jun 2012. <http://www.icoachmath.com/math_dictionary/Rectangular_prism.html>.

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