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# Geometric Shapes

Geometric shapes are figures which can be described using mathematical data, such as equations, and are an important component to the study of geometry. Basically, geometric shapes are the spatial representation of mathematical information and are unrelated to other descriptive data such as location.

The term polygon is used when describing figures which are closed and constructed of lines and points. Polygons are referred to as plane figures because they exist in two dimensions.

There are various different types of polygons and they differ in terms of their number of sides. Squares, triangles and hexagons are all examples of polygons. Additionally, other shapes such as circles which are formed by curves are also polygons. A curve is a geometric shape, but not a polygon because it is not a closed figure. Rather it is used to create polygons such as a circle or an ellipse.

In the study of geometry, analyzing the different properties of geometric shapes is a common practice. All geometric shapes differ in terms of their side lengths, number of vertices and angle measurements, to name a few features. Furthermore, the mathematical principles and theories which relate with different shapes vary and thus, having a broad understanding of geometric shapes is useful.

## Categories within Geometric Shapes

### Triangles, Circles and Rectangles

1. Given a right triangle with legs a and b, and hypotenuse c, find the missing side. A=9, b+12 2.The circle has a circumference of 43.96. Using 3.14 to approximate pi, find the value of x (x+5) 3. Find the area of a rectangle that is 9 km by 7 km.

### The Volume of Prisms and Pyramids

1. A triangular pyramid with a base length of 9 inches, a base height of 10 inches, and a height of 32 inches. Find the volume of the figure described. 2. A square pyramid with a base length of 4 cm and a height of 6 cm resting on top of a 4 cm cube. Find the volume of the figure described. 3. The square pyramid at

### Regular Hexagon Inscribed inside a Square

You have a square of area 5 m^2. You need to construct a regular hexagon inscribed in this square. Show how you can find the length of each side, the angles and area of the hexagon and please show the diagram.

7. Figure 4 shows a survey of a building which forms a quadrilateral ABCD Calculate a) the length of the diagonals - AC and BD b) the area of the plot ABCD Please see attachment for diagrams.

### Maple TA: Calculating an Angle, and the Area

Attached are two maple TA problems I would like to understand which one is a correct answer and all possible additional notes explain why. Question 1: Please select all correct formulas below that can be used for calculating the angle ? between two vectors a,b: (See attached for the equations) Question 2: Consider the

### Symmetries of an origami buckyball torus

Find the symmetries of an origami buckyball torus using the Orbit-Stabilizer Theorem. The torus has 96 vertices, 24 heptagon faces, 48 hexagon faces, 24 pentagon faces. The torus is made from 288 pentagon-hexagon-zig-zag units (PHiZZ units)

### Number of colored hexagons up to cyclic symmetry

If C6 acts on a regular hexagon by rotation and each of the vertices is colored red, blue or green, use the Burnsideââ?¬â?¢s formula to determine how many possible colorings there are up to cyclic symmetry.

### Area of the Parallelogram

One can show that the k-dimensional volume V_k of the parallelopiped generated by the vectors a_1,..,a_k in R^n is given by V_k=square root of (det A^TA), where A=(a_1|â?¦|a_k)_nxk. Use this information to compute the area of the parallelogram generated by a_1=(1,0,1,0,1) and a_2=(1,1,1,1,1) in R^5.

### Length & Width of Recatangular Floor

Set up an equation and solve the following problem. The length of a rectangular floor is 8 meter less than twice its width. If a diagonal of the rectangle is20 meters, find the length and width of the floor. a. width is 12 m, length is 16 m b. width is 24 m, length is 32 m c. width is 14 m, length is 18 m d. width

### Diameter of a non-empty set in a metric space

6. The diameter delta(A) of a non-empty set A in a metric space (X, d) is defined to be delta(A) = sup [x, y BELONGING_TO A] d(x,y). A is said to be bounded if delta(A) < infinity. Show that A SUBSET B implies delta(A) <= delta(B). Please see the attached image for proper description of the question with appropriate sym

### Comparison of prices of circle-shaped pizzas

Medium pizza with a 12 inch diameter sells for \$5, and Large pizza with a 16 inch diameter sells for \$10. Which pizza is the better buy, and why?

### 64 Alternating White and Black Squares - Square Determination

A checkered flag used for racing is a square flag containing 64 alternating white and black squares. How many squares on the checkered flag contain an equal number of white and black squares? Be sure to describe how you arrived at your answer.

### Conic Section: Ellipse

Transform each equation to standard form. Then find the center, foci, major and minor axes, and ends of each latus rectum. Draw the curve 1. 4x^2 + y^2 + 8x - 4y - 8 = 0 2. 16x^2 + 25y^2 + 160x + 200y + 400 = 0 3. 9x^2 + 4y^2 - 36x + 8y + 31 = 0 4. 4x^2 + 9y^2 - 16x + 18y - 11 = 0 5. 25(x + 1)^2 + 169(y - 2)^2 =

### Find the height

Please see attached file for questions with diagrams. 27. The height of the house shown here can be found by comparing its shadow to the shadow cast by a 3-foot stick. Find the height of the house by writing a proportion and solving it. 28. A fire lookout tower provides an excellent view of the surrounding countryside. The

### Concepts of Geometrical Shapes

Give a clarification in the true numbers needed to help with the problems. 1. Equation Number 1 has a degree of 137 not 78 as depicted in the diagram. 2. Equation number 2 has a height if 7 inches not 9 inches as depicted in the diagram.

### Quadrilaterals and Constructing Images under Translation

1. For each of the following, find the image of the given quadrilateral under a translation from A to B: -- 2. Contruct the image BC under the tranalation pictured in the figure by using the following: a. Tracing paper b.Compass and straightedge

### Center of Gravity Proof: Example Problem

Prove the the center of gravity of a lamina in the shape of a parallelogram is at the point of intersection of the diagonals and that it is the same point as the center of gravity of four particles two of mass m at one pair of opposite vertices and two of mass 2m at the other pair of opposite vertices.

### Various problems

1. A plate glass window measures 5 ft by 8 ft. If glass costs \$6 per square foot, how much will it cost to replace the window? A) \$78 B) \$1,440 C) \$240 D) \$480 2. A bedroom is 10 ft by 11 ft. What is its perimeter? A) 21 ft B) 110 ft C) 42 ft D) 55 ft 3. Turner agrees to buy a boat for \$2,800 down and \$129 a month

### Find Area, Chord and Arc in a Circle

The spreadsheet attachment provides all of the givens and we are asked to determine the central angle, length of a chord, and the area of the identified circle segment? See the attached file.

### Length of Hypotenuse

Please solve and explain. 1.A telephone pole 35ft. tall has a guy wire attached to it 5 ft. from the top and tied to a ring on the ground 15 feet from the base of the pole. Assume that an extra 2 feet of wire are needed the wire to the ring and the pole. What length of wire is needed for the job? Give an answer to the near

### Geometrical Constructions - Tangents to a Circle

Let C be a circle with center P and X a point in the exterior of C. Construct the two tangent lines to the circle that pass through X.

### Find the Mass of the Given Shape

1) (From prob_4.doc) Find the mass of the annulus (donut shape) having radius 1<r<2 when the density funciton rho(r,theta) = (1-ar^2) where a is a constant. 2) (From prob_5.doc) Find the total mass of the 3D object when the mass density rho and the object size is rho(r,phi,theta)=r^2sin(theta) where 1<r<2, 0<phi<pi and 0<thet

### Speed, Position and Arc Length

Two identical bugs start moving at the same time on a flat table, each at the same constant speed of 20 cm/min. Assume that initially (i.e. at time t = 0) bug 1 is located at point (1, 1) and bug 2 is located at the point (-1, 1). Assume that units in the xy-plane are measured in meters and time is measured in minutes. Further

### Volume of a Pipeline

The attached document shows a pipeline of 24 inch diameter (approx. 600mm) buried 1 m below the ground. There is a water pipe which prevents the pipe from going horizontally and hence it has to follow one of two pathways ie. either along the dark blue 5 mm diameter curves and exit at the bottom or along the dotted red double 40

### Finding Angles

The distance from A to B is 4 metres. The angles shown are in degrees. What is the angle of alpha?

### Finding the Lengths and Angles of Arcs

Question#1 Two streets meet at an angle of 83.0 degrees. What is the length of the piece of curved curbing at the intersection if it is constructed along the arc of a circle 15.0 ft in radius? Question #2 Through what angle does the drum turn in order to lower the crate 18.5 feet. The drum has a circumference of 2.38 feet

### Real-Life Examples of Parabolic Shapes

Select and discuss two examples of real life objects that incorporate the parabolic shape. Explain why the parabolic shape was used for the objects.

### How many yards would he run?

Suppose at the kickoff of a football game, the receiver catches the football at the left side of the goal line and runs for a touchdown diagonally across the field. How many yards would he run? (A football field is 100 yards long and 160 feet wide).

### Circumference of a Circle and Radius

If the circumference of a circle is 5.68 x 10^10 feet, then what is its radius?

### Similar Shapes: Volume, Height, Circumference and Area

Please complete the attached and explain steps or formulas: 12 their volumes 15 their volumes 20. Two similar cylinders have bases with areas 16 cm sq. 2 and 49 cm sq. 2 . If the larger cylinder has height 21 cm, find the height of the smaller cylinder. 22. their volumes 23. the areas of their bases 25. their volumes 26.