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# Geometry

#### Math Review of Representing Solid Figures

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##### Overview

Three-dimensional solid figures can be represented by the two-dimensional pattern of polygons that create them. The pattern, called a net, is a visual representation that illustrates the formula for the surface area of the three-dimensional figure. If the net were folded, it would produce that figure.

##### Representation of Cylinders

The net for a cylinder consists of two circles adjoining a rectangle. This relates to the formula for the surface area of a cylinder; 2 πr2+ 2πrh. The 2 πr2 is the formula for the area of the two circles. In order for the rectangle to fit around the circumference of the circle, the width of the rectangle is 2πr. The height of the rectangle is the height of the cylinder. Since the formula for the area of a rectangle is length multiplied by width, the area of the rectangle will have the measurement 2πrh.

Representation of Rectangular Prisms

The net for a cube consists of the six square faces that make up the cube. A square has the same length, width, and height. It is a special type of a rectangular prism, also known as a rectangular cuboid. A rectangular prism also has 6 faces, in three parallel pairs that meet at right angles. The top and bottom faces are congruent, as are the two other pairs of opposite sides. The formula for the surface area of a rectangular prism is 2(lw +wh +lh), which the net illustrates perfectly.

##### Representations of Pyramids

Pyramids are solid figures with triangular faces that meet at a single point called an apex, and a polygon base. A tetrahedron is a special type of pyramid with 4 triangular faces, and a regular tetrahedron, with all triangles equilateral and congruent, is a Platonic solid. Another type of pyramid has a square or rectangular base and three triangular sides. The net that illustrates the pyramids has the base bounded by triangles on each side. A regular tetrahedron has a net with all four triangles inside a larger triangle.

##### Representations of Other Figures

Many other solid figures can be represented by their nets. For example, a cone with a circular base is represented by the circular base adjoined by a quarter circle. Solid figures have been extrapolated into more than three dimensions. A tesseract is a four-dimensional figure with three-dimensional faces. It has been used in surreal art, science fiction, music, and popular culture.

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### Overview

Lines representing graphs of linear equations have slopes, defined as the change in y/change in x. Parallel lines have the same slope, but no solutions in common. Perpendicular lines intersect at one point to form right angles.

### Finding Slopes of Parallel Lines

Imagine a line graphing the equation y = 2x +3. Using the slope-intercept formula y = mx +b, the equation has a slope of 2. Suppose another line exists graphing the equation y = 2x +5. Using the slope –intercept formula y = mx +b, that equation also has a slope of 2. The lines have no points in common, and they both have the same slope. Those lines are parallel. Lines that are not vertical are parallel if they have the same slope. Vertical lines, which have an undefined slope, are parallel by definition, as they do not intersect.

### Finding Slopes of Perpendicular Lines

Lines that are perpendicular intersect at one point to form right angles, measuring 90 degrees. Not all lines that intersect are perpendicular. Vertical lines are perpendicular to horizontal lines by definition. For example, lines represented by the equations y = -2 and x = 4 are perpendicular. Lines that are not vertical are perpendicular only if the product of their slopes equals -1. Suppose a line y = 2/3x +1 and another line y= -3/2x +2. They will meet at one point and be perpendicular to each other. The product of their slopes (2/3) (-3/2) equals -1.

### Are Lines Parallel or Perpendicular?

Sometimes math problems present the coordinates of points along a line, rather than the slope-intercept formula. The slopes of the lines can be calculated using the slope formula, then compared. If the slopes are the same, they are parallel. If the product of their slopes equals -1, they are perpendicular. Suppose one line passes through the points (0, 2) and (-3, -3) and another line passes through the points (4, 2) and (1, -3). The slope of the first line is given by the equation for the slope (-3 -2)/ (-3-0) or -5/-3. The slope of the second line is given by the equation (-3-2)/(1-4) or -5/-3. The lines are parallel.

### Geometry Applications

The algebra of slopes of parallel and perpendicular lines can be applied to geometric forms. Since parallel lines have the same slope, lines of a parallelogram can be shown to be parallel. Suppose the coordinates of point A are (0, 2); point B (4, 2); point C (1, -3), and point D (-3, -3). A figure drawn on ABCD is a parallelogram, because lines AD and BC are parallel, and lines AB and CD are parallel.

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### Overview

In the winter sport of ice hockey, players, sticks, and the puck all move at a rapid pace. Sports scientists know that the apparent chaos of the game is governed by math and physics principles such as laws of motion; angles, vectors; transfer of potential energy to kinetic energy; and force, impulse, and collision.

### Three Laws of Motion

The players, sticks, and the puck observe Newton’s Three Laws of Motion like any other moving object. For example, a player standing on the ice will stay in the same position because of inertia, unless something happens to change his position, like a body check. Similarly, a hockey puck will travel in a straight line at the same speed, unless a player strikes it with his stick or a goalie stops it. The Second Law of Motion is described mathematically by the equation F=ma, where F stands for force, m, mass, and a, acceleration. Hockey players apply force to the puck with blows from the hockey stick. A typical 100 lb. force of a slapshot creates enough energy to propel the puck at high velocity toward a goal. In the Third Law, the Law of Conservation of Motion, for every action, there is an equal and opposite reaction, so that the force applied to the puck sends it moving in another direction.

### Angles and Vectors

A hockey rink consists of many geometric shapes, such as face-off circles, a cylindrical puck, and semicircular zones for goals. The angle between the blade of the hockey stick and its shaft is an obtuse angle of about 135°. When the puck hits the board at the side of the rink, the angle of incidence that it travels in equal to the angle of reflection that it makes when it veers away from the board. The moving puck has a velocity vector, as do the moving players and their sticks in constant motion. The vector has both speed and direction, and can be calculated.

### Potential and Kinetic Energy

The Law of Conservation of Energy states that energy cannot be created or destroyed, just change form. Thus, the potential energy from the motion of the hockey stick in the player’s hands is transferred to the kinetic energy of the moving hockey puck as it accelerates. Kinetic energy is energy of motion, while potential energy is the power that will propel the puck into motion.

### Force, Impulse, and Collision

Force is the push or pull needed to accelerate an object (such as a hockey puck) by changing its velocity. It can be constant, or for a short time. During a slapshot, the blade of the hockey stick is in contact with the puck for less than 1/200 of a second. The measure of that impulse, or force multiplied by change in time, shows that brief transfers of energy can result in motion and goals scored. Collisions can be either elastic, where an object retains force and energy, or inelastic, where the object’s energy is dissipated, such as when a goalie traps the oncoming puck, preventing a score.

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### Overview

The matchup between NFC West Champion Seattle Seahawks and AFC East Champion New England Patriots promises thrilling gridiron action during Super Bowl XLIX. Behind the scenes, principles of math and physics determine the shape of the ball itself, the running game, the passing game, and the kicking game.

### Footballs and Ellipses

Many sports from basketball to tennis, golf, and baseball are played with spherical balls, based on circles. Football and rugby are played with balls that are based on ellipses. An ellipse has two axes, a long major axis and a short minor axis. When a solid is made by rotation around the major axis, it is called a prolate spheroid, the shape of a football.

### Passes and Gyroscopes

One of the advantages to an elliptical ball is that it can be thrown accurately for long distances. Skilled quarterbacks such as Russell Wilson or Tom Brady throw accurate spiral passes to their intended receivers. The tighter the spiral, the more the spinning ball can resist drag. It acts like a gyroscope, keeping the ball on track along its intended path. If the pass is wobbly, the ball slows down because there is more area for air resistance.

### Pythagorean Interceptions

Suppose a running back catches the pass and runs downfield. The distance to the end zone is 40 yards. The closest defender to the running back is 30 yards away at a right angle. In order to get close enough to tackle the runner and intercept the ball, the defender will have to run at an angle. The path will resemble a right triangle, and the distance the defender will have to run can be solved by using the Pythagorean Theorem. Let the side of triangle for the ball carrier be represented by a, and the side of the triangle from the carrier to the defender by represented by b. The path to intercept can be represented by c. Using the Pythagorean Theorem, a2 +b2= c2. In the example, 1600 +900 = 2500. The defender will have to run a 50 yard diagonal to catch up to the ball carrier.

### Parabolic Punts

When a kicker punts the football, the ball follows a parabolic arc. It follows a vector with speed and distance. At the highest point of the parabola, the ball reaches its maximum height and speed. It begins to decline due to the force of gravity. The greater the force of the kick, the higher the highest point will be, and the farther the ball will travel before it bounces.

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### Overview

Solid objects have both volume and surface area. Formulas for calculations can be expressed using algebra, especially if the measurements of sides are stated by variables. The ratio of volume to surface area is important in applications in chemistry and biology.

### Measurement of Volume

The volume of a solid is the amount of space that it contains. For example, the volume of a rectangular prism is measured as its length times width times height. In math language, V=lwh. The volume of a cylinder is π times the radius squared times the height of the cylinder, or V=πr2h. The volume of a cone is 1/3 πr2h, very similar to the formula of the volume of a cylinder. The volume of a pyramid is 1/3 lwh, which is similar to the formula for the volume of a rectangular prism, like a cube or a brick.

### Measurement of Surface Area

The surface area of a solid object is the total area contained in the two-dimensional surfaces of the solid object. (Think of the amount of wrap it would take to completely cover all surfaces without overlapping.) For example, the surface area of a rectangular prism is 2(lw +wh +lh). A cube has 6 faces, a top, bottom, and 4 sides. The surface area of a cylinder has 2 πr2+ 2πrh, because a cylinder has a top circle, a bottom circle, and the circular surface. The surface areas of cones and pyramids are more complex to calculate and depend on the orientation of the object.

### Expressions Using Polynomials

Suppose the solid figure has measurements expressed in variables rather than constant numbers. For example, a rectangular prism has a length of c+1, a width of 2c, and a height of c-5. The volume of the rectangular prism would be (c +1)(2c)(c-5). The expression could be rearranged to (2c)[(c +1)(c-5)] using the Commutative and Associative Properties. The expression (c +1)(c-5) could be expanded to c2-4c -5. Multiply each part by 2c for 2c3-8c2 -10c.

### Applications of Surface Area

The surface area of an object is important to various types of chemical reactions. For example, iron ground into a fine powder has a larger surface area than the same amount of iron in a block. It catches fire easily, while the block of iron is highly stable.

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### Overview

Students can connect algebra to geometry by expressing geometric inequalities in compound inequality form. Since the sum of the lengths of any two sides of a triangle is greater than the length of the third side, there are three inequality statements. All three inequality statements must be true for three line segments to form a triangle.

### The Triangle Inequality

The perimeter of any polygon is the sum of the measurements of its sides. Therefore, the perimeter of a triangle is the sum of the measurements of its three sides. Suppose a triangle has 3 sides with measurements a, b, and c. In Euclidean geometry, there are 3 basic inequality statements for any triangle: a +b > c; b +c > a; a +c > b.

### Right Triangles

In order for a triangle to be a right triangle, one angle must be a 90 degree angle. The sides of the triangle are in special relationship with one another, expressed by the Pythagorean Theorem, such that a2+ b2 = c2. Right triangles are a special case of the Triangle Inequality, such that the measure of the hypotenuse c is greater than the measure of either leg a or leg b. However, the measure of c must also be less than the sum of both legs. Suppose an isosceles triangle ABC is constructed so that AB and BC are equal. By definition, the base angles B and C are also equal. If an altitude CD is constructed, it will divide ABC into two right triangles, ACD and CBD. Euclid showed that both AD and CD are shorter than the hypotenuse AC. According to the Triangle Inequality, the measure of AD + CD must be greater than AC.

### Non-Triangles

A figure made up of three line segments is not a triangle if any of the inequalities are untrue. Therefore, all three conditions must be satisfied such that a + b >c; a + c >b, and b + c =a. If any of the elements are untrue, the entire statement is untrue. Students can use the Triangle Inequality as a test for geometric figures, if given the lengths of any three line segments.

### Special Triangles and the Triangle Inequality

One of the most famous of the special triangles is the 3-4-5 triangle. It got its name because one side measures 3 units, one side measures 4 units, and one side measures 5 units. It is also a right triangle, and demonstrates the Pythagorean Theorem, because 32+ 42= 52. It also follows the Triangle Inequality because 3 +4 & gt;5, 3 + 5 >4, and 4 + 5 >3. The sides of the 3-4-5 triangle follow an arithmetic progression. In algebraic terms, sides follow a pattern of a, a + d, and a + 2d. Let a be 3, then 3 + 1 is 4, and 3 + (2·1) = 5. It can be shown that any triangle that follows an arithmetic progression, and fulfills all three inequalities of the Triangle Inequality is similar to the 3-4-5 triangle. Suppose that three line segments follow the arithmetic progression 6, 7, 8. They will form a triangle, because 6 + 7 >8; 7 + 8 >6; 6 + 8 >7. They are also similar to 3, 4, 5 because 6-3 is 3, 7-3 is 4 and 8-3 is 5.

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### Overview

The relationship of the length of the legs of the right triangle to the hypotenuse is expressed by one of the most famous equations in geometry, the Pythagorean Theorem. Deriving the measurement of the sides of the right triangle was one of the earliest uses of irrational numbers.

### Right Triangles

Right triangles have one 90-degree angle. The legs of the right triangle form the 90 degree angle, and the side opposite the 90-degree angle is the hypotenuse. The relationship between the lengths of the sides is given by the Pythagorean Theorem, a2 + b2 = c2. That means that if the length of the hypotenuse and one leg is known, the length of the other leg can be derived. Suppose that the length of the hypotenuse is 5 cm and one leg is 3 cm. Using the Pythagorean Theorem, 32 + b2 = 52, or 9 + b2 = 25. So 25 – 9 = b2, or 16 = b2. Taking the square root of both sides, b = 4 cm.

Figure 1: The Pythagorean Theorem.

### Irrational Numbers

The followers of Pythagoras soon found that there was a big scar in the perfect geometric world. This was such a shock to them that they took out one Hippasus of Metapontum (who used irrational numbers) and drowned him at sea, or so the story goes. They couldn’t drown the idea so quietly, because the measurement of the side is not always a rational number. Suppose one leg of the triangle measures 10 feet and the other leg measures 15 feet. What is the length of the hypotenuse? 102 or 100 + 152 or 225 equals 325. The square root of 325 (in symbol form √325) is not a perfect square. It is a little more than 18, 18.03 to two decimal places.

### Using Radicals to Measure Right Triangles

Right triangles are found all over, in the angle that a ladder makes with a building, the height of a tree, baseball diamonds and soccer fields. The applications of the Pythagorean Theorem and right triangles can be used to approximate and measure the lengths of the sides of these triangles. (Trigonometry is an entire branch of mathematics that gives more detail.) For example, suppose a ladder is 28 feet long. When the ladder is used against a building, it forms a right triangle. If the ladder is placed so the distance between the wall of the building and the base of the ladder is 2 feet, how high up will the ladder reach? One leg of the triangle is 2 feet, and the hypotenuse is 28 feet long. Using the Pythagorean Theorem, 22 + b2 = 282, or 4 + b2 = 784. If b2 = 784 – 4 = 780, then b = √780, or 27.93.

Figure 2: A ladder against the side of a burning building forms a real-life right triangle.

### Right Triangles Made of Air

Sometimes the right triangle is actually an “air triangle”. The applications of the Pythagorean Theorem and radical expressions still hold. Suppose a plane is at a height of 5000 feet when it approaches an airport. At 5000 feet, the line of sight between the plane and the terminal is 38,000 feet. How far is the horizontal distance between the plane and the airport? This is truly an “air triangle”, as one leg of the triangle is the distance between the plane and the ground, or 5000 feet. Let that be a2. The line of sight between the plane and a point of ground at the terminal is the hypotenuse, or 38,000 feet. So 38000-5000 will be the other leg of the air triangle, or 33000. The square root of 33000 is 181.66.

Figure 3: Right triangles abound in the friendly skies.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Omaha, NE: visit Tutoring in Omaha, NE

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### Overview

Roulettes are special types of curves that are made when one curve is rotated around another by a fixed point. They are in Euclidean space, and can be described by calculus equations. Roulettes, cycloids, and other variations of circular movement have applications to pendulums, gears and subatomic particles.

### General Definition of a Roulette

A roulette is a type of curve. It is the path of a fixed point in relation to one curve as that curve travels along a straight line or another curve. The simplest form of a roulette is a cycloid. Imagine a circle that looks like a wheel with one spoke from the center to the edge of the wheel, one radius long. The path traced as that circle travels along the line is a cycloid.

### The Cycloid Family

If the roulette is formed by a circle travelling outside the circumference of another circle it is called an epicycloid, and if the circle is traveling inside the circumference it is called a hypocycloid. Bernoulli in the late 17th century proposed a challenge, which was solved by him, Newton, and Leibniz. The challenge was to find the fastest path a point will take to travel moving without friction, using gravity.

### Playing with Cycloids

Mathematicians play with cycloids using differential calculus equations. The path a cycloid travels can be measured by an equation involving the differential of x divided by the differential of y all squared in relation to the radius of the circle.  Many children of all ages have played with a toy called a Spirograph that essentially generates cycloids of various types, using pens of different colors.

### Applications

Huygens, living in the late 17th century, used the cycloid to make a pendulum that would be the most efficient type ever devised, in order to improve timekeeping and measurements aboard ship. The motions of gears also follow paths described by the cycloid family. In order for gears to regulate machinery, they need to touch and move freely, along cycloid paths. Engineers calculate the speed of gears; the location, number, and size of teeth (fixed points); and the size of the gears themselves. The path of electrons in electric and magnetic fields also follow cycloid movements.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Savage, MN: visit: Tutoring in Savage, MN

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### Overview

There are infinitely many regular polygons. However, there are a fixed number of regular polyhedra, called Platonic solids. Mathematicians use the definitions of the regular polygons and the characteristics of solid objects to arrive at their conclusions.

### Characteristics of Regular Polyhedra

Regular polygons, such as equilateral triangles, squares, equilateral pentagons, hexagons, heptagons, octagons, and so on, have all sides and angles equal. There are an infinite number of them in two-dimensional space. In three-dimensional space, solids with regular polygon faces are called regular polyhedra.

### Triangular Faces

In order for a solid to have regular triangular faces, the vertices of the triangles must meet at a point to make a corner. If three triangles meet at each vertex, there are 4 faces, called a tetrahedron. If four triangles meet at each vertex, there are 8 faces, called an octahedron. If five triangles meet at each vertex, there are 20 faces, called an icosahedron. No more triangles can form a corner at a vertex.

### Square and Pentagonal Faces

There is only one regular solid made of squares, the cube. Only 3 squares can meet at each point to form a cube. A cube has 6 faces, a top, bottom, and 4 sides. A pentagon has five sides. A regular pentagon has angles that have the same measurement, with sides that are the same length. If three pentagons meet at each vertex, there are 12 faces, called a dodecahedron.

### Other Polygons and Other Dimensions

Mathematicians since Plato have shown that there are only those five regular polyhedra in classical three-dimensional space. In higher dimensions than 3, analogues to the polyhedra are called polytopes. Their faces also meet at vertices, and they are symmetrical along various axes. In the 19th century, a mathematician found that there were 6 regular polytopes in four dimensions. One of those figures is a four-dimensional cube, or a cube of a cube, called a tesseract, familiar to readers everywhere. (The cube of a cube of a cube is a penteract.) In dimensions higher than 4, there are only 3 regular polytopes, that correspond to the tetrahedron, cube, and octahedron. The n-dimensional tetrahedron is called a simplex. The n-dimensional cube is also called a hypercube, familiar in science fiction. The n-dimensional octahedron is called a cross-polytope with multiple cells.

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### Overview

The quantity represented by the Greek letter tau (6.283185…) is the proportion of the circumference of a circle to its radius. The use of tau simplifies the relationships between angles and circles. These relationships underlie other branches of mathematics and applications to science.

### Special Angles and Tau

The relationship of tau and the circle is such that tau is one turn of the circle. In fact, the Greek letter tau is the first letter in the word tornos, which the English word “turn” is from. Recall the special angles of 30o, 45o, 60o, 90o, 135o, 180o, and 270o with relationship to the circle. Those special angles can be represented in radians using tau τ, in such a way that their measure is expressed as a fraction of the entire circle.

### Tau and the Unit Circle

The unit circle has a radius as 1, and the most fundamental relationships in trigonometry set sin θ and cos θ as points on the unit circle. Therefore, if the sine wave starts at zero, the value in Cartesian coordinates is (1, 0). At the highest point of the wave, sin θ has coordinates of (0, 1). The sine wave is again at equilibrium at coordinates (0, 0) and at its lowest point at coordinates (-1, 0). The cosine wave is the opposite of the sine wave, but it also has a period equal to the circle constant.

### The Tau Manifesto

Mathematician and physicist Michael Hartl is one of the leaders of the Tau Day movement, to show the advantages of a widespread adoption of tau as the circle constant. The constant τ is related to the circle in a more intimate way than π, and it is more natural to use, in special angles, the unit circle, and other mathematical expressions.

### Tau Is Easier than Pi

The tau constant simplifies special angles, sines, and cosines. Tau is exactly equal to 2π, so it can be used in any equation or mathematical expression that uses 2π. For example, the normal distribution has a factor of 2π in its definition. Some advanced mathematics equations that refer to space, time, and polar coordinates as integrals go from a limit of 0 to a limit of 2π, or τ. Even Euler’s formula, e=cos θ + i sin θ, becomes Euler’s identity when the value of θ is set at τ. Then, e =1. Without going into the very complex mathematical argument, the small letter e stands for the natural logarithm and the small i for the imaginary factor in complex numbers. The number 1 is unity, and τ brings it all together.

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