Brief Project Overveiw
In this project we learned many ways to measure our world. We learned the Pythagorean Theorem and then went into the distance formula, the equation of a circle, the unit circle, the definition of sine, cosine and tangent, right-triangle trigonometry, area of polygons, area of a circle, and then volumes of many different shapes. We then got to make our own volume projects to complete.
The Mathematical Consepts
Pythagorean Theorem
The Pythagorean Theorem is the equation used to solve for the hypotenuse of a right triangle. The equation is a^2 + b^2 = c^2. It was found by Pythagoras.
The Distance Formula
The distance formula is the formula to tell the distance of a line on a grid. The formula is the square root of the (second x subtracted by the first x)squared plus (second y subtracted first y) squared. The formula was derived from the Pythagorean theorem and using it to find line lengths from the center point. Then they added the second point to put that point at the center from wherever it was on the grid.
Equation of a Circle
The equation of a circle is a equation that tells you the radius of a circle by using the center point and a point on the edge of the circle. It also resembles the distance formula.
The Unit Circle
The unit circle is a circle with a radius of one. It is used to help make simpler calculations in trig by using the unit circle to find the side and then multiplying it by the scale factor to get the side of the triangle you want. It uses proportions. and the angle used is called Theta.
Sine, Cosine, and Tangent
Sine, cosine, and tangent are all relationships between two sides on a triangle using a angle to tell what triangle it is. Sine is the ratio between the side opposite to the angle and the hypotenuse. Cosine is the ratio between the side adjacent to the angle and the hypotenuses. Tangent is the ratio between the side opposite of the angle and the side adjacent to the angle. sin(θ), cos(θ), and tan(θ).
Right-Triangle Trigonometry
Right triangle trigonometry is using sine, cosine, and tangent to find different side lengths on a right triangle. You use the ratios in proportions so you can find the missing sides. 98(hyp)*sine(50)= opp.
Area of Polygons
This one is pretty self explanatory. We were taught the different equations to get an area. First was a square/rectangle with the equation length times width. Next, came the equation for a triangle base times height times a half. that comes from the square/rectangle equation because a triangle is half of a square/rectangle. We also did a POW where we found the area of any shape using a grid and things called interior pegs and border pegs. The area equation for any regular polygon is long equation but the steps are to find the angles and make triangle using the center and the veritcies as points on the triangle and then use trig to find the height and find the area.
Area of a Circle
The area of a circle equation is πr^2. There are a few different ways to find this equation and one is inscribing regular polygons inside a circle. as you got closer to a circle with the polygons you know the area of the polygon is the number of sides multiplied by the triangles you would get like I states previously. that gives you n(# of sides) #(1/2bh). this can be changed into 1/2h*(nb) where nb is the perimeter. Now this equation can be roughly put as 1/2r*(2πr). simplified it is πr^2.
Volume of Different Shapes
We were taught many different volume equations like the one for any prism V=b*h. Base is whatever shape the base is be it triangle or circle's area. We got that by the volume of a cube/rectangular prism witch is lwh or b(lw)h. Then we found the volume of a pyramid was 1/3bh. It is that because three pyramids make a square. the same formula goes for cones. Next was a sphere which is 4/3πr^3. We watched a video explaining how it was found and it had something to do with cylinders the spheres and 2 cones.
Cavalieri's Principle
Cavalieri's Principle states that if two objects with equal height have the same cross section then they have the same volume. The simple way to explain it is you have a stack of coins that has a certain volume. Now if you put a slant in the coins you will still have the same volume as before because you didn't add anything. In the original description of the principle the cross section would be like taking a knife and cutting the two objects at the same height. Using that you can see if they both make the same shape.
My Volume Project (DYOP)
Description
The object I chose to measure is a water tower near me. The way I measured it was by using points a bit away to find the height and the radius. Then I solved the equation for the cylinder.
The Math
Distance measurements: I measured those angles and used the law of sines. The equation was sin(93)/(sin(3)/30). After I found that the C line is 572.43 ft. Height measurement: I measured the angle and used the law of cosines to find that the height was 69.89 ft. the equation was sqrt of 572.43^2+572.43^2-(2*572.43^2)*cos(7). Radius measurements: I measured the angle and used this equation (572.43*tan(10) and found that the radius to be 100.94 ft. The volume is 2237016 ft^3. 69.89*𝝅100.942.
DYOP Relfection
This project was really fun when I actually started doing it. I had to do it at home so I had not much to do in school. I went on a hike to get to the place where I would take the measurements. I had to use a different water tower because when we got to the place where the first one was they had removed it. This project was just as fun as the other ones we have done. Some challenges I face was finding what water tower to measure. I had to change because as I said before it was being taken out. I wish I had a better way to measure the degree because a small protractor may not have been the most accurate.
Reflection for Measuring Your World
This was a fun project to do. I learned some new things from this project. One would be arcsine, arccosine, and arctangent. They help you get the angle using the side lengths. There were many things that just felt like a repeat to me like sine. cosine, and tangent. For the things I already knew I had to use the habit describe and articulate to tell the other people at my table what I did. For the things I didn't know I used the seek why and prove but less of the proving. I just had to figure out how the equation or tool worked then I could use it. It was like this for the time heading up to the DYOP. However during the DYOP i used the habit Be Confident, Patient and Persistent. I had to be confident that my calculations were correct. When I did the calculations I had some conflicting results with the measurement on google maps that I didn't know how to change because I would retry the calculations and I would be right. But, even before that I had to be confident in my measuring and steps to find the answer. I didn't have the most accurate of tools so I took the measurements as best I could. I had to be patient and not just rush through my measurements. Overall, I had fun in this project and really liked learning more trig.