Special rules for 30-60-90 Triangle

Video Special Rules 30-60-90 Triangle

In this video you will learn...

 

• How to identify the short leg,long leg, and the hypothesis
• How to calculate the short leg given the long leg and hypothesis
• How to find the length of tegth of the long leg given the short leg in a 30-60-90 Triangle
• How to find the hypothenuse given the short leg
• Given the long leg of a 30-60-90 triangle how to find the two other legs

Transcript of the video

 

30=60=90 Triangles

Hi Welcome to MooMooMath. Today we are going to talk about 30-60-90 Triangles. Now there are a set of rules so let me go ahead and share these rules with you, and then we will look at the three different directions that you can use to solve these. The rules for 30-60-90 triangles are as follows. The short leg is always opposite the 30 degree angle. We will call the short leg S. The longer leg is always opposite the 60 degree angle. To find its length takes the short side times the square root of three. Our hypotenuse is always opposite the right angle. To find the hypotenuse take the short leg and multiple it by two. So our short leg is our starting point. Short leg is S Long leg is S times square root of three and hypotenuse is 2 times the short side. Or double the short leg. Let’s go back and work one of these. If we have the short leg all the rules are set. I double it to get the hypotenuse. So I take 2 times 3 so my hypotenuse is 6. Now to get the longer leg I take the short leg times the square root of 3. So I take 3 times the square root of 3 on a calculator or you can leave it as a rational answer. Let’s look at the second method. If given the hypotenuse and you have to figure out the other two sides. The easiest one to get is always the short leg because it is half the hypotenuse. You have a one to two ratio. So if my hypotenuse is 10 then half of that is 5. Once you get the short leg you can get the longer leg, which is the short leg times the square root of three. So it is just five square root three. Now let’s do the hardest direction. When you have the long leg and you’re trying to find the other two sides this is the most difficult direction because you are using the square root of three. Now to go from the short side to long side you multiple by the square root of three. So to go from the long to the short you would divide by the square root of three. So I am taking six divided by the square root of three to get the short leg. To rationalize that multiple the bottom by root three and the top by root three. Square root of three times the square root of three is three. So I have 6 square three over three and the coefficients divide so I have two square root three. Now that is your short leg. You always get the short leg first.  So you double that to get the hypotenuse so two times two is four and the radical stays the same so your hypotenuse is four square root three. So those are your three directions.  Hope this was helpful

How to find the surface area of a prism

How to find the surface area of a prism

Surface Area of a Prism

First, what is a prism? A prism is a rectangular solid with a length, a width, and a height.

To find the surface area you are trying to figure out the total area of the surface of the prism.

The formula for finding the surface area equals

2* (length *width) + 2(length*height) + 2*(height * width) = Surface area of a prism

Find the surface area of a prism with a length of 6 units, a height of3 units and a width of 4 units.

Step 1. Plug in the appropriate units in the formula

              2* (6*4) + 2(6*3) + 2( 3*4)

                   (L*W)      ( L*H)      (H*W)

Step 2. 2*24 + 2*18 + 2*12

Step 3. 48+36+24 = 108 units squared 

Surface area is always squared

 

How to find the volume of a sphere.

How to find the volume of a sphere

What is a sphere? You can think of a sphere as a circle in three dimensions. A sphere is perfectly symmetrical, does not have any vertices or corners, no edges. In addition, all of the points on a sphere are the same distance from the center.

Video  Volume of a sphere

The formula for finding the volume of a sphere equals 4⁄3 πr^3  r =radius

 

 

 

Step 1. Plug 6 in the formula for the radius

 

Step 2. 4⁄3 π6^3 = 4⁄3 π216

 

Step 3. Multiply 4⁄3 x 216=( 4*216 )/3

 

Step 4 864/3 = 288π units^3

Volume of a Rectangular Prism

How to find the volume of a rectangular prism ?

Video Volume Rectangular Prism

 

How do you find the volume of a rectangular prism? A rectangular prism is a three dimensional shape with two rectangular parallel bases. The formula for finding the volume of a rectangular prism is base area times the height. ( Ba * h )

Problem 1. Find the volume of a rectangular prism that has a base of 4 units and 7 units, and a height of 5 units.

 

 

 

Step 1. Use the formula for finding volume which equals V= Ba * h

                                                                                           Ba = Base Area

                                                                                           H = height

Step 2. Find the base area of a rectangle Ba = Length x Width

              4* 7 = 28 units

Step 3. Multiply the Ba * height

               28 * 7 = 140 units cubed

 

 

Finding the Perimeter of an Equilateral Triangle

Finding the perimeter of an equilateral triangle

First, an equilateral triangle is a triangle that has three equal sides. In order to find the perimeter you can add the three sides together or you can just three times the length of one side. 

Video Perimeter of an Equilateral Triangle

Problem 1. Find the perimeter of an equilateral triangle with a side of twelve units.

 

 

Step 1. 3*12 = 36 units

 

Problem 2. Find the perimeter of an equilateral triangle with an altitude of 12 units.

 

 

 

Step 1. Find the length of one side. The altitude creates a 30-60-90 triangle. The altitude becomes the long side of a 30-60-90 triangle. 

 

Step 2. In order to get the side length use (long side of 30-60-90 triangle)/√3 which equals 12/√3

For a review of 30-60-90 rules check here

 Step 3. 12/√3 = 4√3 which is the length of half of the equilateral side length

 Step 4. In order to get the entire length take 2* 4√3 =

Step 5. Now you can use 3 * the side length or 3* 8√3= 24√(3 ) units

 

 

 

Volume of a pyramid

Volume of a pyramid

Volume of a pyramid video

What is a pyramid? A pyramid is a three dimensional figure that has a base and an apex which you can think of as a point.

The formula for volume equals 1/3 * base * height

The base area will depend on the shape of the base.

For example if the shape of the base is a square you would just multiple length times width

If the shape is a triangle you would take ½ base * height to get the base and then multiply this by the height of the triangle.

Check here for a list of base area formulas.

Problem 1. Find the volume of a pyramid with a square base that has a side of 4 units and a height of 5 units.

 

Step 1. Find the base area 4 * 4 = 16 units

Step 2. Multiply base area times height = 16* 5 = 80 units

Step 3. Multiply this by 1/3 ( remember the formula is 1/3 * base * height = 1/3*80 = 26.6 units cubed

Problem 2. Find the volume of a pyramid with a triangle base with a base of 6 units and a height base of 8 units and a pyramid height of 7 units

 

 

 

 

Step 1. Find the base area using the formula ½ base * height

Step 2. ½ * 6 * 8 = 24 units

Step 3. Find the volume using 1/3 * base * height3

Step 4. 1/3 * 24 * 7 = 8*7

Step 5. 8*7 = 56 units cubed

How to find the volume of a prism

Volume of a Prism

Here is the problem worked on video

What is a prism? A prism is a three dimensional figure with parallel bases.

The formula for finding the volume of a prism is just length X width X height

Let’s look at a problem

What is the volume of a prism with a base of 6 by 7 units and a height of 9 units?

 

Step 1 Plug the units into the formula l * w * h    6*7*9

Step 2. Multiply 6*7*9 = 42*9

            42*9 = 378 units cubed (volume is always cubed)

 

Now a prism can be have different shaped bases. The key to finding the volume is to find the base area formula for that shape and then multiply this by the height.

Check here for a list of most base area formulas

For example, if you had a triangle shaped prism you would use ½ base x length x height to find the volume of the triangular prism

 

 

Finding the volume of a Cone given the slant height

           Finding the Volume of a Cone given the Slant Height

 Here is the video that works the volume of a cone using the slant height

Problem 1 Find the volume of a cone with a slant height of 9 units and a diameter of 12 units.

 

 

 

 

 

 

Step 1 Find the radius by taking ½ of the diameter

  ½ * 12 =6 units

 

Step 2 Notice that the slant height is part of a right triangle. We need the height to figure out the volume. The formula for volume = πr^2*h

 

 

 

 

 

 

 

Step 3. We can use the Pythagorean Theorem to find the height.

The radius becomes the leg of the right triangle

The slant height becomes the hypotenuse of the right triangle

So for the height I will use a^2+ 6^2= 9^2

Step 3a a^2+ 36= 81 

 

Step 3b a^2= 45  

 

Step 3c  a= √(45 ) this simplifies to 3√5 

 

Step 4 So now use the volume formula πr^2*h 

 

Step 4a  π6^2*3√5

 

Step 4b  36π* 3√5 

 

Step 4c 108√5 π units^3 is the volume

Finding the volume of a cone

The formula for the volume of a cone = 1⁄3 πr^2*h

h =height

r = radius

 Please note πr^2*h is actually the volume of a cylinder that has the same height and radius. However it takes three cones to fill up a cylinder so that is why you multiply by 1/3

 Video for Volume of a Cone

Problem 1. Find the volume of a cone with a height of 10 and a radius of 6 units.

 

 Click the cone above for additional information

Step 1.  Plug your numbers into the formula. 

              1⁄3 π6^2*10  

 

Step 2 Simplify 1⁄3 π6^2*10  

 

Step 2b 1⁄3 π36*10

 

Step 3 1⁄3 x 36⁄1 π x 10  

 

Step 3b 12π x 10=120π units^3 (Please note volume is always cubed)

Volume of a Cube

Finding the Volume of a Cube

Volume of a Cube Video

The formula for the volume of a cube equals one side raised to the third power.

Volume = s^3

 

Problem 1 What is the volume of a cube with a side length of 3 units?

 

 

 

 

Step 1 3*3*3 or 3^3 = 27 units ^3

 

Problem 2 This problem is slightly more challenging.

Find the volume of a cube with a diagonal length of 6 units.

 

 

 

 Step 1. Use your 45-45-90 rules of a triangle to find the length of one side.

 

Step 2. The diagonal cuts the side of the cube into two 45 degree angles. See below.  

 

 

 

 Step 3. The side length will equal Hypotenuse  √2 = 6√2

 

Step 4 Rationalize. 6/√2 * √2/√2 = (6√2)/2 =3√2 = one side

 

Step 5 Now I just need to cube the side (3√2 )^3 = 3*3*3 *√2*√2*√2

3*3*3 *√2*√2*√2 = 27*2√2

27*2√2 = 54√(2 ) units^3

You may also enjoy...

Volume of a cylinder video

Volume of a Cylinder

Finding the volume of a cylinder

 Video Volume of a Cylinder

The formula for finding the volume of a cylinder equals πr^(2 )* h  

The πr^(2 ) equals the area of the base and the h equals the height of the cylinder

 

 

Problem 1 Find the volume of the cylinder that has a diameter of 6 units 

                   and a height of 8 units.

 Step 1 Find the radius by dividing the diameter in half. 

            6 /2 = 3 units which is r or the radius

 Step 2. Plug the radius into πr^(2 ) which equals π3^(2 ) 

             9π = base area of the cylinder

 Step 3. 9π * height which equals 9π * 8 = 72π units^3 (remember that       

             volume is always units cubed)

How to find the Area of a Parallelogram

How to find the Area of a Parallelogram

Video How to find the Area of a Parallelogram

 

The formula for finding the area of a parallelogram equals base times height or A=b*h

Please note, the height is not the length of a side but is the distance from base to base. Please see the drawing below.

 

 

 

 

 Problem 1. What is the area of a parallelogram with a base of 8 units and sides of 5 units and a height of 4 units? 

 

 Step 1. Multiply the base of 8 units times the height of 4 units.

 Step 2. 8*4 = 32 units squared

 

Problem 2. What is the area of a parallelogram that has a side of 6 units, a base of 10 units and an angle measure of 60 degrees?

 

 

 

 Step 1. Find the altitude. If you draw a vertex straight down it creates a triangle. See picture below. The triangle is a 30-60-90 triangle. I can use the 30-60-90 rules to find the height of the parallelogram.

The rules of a 30-60-90 are as follows:

 Short leg =x 

Long leg =  x√3

Hypotenuse = 2x

 

 

 

Step 2. The length of the side leg equals 6 and is my hypotenuse in my triangle   

Therefore, 6 =2x so x =3

 Step 3. Now that I know x I can find the height by finding the length of the long leg 

long leg=3√3 

 Step 4. Use area equals base * height  or A =b*h

            10 * 3√3 = 30√3 units^2

 

Top Ten Properties of a Parallelogram

Top Ten Properties of a Parallelogram

1. A parallelogram is a quadrilateral.

2. The opposite sides of a parallelogram are parallel.

3. The opposite angles of a parallelogram are congruent.

4. The opposite sides of a parallelogram are congruent.

5. The perimeter of a parallelogram equals 2 (base + side)

6. The area of a parallelogram equals base * height

7. The adjacent angles of a parallelogram are supplemental.

8. The diagonals of a parallelogram bisect each other. (In other words the cut each other exactly in half.

9. A parallelogram has four sides

10. The sides of a parallelogram must be straight.

Video Parallelogram

Video Area of a Parallelogram

Video Perimeter of a Parallelogram

Top Ten Properties of a Rhombus

Top Ten Properties of a Rhombus

1. A rhombus falls in the quadrilateral family so it has four sides.

2. A rhombus is also a parallelogram and has the properties of a parallelogram

3. A rhombus has congruent opposite sides that are parallel

4. All four sides of a rhombus are congruent

5. Opposite angles in a rhombus are congruent.

6. The diagonals of a rhombus are perpendicular.

7. The diagonals cross and make four right angles.

8. The perimeter of a rhombus equals 4 x one side

9. The area of a rhombus equals ½(Diagonal 1* Diagonal 2) or Base * Height (not the length of the side but the altitude)

10. The diagonals create four congruent triangles that reflect upon each other

Video Property of a Rhombus

Video Area of a Rhombus

Video Perimeter of a Rhombus

.