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Geometry Formulas: Surface Area & Volume

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**A formula is just a set of instructions**

A formula is just a set of instructions. It tells you exactly what to do! All you have to do is look at the picture and identify the parts. Substitute numbers for the variables and do the math. That’s it!

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**Rectangle A = b(h) Triangle A = b(h) 2 Circle A = π r² radius**

Let’s start in the beginning… Before you can do surface area or volume, you have to know the following formulas. Rectangle A = b(h) Triangle A = b(h) 2 Circle A = π r² radius C = πd diameter

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**VOLUME FORMULAS Rectangular Prism v=l x w x h (lwh) Cube v= s ^3**

Cylinder v= x r^2 x h PI =3.14

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TRIANGLES You can tell the base and height of a triangle by finding the right angle:

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CIRCLES You must know the difference between RADIUS and DIAMETER. r d

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**Either method will gve you the same answer.**

Let’s start with a rectangular prism. Surface area can be done using the formula SA = 2 lw + 2 wl + 2 lw OR Either method will gve you the same answer. you can find the area for each surface and add them up. Volume of a rectangular prism is V = lwh

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Example: 7 cm 4 cm 8 cm Front/back 2(8)(4) = 64 Left/right 2(4)(7) = 56 Top/bottom 2(8)(7) = 112 Add them up! SA = 232 cm² V = lwh V = 8(4)(7) V = 224 cm³

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To find the surface area of a triangular prism you need to be able to imagine that you can take the prism apart like so: Notice there are TWO congruent triangles and THREE rectangles. The rectangles may or may not all be the same. Find each area, then add.

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**3. The back rectangle is different**

Find the AREA of each SURFACE 1. Top or bottom triangle: A = ½ bh A = ½ (6)(6) A = 18 2. The two dark sides are the same. A = lw A = 6(9) A = 54 Example: 8mm 9mm 6 mm mm 3. The back rectangle is different A = lw A = 8(9) A = 72 ADD THEM ALL UP! SA = 216 mm²

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**SURFACE AREA of a CYLINDER.**

Imagine that you can open up a cylinder like so: You can see that the surface is made up of two circles and a rectangle. The length of the rectangle is the same as the circumference of the circle!

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**EXAMPLE: Round to the nearest TENTH.**

Top or bottom circle A = πr² A = π(3.1)² A = π(9.61) A = 30.2 Rectangle C = length C = π d C = π(6.2) C = 19.5 Now the area A = lw A = 19.5(12) A = 234 Now add: = SA = in²

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**There is also a formula to find surface area of a cylinder.**

Some people find this way easier: SA = 2πrh + 2πr² SA = 2π(3.1)(12) + 2π(3.1)² SA = 2π (37.2) + 2π(9.61) SA = π(74.4) + π(19.2) SA = SA = in² The answers are REALLY close, but not exactly the same. That’s because we rounded in the problem.

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**The formula tells you what to do!!!!**

Find the radius and height of the cylinder. Then “Plug and Chug”… Just plug in the numbers then do the math. Remember the order of operations and you’re ready to go. The formula tells you what to do!!!! 2πrh + 2πr² means multiply 2(π)(r)(h) + 2(π)(r)(r)

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**Cylinder V = (πr²)(H) Volume of Prisms or Cylinders**

You already know how to find the volume of a rectangular prism: V = lwh The new formulas you need are: Triangular Prism V = (½ bh)(H) h = the height of the triangle and H = the height of the cylinder Cylinder V = (πr²)(H)

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**V = 162 mm³ V = (½ bh)(H) V = ½(6)(6)(9) Volume of a Triangular Prism**

We used this drawing for our surface area example. Now we will find the volume. V = (½ bh)(H) V = ½(6)(6)(9) V = 162 mm³ This is a right triangle, so the sides are also the base and height. Height of the prism

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**V = 864 cm³ V = (½ bh)(H) V = (½)(12)(8)(18) Try one:**

Can you see the triangular bases? V = (½ bh)(H) V = (½)(12)(8)(18) V = 864 cm³ Notice the prism is on its side. 18 cm is the HEIGHT of the prism. Picture if you turned it upward and you can see why it’s called “height”.

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**V = (πr²)(H) V = (π)(3.1²)(12) V = (π)(3.1)(3.1)(12) V = 396.3 in³**

Volume of a Cylinder We used this drawing for our surface area example. Now we will find the volume. V = (πr²)(H) V = (π)(3.1²)(12) V = (π)(3.1)(3.1)(12) V = in³ optional step!

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**V = (πr²)(H) V = (π)(4²)(10) V = (π)(16)(10) V = 502.7 m³ Try one:**

d = 8 m V = (πr²)(H) V = (π)(4²)(10) V = (π)(16)(10) V = m³ Since d = 8, then r = 4 r² = 4² = 4(4) = 16

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**Here are the formulas you will need to know:**

A = lw SA = 2πrh + 2πr² A = ½ bh V = (½ bh)(H) A = π r² V = (πr²)(H) C = πd and how to find the surface area of a prism by adding up the areas of all the surfaces

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Surface Area of Prisms and Cylinders Retrieved from

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