Geometry Formulas Surface Area Volume A formula is

Geometry Formulas: Surface Area & Volume

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!

Let’s start in the beginning… Before you can do surface area or volume, you have to know the following formulas. Rectangle A = lw Triangle A = ½ bh Circle A = π r² C = πd

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

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

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

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

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.

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

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!

EXAMPLE: Round to the nearest TENTH. Top or bottom circle Rectangle A = πr² C = length A = π(3. 1)² C=πd A = π(9. 61) C = π(6. 2) A = 30. 2 C = 19. 5 Now add: 30. 2 + 234 = Now the area A = lw A = 19. 5(12) A = 234 SA = 294. 4 in²

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 = 233. 7 + 60. 4 SA = 294. 1 in² The answers are REALLY close, but not exactly the same. That’s because we rounded in the problem.

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)

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)

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) This is a right triangle, so the sides are also the base and height. V = 162 mm³ Height of the prism

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”.

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) optional step! V = (π)(3. 1)(12) V = 396. 3 in³

Try one: 10 m V = (πr²)(H) d=8 m V = (π)(4²)(10) V = (π)(16)(10) Since d = 8, then r = 4 r² = 4(4) = 16 V = 502. 7 m³

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