O is the center of the circle. The radius is 3.
What is the diameter?
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What is the circumference?
What is the area?
Circles
Words of "Wiz-dom"—The key to circle problems is the radius. All other measurements (diameter, circumference, and area) rely on it. So if you are given the radius, you can calculate everything else. If you are given anything else, you can calculate the radius. Using the formulas at the beginning of a math section, do the following review problems.
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O is the center of the circle. The radius is 3. What is the diameter? 6 What is the circumference?
What is the area?
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Words of "Wiz-dom"—The other issue the test writer wants to test is whether you understand that there are 360 degrees in a circle. Of course, this leads to applying rules related to portions of the area of a circle and the arcs of a circumference based on the number of degrees where two radii intersect. The formula looks like the following:
Answer the following review questions.
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Circle O has a radius of 8. What is the diameter? 16 What is the circumference?
What is the area?
What is the length of arc ABC?
What is the area of the wedge OABC?
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Sample Questions
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13 |
If the diameter of the circle O is 8, what is the area of the shaded region? |
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(A) |
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(B) |
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(C) |
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(D) |
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(E) |
The answer is C.
Since this is a circle question, you know the key is the radius. Since the test writer gave you the diameter of 8, you know the radius is 4, so the area of the circle is 
. The area that interests you is defined by a right angle so that is 1/4 of the total area or 
. When you subtract the area of the triangle (
) that leaves the shaded region.
Many students get as far as 
but don’t see their answer. In this case, they may recognize it as C but on the test the reduced answer may not be as obvious. Be sure to remember to reduce your answer whenever you can. It is a real shame to get the right answer but not get credit due to the failure to reduce.
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5 |
In the diagram above, the area of the circle is |
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(A) 1 |
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(B) 2 |
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(C) 4 |
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(D) 8 |
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(E) 12 |
The answer is C.
It helps to draw a radius in this diagram (Pillar II: Restating the given information). We are told in the narrative that the area is 
so the radius must be 1. That means the side of the square is 2, resulting in an area of 4.
This is an easy one to guess without doing any work. The area of the square has to be slightly more than or 3.14. That certainly eliminates A and B. The square isn’t over twice as big as the circle. So that eliminates D and E, leaving only C!
Another wizardly technique would be to start with answer C and plug in. If the area of the square is 4, then each side has to be 2. If the side is 2, then the radius has to be 1. The area of a circle with a radius of 1 is . C has to be right.
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12 |
A 16-inch (diameter) pizza is cut into 8 equal slices. What is the perimeter of the one slice that Ms. Townsend ate? |
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(A) |
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(B) |
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(C) |
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(D) |
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(E) |
The answer is D.
One slice has a perimeter of two radii (16) plus one-eighth of the circumference 
. So the calculation leaves us with 
. This has to be reduced to find the answer the test writer wants, answer D.
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19 |
A 16-inch (diameter) pizza is cut into 8 equal slices. What is the sum of the angles at the center of the pizza for the three pieces that were eaten by The Wiz? |
The answer is E.
Dividing a circle into 8 equal slices leaves 45 degrees for each slice. Three slices at 45 degrees each is a total of 135, answer E.
Another way to do the problem is to multiply 3/8 times 360.
Solids
Words of "Wiz-dom"—You’ll only see two possible shapes of solids. The most common is a rectangular (or cubic) solid. The formula for its volume is in the "Reference Information" at the beginning of the math section. Keep in mind that its surface area and perimeters of individual faces are the same as rectangles and squares. The less common type is a right cylinder. That’s a can! They are simply a pair of identical circles connected by a rectangle that has been rolled into a tube. Prove it to yourself by cutting a paper towel or toilet paper roll. You’ll also find the formula for the volume at the beginning of a math section. Having said there are only two possible shapes, the test writer has been known to sneak cones onto the test. Don’t panic. Relax. The questions will be about the circle on the bottom (so it’s a circle question in disguise) or a proportion problem. I’ve never seen a volume or surface area question that involves the side of the cone. We’ll do one of these to ease your concerns.
Rectangular Solids
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The following questions relate to the solid to the left for which the height is 3, the length is 6, and the width is 4. What is the volume of the solid? V=lwh=6x4x3=72 What is the surface area of the solid? A=2lw+2lh+2wh=108 What is the length of the diagonal AB of the end? 5. Notice that 3:4:5 triangles can show up in some funny places! If you labeled the diagram with the values provided, it was easier to see. What is the perimeter of the top (or bottom)? 20. Just for review. What is the ratio of the height to the length? 1:2. Be sure to reduce! If an ant walks only on the edges, what is the length of the shortest route from A to C? 6+4+3=13 |
Right Cylinders
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The following questions relate to the solid to the left for which the height is 8 and the radius is 3. What is the volume?
What is the surface area? There are two circles and a rectangle. The circles are relatively easy but remember to double the calculation of the surface area of one of them because you have a top and a bottom. That gives you What is the circumference?
What is the shortest distance from A to B (points exactly opposite one another?) 10. This is another sneaky review of 3:4:5 right triangles hidden as a multiple of 2 in the shape of a right cylinder. On a real can, the ant has to run across the diameter (6) and down the side (8). If he could have flown, it would have been 10! (On the other hand, if you drew the hypotenuse from A to B, he could slide down it!) If an ant runs around the top rim, then straight down the side, then runs around the bottom rim, how far did she go?
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Sample Questions
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5 |
If the length of the side of a cube is doubled, what effect does it have on the surface area of the cube? |
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(A) 2 times as large |
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(B) 4 times as large |
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(C) 12 times as large |
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(D) 24 times as large |
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(E) 48 times as large |
The answer is B.
Use the Nike approach—Just Do It! An easy way to start is with a cube with an edge of 1. Each side has an area of 1. Since a cube has six sides, it has a total surface area of 6. Doubling the length of the edge to 2, yields a side of 4; times the 6 sides is 24. 24/6 is 4 times as large, answer B.
An easier way is to realize that each side is 4 times larger since the increase in size is the square of the increase. (The fact that there are 6 sides is irrelevant.) In other words if the edge had been tripled the increase in area would have been 9 times.
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If the height of a right cylinder is tripled and its radius is doubled, how much larger is the volume? |
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(A) 5 times larger |
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(B) 6 times larger |
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(C) 12 times larger |
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(D) 16 times larger |
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(E) 24 times larger |
The answer is C.
The height increase raises the volume by three times. The increase in the radius makes it four times as large. Three times four is 12 or C.
Use Nike again. Substituting values works easily. Start with a height of 1 and a radius of one. That is a volume of 
. Make the increases, raising the radius to 2 (the area of the circle is now 
) and the height to 3. The volume of the new cylinder is 
, or 12 times as large.
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A painter is going to paint the walls of The Wiz’s cogitating room that is 12 feet wide and 18 feet long. The walls are 8 feet high. He can cover 400 square feet of surface area with each gallon of paint. If he can only buy whole gallons of paint, how many gallons does he need to purchase? |
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(A) 5 |
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(B) 4 |
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(C) 3 |
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(D) 2 |
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(E) 1 |
The answer is D.
The painter has two walls that are 12 by 8 (192 square feet) and two that are 18 by 8 (288 square feet) for a total of 480 square feet. He can’t buy a fraction of a gallon so he needs to buy two gallons, answer D.
Don’t fall in the trap of rounding down!
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What is the most number of small cubes with an edge of 15 that will fit in a large cube with an edge of 75? |
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(A) 250 |
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(B) 125 |
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(C) 100 |
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(D) 25 |
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(E) 5 |
The answer is D.
The large cube is 5 times as large as a small one. 5 cubed is 125, D.
You could envision (or draw) these figures. The large cube would hold 5 small cubes along the length and along the width and they can be stacked 5 high.
An interesting thing to notice is that the ratio is 1:5. So you could have used cubes that were 1 and 5 rather than the larger numbers of 15 and 75. It all comes out the same.
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A cube has a volume of 27. What is its surface area? |
The answer is 54.
The cube root of 27 is 3. That means the cube has an edge length of 3. Each of the six surfaces has an area of 9. So 9 times 6 is 54.
Don’t let the students go away thinking the surface area is twice the volume. Show them that this doesn’t work by demonstrating a cube with edge of 1. Repeat it doing an edge of 2. If you have time, show them that the ratios get more even as you go from 1 to 6. At an edge of 6, the volumes and surface areas are equal. At seven and above, the volume is larger than the surface area.
The answer is 54.
The cube root of 27 is 3. That means the cube has an edge length of 3. Each of the six surfaces has an area of 9. So 9 times 6 is 54.
The surface area is not always twice the volume. Prove it to yourself with a cube with an edge of 1. It has a volume of 1 and a surface area of 6. Repeat it doing an edge of 2. You’ll see a pattern that the ratio between volume and surface area gets more even as you go from 1 to 6. At an edge of 6, the volumes and surface areas are equal. At seven and above, the volume is larger than the surface area. Another Wizardly pattern in the world of math!
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A rectangular solid has a length of 12, a height of 8, and a width of 6. What is the minimum length of tape needed to cover all of the edges without overlap? |
The answer is 104.
Remember to draw diagrams when the test writer doesn’t provide one.
There are two identical perimeters to calculate. The bottom and top work well to give 72. Then there are 4 heights of 8 for 32 more. The total is 104.
Don’t calculate 6 perimeters, one for each face, and add them together. You need to remember that there are common edges.
. What is the area of the square?
. Now you have to add the area of the sides which is the height times the circumference of the circle (
). So the answer is 
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. She ran two circumferences plus the height. Remember to reduce!!