# HSPT Math : Geometry

## Example Questions

A manufacturer makes wooden circles out of square blocks of wood. If the wood costs $0.25 per square inch, what is the minimum waste cost possible for cutting a circle with a radius of 44 in.? Possible Answers: 1936 dollars 7744 – 1936π dollars 1936 – 484π dollars 1936π dollars 5808 dollars Correct answer: 1936 – 484π dollars Explanation: The smallest block from which a circle could be made would be a square that perfectly matches the diameter of the given circle. (This is presuming we have perfectly calibrated equipment.) Such a square would have dimensions equal to the diameter of the circle, meaning it would have sides of 88 inches for our problem. Its total area would be 88 * 88 or 7744 in2. Now, the waste amount would be the "corners" remaining after the circle was cut. The area of the circle is πr2 or π * 442 = 1936π in2. Therefore, the area remaining would be 7744 – 1936π. The cost of the waste would be 0.25 * (7744 – 1936π). This is not an option for our answers, so let us simplify a bit. We can factor out a common 4 from our subtraction. This would give us: 0.25 * 4 * (1936 – 484π). Since 0.25 is equal to 1/4, 0.25 * 4 = 1. Therefore, our final answer is: 1936 – 484π dollars. ### Example Question #21 : Radius A manufacturer makes wooden circles out of square blocks of wood. If the wood costs$0.20 per square inch, what is the minimum waste cost possible for cutting a circle with a radius of 25 in.?

500 dollars

625 dollars

500 - 125π dollars

625 - 25π dollars

2500 - 625π dollars

500 - 125π dollars

Explanation:

The smallest block from which a circle could be made would be a square that perfectly matches the diameter of the given circle. (This is presuming we have perfectly calibrated equipment.) Such a square would have dimensions equal to the diameter of the circle, meaning it would have sides of 50 inches for our problem. Its total area would be 50 * 50 or 2500 in2.

Now, the waste amount would be the "corners" remaining after the circle was cut. The area of the circle is πr2 or π * 252 = 625π in2. Therefore, the area remaining would be 2500 - 625π. The cost of the waste would be 0.2 * (2500 – 625π). This is not an option for our answers, so let us simplify a bit. We can factor out a common 5 from our subtraction. This would give us: 0.2 * 5 * (500 – 125π). Since 0.2 is equal to 1/5, 0.2 * 625 = 125. Therefore, our final answer is: 500 – 125π dollars.

### Example Question #21 : Basic Geometry

Mary has a decorative plate with a diameter of ten inches. She places the plate on a rectangular placemat with a length of 18 inches and a width of 12 inches. How much of the placemat is visible?

Explanation:

First we will calculate the total area of the placemat:

Next we will calculate the area of the circular place

And

So

We will subtract the area of the plate from the total area

### Example Question #21 : Circles

Allen was running around the park when he lost his keys.  He was running around aimlessly for the past 30 minutes.  When he checked 10 minutes ago, he still had his keys.  Allen guesses that he has been running at about 3m/s.

If Allen can check 1 square kilometer per hour, what is the longest it will take him to find his keys?

Explanation:

Allen has been running for 10 minutes since he lost his keys at 3m/s.  This gives us a maximum distance of  from his current location.  If we move 1800m in all directions, this gives us a circle with radius of 1800m.  The area of this circle is

Our answer, however, is asked for in kilometers.  1800m=1.8km, so our actual area will be  square kilometers.  Since he can search 1 per hour, it will take him at most 10.2 hours to find his keys.

### Example Question #11 : How To Find The Area Of A Figure

Find the area of a square if the length is .

Explanation:

The area of a square is:

Substitute the length and simplify.

### Example Question #12 : How To Find The Area Of A Figure

A square has area 64. Which of the following gives 25% of the length of one of its diagonals?

Explanation:

A square with area 64 has as the length of one of its sides the square root of this area, which is . The length of a diagonal of a square is  times this sidelength, so the square has diagonals of length .

25% of this is

.

### Example Question #13 : How To Find The Area Of A Figure

Your backyard is  wide and  long, what is its area?

Explanation:

The area of a rectangle is

So for this you just multiple those two values together to get,

.

Remeber that the units of area are squared.

### Example Question #11 : How To Find The Area Of A Figure

If a trianlge has a base of  and a height of , the area is . True or false?

False

True

False

Explanation:

The area of a triangle is

.

is equal to the base multiplied by the height.

So you still need to multiple by  or divide by .

This gives you an actual area of .

Therefore the statement is false.

### Example Question #14 : How To Find The Area Of A Figure

A blindfolded man throws a dart at the above target. Assuming the dart hits the target, and that no skill is involved, give the probability that the dart will land inside a blue region.

Explanation:

The reasoning for the answer can best be seen by adding two more lines as shown in purple in the diagram below:

The target is now divided into sixteen squares of equal size, four of which are blue. This makes the probability that the dart will land in the blue region

.

### Example Question #15 : How To Find The Area Of A Figure

Above is a target. The radius of the smaller quarter-circles is half that of the larger quarter-circles.

A blindfolded man throws a dart at the above target. Assuming the dart hits the target, and that no skill is involved, give the odds against the dart landing in the yellow region.

39 to 1

13 to 1

14 to 1

40 to 1

39 to 1

Explanation:

Call the radius of one of the smaller quarter-circles 1 (the reasoning is independent of the actual radius). Then the area of each quarter-circle is

.

Each of the four wedges of one such quarter-circle has area

.

The yellow region is one such wedge.

The radius of each of the larger quarter-circles is 2, so the area of each is

The total area of the target is

Therefore, the yellow wedge is

of the target, and the odds against the dart landing in that region are 39 to 1.