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SSAT Upper Level Math : Geometry

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All SSAT Upper Level Math Resources

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Example Questions

Example Question #41 : Geometry

Give the circumference of a circle that circumscribes a 30-60-90 triangle whose shorter leg has length $6\frac{1}{2}$ .

Possible Answers:

$\frac{13 \pi}{2}$

$\frac{13 \pi\sqrt{2}}{2}$

$26 \pi$

$13 \pi$

$13 \pi\sqrt{2}$

Correct answer:

$13 \pi$

Explanation:

If a right triangle is inscribed inside a circle, then the arc intercepted by the right angle is a semicircle, making the hypotenuse of triangle a diameter.

The length of a hypotenuse of a 30-60-90 triangle is twice that of its short leg, so the hypotenuse of this triangle will be twice $6\frac{1}{2}$ , or

$6\frac{1}{2} \times 2 = 13$ .

This is the diameter, also, so the circumference is $\pi$ times this, or $C =13 \pi$ .

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Example Question #42 : Geometry

Find the circumference of a circle with a diameter of 14.

Possible Answers:

$14\pi$

$196\pi$

$49\pi$

$7\pi$

$28\pi$

Correct answer:

$14\pi$

Explanation:

Write the formula to find the circumference of a circle.

$C=2\pi r=\pi D$

Substitute the diameter into the formula.

$c=\pi(14)= 14\pi$

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Example Question #11 : How To Find The Circumference Of A Circle

Find the circumference of the circle with an area of $9\pi$ .

Possible Answers:

$18\pi^2$

$9\pi$

$6\pi$

$81\pi$

$36\pi$

Correct answer:

$6\pi$

Explanation:

Write the formula for the area of the circle. The radius is needed to find the circumference of the circle.

$A=\pi r^2$

Substitute the area.

$9\pi=\pi r^2$

r^2=9

r=3

The radius of the circle is 3. Substitute this in the circumference formula.

$C=2\pi r= 2\pi(3)=6\pi$

The correct answer is: $6\pi$

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Example Question #15 : How To Find The Circumference Of A Circle

Track

The track at Cesar Chavez High School is a perfect circle of radius 500 feet, and is shown in the above figure. Rafael starts at point C, runs around the track clockwise twice, and continues to run clockwise until he makes it to point D. Which of the following comes closest to the number of miles Rafael has run?

Possible Answers:

$2\frac{1}{2} \textup{ mi}$

$1\frac{1}{2} \textup{ mi}$

$2 \textup{ mi}$

$\frac{1}{2} \textup{ mi}$

$1 \textup{ mi}$

Correct answer:

$1\frac{1}{2} \textup{ mi}$

Explanation:

The circumference of a circle with radius 500 feet is

$C = 2 \pi r \approx 2 \cdot 3.14 \cdot 500 = 3,140$ feet.

Rafael runs this distance twice, then he runs from Point C to D, which is about one-fifth of this distance. Therefore, Rafael's run will be about

$3,140 \cdot 2 \frac{1}{5} = 6,908$ feet.

Divide by 5,280 to convert to miles:

$6,908 \div 5,280 \approx 1.3$ miles.

Of the given choices, $1 \frac{1}{2 }$ miles is the closest to the correct length.

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Example Question #16 : How To Find The Circumference Of A Circle

Track

The track at Eisenhower High School is shown above; it is comprised of a square and a semicircle.

Mike begins at Point A, runs five times around the track clockwise, and continues further until he reaches Point C. Which of the following comes closest to the distance Mike runs?

Possible Answers:

$3 \textup{ mi }$

$3\frac{1}{2} \textup{ mi }$

$2 \textup{ mi}$

$4 \textup{ mi}$

$2\frac{1}{2} \textup{ mi }$

Correct answer:

$2\frac{1}{2} \textup{ mi }$

Explanation:

First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 500 feet; this length is

$\frac{1}{2} C = \frac{1}{2} \cdot \pi d \approx \frac{1}{2} \cdot 3.14 \cdot 500 = 785$ feet.

The distance around the track is about

500 + 785 + 500 +500 = 2,285 feet.

Mike runs around the track five times, which is a distance of about

$2,285 \times 5 = 11,425$ feet.

He then proceeds to Point B, which is an additional 500 feet, and to Point C, which is half the semicircle, or

$\frac{1}{2} \cdot 785 \approx 393$ feet.

Therefore, Mike runs about

11,425 + 500 + 393 = 12,318 feet.

Divide by 5,280 to convert to miles:

$12,318 \div 5,280 \approx 2.3$ miles.

Of the choices, the closest is $2\frac{1}{2}$ miles.

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Example Question #17 : How To Find The Circumference Of A Circle

Track

The track at Douglas MacArthur High School is shown above; it is the composite of a square and a semicircle.

Jennifer wants to run one mile. If she begins at Point A and begins running clockwise, where will she be when she is finished?

Possible Answers:

Between Point E and Point A

Between Point C and Point D

Between Point D and Point E

Between Point B and Point C

Between Point A and Point B

Correct answer:

Between Point E and Point A

Explanation:

First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 400 feet; this length is

$\frac{1}{2} C = \frac{1}{2} \cdot \pi d \approx \frac{1}{2} \cdot 3.14 \cdot 400 = 628$ feet.

The distance around the track is about

400 + 628 + 400 + 400 = 1,828 feet.

Divide this into 5,280 feet (one mile):

$5,280 \div 1,828 = 2\textup{ R }1,624$

This means that Jennifer will run two times around the track to return to Point A. She will have 1,624 feet to go, so she will do the following:

She will run 400 feet from A to B, leaving 1,624-400 = 1,224 feet;

She will run 628 feet from B to D, leaving 1,224 - 628 = 596 feet;

She will run 400 feet from D to E, leaving 596-400 = 196 feet;

She will then run the remaining 196 feet from Point E toward Point A.

The correct response is that she will be between Point E and Point A.

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Example Question #44 : Area And Circumference Of A Circle

Track

The track at Madison High School is a perfect circle of radius 600 feet, and is shown in the above figure. Boris wants to run around the track for one mile. If Boris starts at point A and runs clockwise, which of the following is closest to the point at which he will stop running?

(Assume the five points are evenly spaced)

Possible Answers:

Correct answer:

Explanation:

A circle of radius 600 feet will have a circumference of

$C = 2 \pi r \approx 2 \cdot 3.14 \cdot 600 = 3,768$ feet.

Boris will run one mile, or 5,280 feet, which will be about

$\frac{5,280}{3,768} \approx 1.4$ tiimes the circumference of the track, or, equivalently, once around the track, plus another two-fifths of the circle. This means he will end up closest to point C.

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Example Question #43 : Geometry

If you have a circular yard and need to put up a fence around the outside, you would use the formula $\pi r^{2}$ to figure out the amount of fence you need.

Possible Answers:

False

True

Correct answer:

False

Explanation:

To figure out the amount of the fence around a circular yard, you need to find the circumference of the yard. The equation for the circumference of a cirle is $2\pi r$ and not $\pi r^{2}$ .

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Example Question #44 : Geometry

Find the circumference of a circle with a diameter of 12.

Possible Answers:

$24\Pi$

$18\Pi$

$12\Pi$

$6\Pi$

Correct answer:

$12\Pi$

Explanation:

There are two possible formulas for finding the circumference of a circle. They are as follows:

$C=2\Pi \cdot r$

And:

$C=\Pi \cdot d$

Where C is circumference, r is radius, and d is diameter.

For this problem, we are given the diameter is 12, meaning we can plug our numbers into the second equation. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.

$C=\Pi \cdot 12= 12\Pi$

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Example Question #45 : Geometry

Find the circumference of a circle with a diameter of 6.

Possible Answers:

$3\Pi$

$36\Pi$

$6\Pi$

$12\Pi$

Correct answer:

$6\Pi$

Explanation:

There are two possible formulas for finding the circumference of a circle. They are as follows:

$C=2\Pi \cdot r$

And:

$C=\Pi \cdot d$

Where C is circumference, r is radius, and d is diameter.

For this problem, we are given the diameter is 6, meaning we can plug our numbers into the second equation. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.

$C=\Pi \cdot 6= 6\Pi$

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All SSAT Upper Level Math Resources

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SSAT Upper Level Math : Geometry

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #41 : Geometry

Example Question #42 : Geometry

Example Question #11 : How To Find The Circumference Of A Circle

Example Question #15 : How To Find The Circumference Of A Circle

Example Question #16 : How To Find The Circumference Of A Circle

Example Question #17 : How To Find The Circumference Of A Circle

Example Question #44 : Area And Circumference Of A Circle

Example Question #43 : Geometry

Example Question #44 : Geometry

Example Question #45 : Geometry

All SSAT Upper Level Math Resources

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