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SSAT Upper Level Math : Geometry

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

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Example Questions

Example Question #3 : How To Find The Volume Of A Cylinder

We have two right cylinders. The radius of the base Cylinder 1 is $\sqrt{3}$ times more than that of Cylinder 2, and the height of Cylinder 2 is 4 times more than the height of Cylinder 1. The volume of Cylinder 1 is what fraction of the volume of Cylinder 2?

Possible Answers:

$V_{1}=\frac{2}{3}V_{2}$

$V_{1}=V_{2}$

$V_{1}=\frac{3}{2}V_{2}$

$V_{1}=\frac{3}{4}V_{2}$

$V_{1}=\frac{4}{3}V_{2}$

Correct answer:

$V_{1}=\frac{3}{4}V_{2}$

Explanation:

The volume of a cylinder is:

$V=\pi r^2h$

where is the volume of the cylinder, is the radius of the circular end of the cylinder, and is the height of the cylinder.

So we can write:

$V_{1}=\pi (r_{1})^2h_{1}$

and

$V_{2}=\pi (r_{2})^2h_{2}$

Now we can summarize the given information:

$r_{1}=\sqrt{3}\cdot r_{2}$

$h_{2}=4\cdot h_{1}\Rightarrow h_{1}=\frac{h_{2}}{4}$

Now substitute them in the $V_{1}$ formula:

$V_{1}=\pi (\sqrt{3}r_{2})^2\times \frac{h_{2}}{4}\Rightarrow V_{1}=\frac{3}{4}\pi (r_{2})^2\times h_{2}\Rightarrow V_{1}=\frac{3}{4}V_{2}$

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Example Question #1 : Know And Use The Formulas For The Volumes Of Cones, Cylinders, And Spheres: Ccss.Math.Content.8.G.C.9

Two right cylinders have the same height. The radius of the base of the first cylinder is two times more than that of the second cylinder. Compare the volume of the two cylinders.

Possible Answers:

$V_{1}=3\cdot V_{2}$

$V_{1}=V_{2}$

$V_{1}=2\cdot V_{2}$

$V_{1}=\sqrt{2}\cdot V_{2}$

$V_{1}=4\cdot V_{2}$

Correct answer:

$V_{1}=4\cdot V_{2}$

Explanation:

The volume of a cylinder is:

$V=\pi r^2h$

where is the radius of the circular end of the cylinder and is the height of the cylinder. So we can write:

$V_{1}=\pi (r_{1})^2h_{1}$

$V_{2}=\pi (r_{2})^2h_{2}$

We know that

$h_{1}=h_{2}$

and

$r_{1}=2r_{2}$ .

So we can write:

$V_{1}=\pi (r_{1})^2h_{1}=\pi (2r_{2})^2\cdot h_{2}=4\pi (r_{2})^2h_{2}\Rightarrow V_{1}=4V_{2}$

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Example Question #1 : How To Find The Volume Of A Cylinder

A cylinder has a diameter of inches and a height of inches. Find the volume, in cubic inches, of this cylinder.

Possible Answers:

98.3 in^3

402.1 in^3

308.9 in^3

100.5 in^3

Correct answer:

100.5 in^3

Explanation:

$\text{Volume of cylinder}=\pi \times \text{radius}^2 \times \text{height}$

Since we are given the diameter, divide that value in half to find the radius.

$\text{radius}=8\div2=4$

Now plug this value into the equation for the volume of a cylinder.

$\text{Volume}=\pi \times 4^4 \times 2$

$\text{Volume}=\pi \times 32=100.5$ in^3

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Example Question #1993 : Hspt Mathematics

Find the volume, in cubic inches, of a cylinder that has a radius of inches and a height of inches.

Possible Answers:

2695.3 in^3

737.5 in^3

4456.1 in^3

3308.1 in^3

Correct answer:

3308.1 in^3

Explanation:

The formula to find the volume of a cylinder is $\pi r^2\times height$ .

Now, plug in the given numbers into this equation.

$\text{Volume}=\pi \times 9^2 \times 13$

$\text{Volume}=\pi \times 1053=3308.1$ in^3

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Example Question #731 : Geometry

Find the volume of the cylinder if the circular base has an area of $\pi$ , and the height of the cylinder is also $\pi$ .

Possible Answers:

$\frac{\pi^3}{4}$

$\frac{\pi^3}{2}$

$\pi^3$

$\frac{\pi^2}{2}$

$\pi^2$

Correct answer:

$\pi^2$

Explanation:

Write the formula for volume of a cylinder.

$V=\pi r^2 h$

Remember that area of a circle is $A=\pi r^2$

Since the area of the circle is known, substitute the area into the formula.

$V=\pi(\pi)= \pi^2$

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Example Question #1 : How To Find The Perimeter Of A Hexagon

A regular hexagon has perimeter 9 meters. Give the length of one side in millimeters.

Possible Answers:

$1,800\textrm{ mm }$

$2,000\textrm{ mm }$

$1,200\textrm{ mm }$

$1,500 \textrm{ mm }$

$2,400 \textrm{ mm }$

Correct answer:

$1,500 \textrm{ mm }$

Explanation:

One meter is equal to 1,000 millimeters, so the perimeter of 9 meters can be expressed as:

9 meters = $9 \times 1,000 = 9,000$ millimeters.

Since the six sides of a regular hexagon are congruent, divide by 6:

$9,000 \div 6 = 1,500$ millimeters.

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Example Question #1 : Perimeter Of Polygons

A hexagon with perimeter 60 has four congruent sides of length t+1 . Its other two sides are congruent to each other. Give the length of each of those other sides in terms of .

Possible Answers:

28-2t

28+2t

56-2t

56+2t

28-t

Correct answer:

28-2t

Explanation:

The perimeter of a polygon is the sum of the lengths of its sides. Let:

Length of one of those other two sides

Now we can set up an equation and solve it for in terms of :

2x+4(t+1)=60

$\Rightarrow 2x+4t+4=60$

$\Rightarrow 2x=60-4-4t$

$\Rightarrow 2x=56-4t$

$\Rightarrow x=28-2t$

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Example Question #2 : How To Find The Perimeter Of A Hexagon

Two sides of a hexagon have a length of , two other sides have the length of t-1 , and the rest of the sides have the length of t+1 . Give the perimeter of the hexagon.

Possible Answers:

6t+6

6t-6

8t+8

Correct answer:

Explanation:

The perimeter of a polygon is the sum of the lengths of its sides. So we can write:

$Perimeter=2\left [ t+(t+1)+(t-1) \right ]=2(3t)=6t$

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Example Question #4 : How To Find The Perimeter Of A Hexagon

A regular hexagon has perimeter 15 feet. Give the length of one side in inches.

Possible Answers:

Correct answer:

Explanation:

As the six sides of a regular hexagon are congruent, we can write:

$Perimeter=6a=15\Rightarrow a=2.5$ feet; is the length of each side.

One feet is equal to 12 inches, so we can write:

$a=2.5\times 12=30$ inches

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Example Question #3 : How To Find The Perimeter Of A Hexagon

Each interior angle of a hexagon is 120 degrees and the perimeter of the hexagon is 120 inches. Find the length of each side of the hexagon.

Possible Answers:

$20\ inches$

$60\ inches$

$30\ inches$

$25\ inches$

$10\ inches$

Correct answer:

$20\ inches$

Explanation:

Since each interior angle of a hexagon is 120 degrees, we have a regular hexagon with identical side lengths. And we know that the perimeter of a polygon is the sum of the lengths of its sides. So we can write:

$Perimeter=6a=120\Rightarrow a=20$ inches

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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 #3 : How To Find The Volume Of A Cylinder

Example Question #1 : Know And Use The Formulas For The Volumes Of Cones, Cylinders, And Spheres: Ccss.Math.Content.8.G.C.9

Example Question #1 : How To Find The Volume Of A Cylinder

Example Question #1993 : Hspt Mathematics

Example Question #731 : Geometry

Example Question #1 : How To Find The Perimeter Of A Hexagon

Example Question #1 : Perimeter Of Polygons

Example Question #2 : How To Find The Perimeter Of A Hexagon

Example Question #4 : How To Find The Perimeter Of A Hexagon

Example Question #3 : How To Find The Perimeter Of A Hexagon

All SSAT Upper Level Math Resources

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