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ISEE Upper Level Math : Geometry

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Example Questions

Example Question #101 : Solid Geometry

Find the volume of a cone with a base diameter of 2, and a height of 10.

Possible Answers:

$10\pi$

$\frac{10}{3}\pi$

$\frac{5}{3}\pi$

$\frac{40}{3}\pi$

$\frac{20}{3}\pi$

Correct answer:

$\frac{10}{3}\pi$

Explanation:

Write the formula to find the volume of a cone.

$V=\frac{1}{3}\pi r^2h$

The diameter is 2, which means the radius is 1. Substitute the known dimensions.

$V=\frac{1}{3}\pi (1)^2(10)$

The answer is: $\frac{10}{3}\pi$

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Example Question #101 : Solid Geometry

A cone has height 240 centimeters; its base has radius 80 centimeters. Give its volume in cubic meters.

Possible Answers:

$0.768 \pi \textrm{ m}^{3}$

$1.536 \pi \textrm{ m}^{3}$

$0.512 \pi \textrm{ m}^{3}$

$4.608 \pi \textrm{ m}^{3}$

Correct answer:

$0.512 \pi \textrm{ m}^{3}$

Explanation:

Convert both dimensions from centimeters to meters by dividing by 100:

Height: 240 centimeters = $240 \div 100 = 2.4$ meters.

Radius: 80 centimeters = $80 \div 100 = 0.8$ meters.

Substitute r = 0.8, h = 2.4 in the volume formula:

$V = \frac{1}{3} \pi r^{2}h$

$V = \frac{1}{3} \cdot \pi \cdot 0.8 ^{2} \cdot 2.4$

$V = \frac{1}{3}\cdot 0.8 \cdot 0.8 \cdot 2.4 \cdot \pi$

$V =0.512 \pi$

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Example Question #102 : Solid Geometry

What is the surface area of a cone with the following measurements?

r=4;d=8;h=10

Possible Answers:

$5+8\pi$

$5+64\pi$

$5+4\pi$

$5+16\pi$

Correct answer:

$5+16\pi$

Explanation:

The surface area of a cone can be found with the following equation:

$SA=\frac{1}{2}h+r^2\pi$

$SA=\frac{1}{2}(10)+4^2\pi=5+16\pi$

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

The axle for a toy car has a length of 4 inches and a diameter of a quarter inch. What is the volume of the axle? Assume it is a cylinder.

Possible Answers:

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

$16\pi in^3$

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

$\frac{\pi}{16}in^3$

Correct answer:

$\frac{\pi}{16}in^3$

Explanation:

The axle for a toy car has a length of 4 inches and a diameter of a quarter inch. What is the volume of the axle? Assume it is a cylinder.

Use the following formula for volume of a cylinder

$V_{cylinder}=\pi r^2h$

Where r and h are our radius and height, respectively.

In this case, we first need to change our diameter to radius. Because our diameter is one quarter of an inch, our radius will be one eighth of an inch.

Plug it in to get:

$V_{cylinder}=\pi (\frac{1}{8})^2*4=\pi*\frac{1}{64}in^2*4in=\frac{4\pi}{64}in^3$

Simplify to get:

$\frac{4\pi}{64}in^3=\frac{\pi}{16}in^3$

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Example Question #2 : Cylinders

You are visiting the drive-through at the bank. You put you money in a cylindrical tube with a height of 8 inches and a radius of 2 inches. What is the volume of the tube?

Possible Answers:

Not enough information

$16\pi in^3$

$64\pi in^3$

$32\pi in^3$

Correct answer:

$32\pi in^3$

Explanation:

You are visiting the drive-through at the bank. You put you money in a cylindrical tube with a height of 8 inches and a radius of 2 inches. What is the volume of the tube?

Begin with the formula for volume of a cylinder:

$V_{cylinder}=\pi r^2h$

We have r and h, which are our radius and height, respectively. Plug them in and solve

$V_{cylinder}=\pi (2in)^2*8in=\pi*32in^2$

Making our answer:

$32\pi in^2$

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

Find the volume of a cylinder with the following measurements:

diameter = 8cm
height = 5cm

Possible Answers:

$80\pi \text{ cm}^3$

$60\pi \text{ cm}^3$

$20\pi \text{ cm}^3$

$40\pi \text{ cm}^3$

$100\pi \text{ cm}^3$

Correct answer:

$80\pi \text{ cm}^3$

Explanation:

To find the volume of a cylinder, we will use the following formula:

$V = \pi \cdot r^2 \cdot h$

where r is the radius and h is the height of the cylinder.

Now, we know the diameter of the cylinder is 8cm. We know that the diameter is two times the radius. Therefore, the radius is 4cm.

We also know the height of the cylinder is 5cm.

Knowing this, we can substitute into the formula.

$V = \pi \cdot (4\text{cm})^2 \cdot 5\text{cm}$

$V = \pi \cdot 16\text{cm}^2 \cdot 5\text{cm}$

$V = \pi \cdot 80\text{cm}^3$

$V = 80\pi \text{ cm}^3$

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Example Question #1 : Cylinders

Find the volume of a cylinder with a diameter of 8in and a height of 7in.

Possible Answers:

$112\pi \text{ in}^3$

$121\pi \text{ in}^3$

$78\pi \text{ in}^3$

$68\pi \text{ in}^3$

$56\pi \text{ in}^3$

Correct answer:

$112\pi \text{ in}^3$

Explanation:

To find the volume of a cylinder, we will use the following formula:

$V = \pi r^2 h$

where r is the radius and h is the height of the cylinder.

Now, we know the diameter of the cylinder is 8in. We also know the diameter is two times the radius. Therefore, the radius is 4in.

We also know the height of the cylinder is 7in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot(4\text{in})^2 \cdot 7\text{in}$

$V = \pi \cdot 16\text{in}^2 \cdot 7\text{in}$

$V = \pi \cdot 112\text{in}^3$

$V = 112\pi \text{ in}^3$

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Example Question #2 : Cylinders

Find the volume of a cylinder with a diameter of 14in and a height of 12in.

Possible Answers:

$273\pi \text{ in}^3$

$168\pi \text{ in}^3$

$588\pi \text{ in}^3$

$456\pi \text{ in}^3$

$336\pi \text{ in}^3$

Correct answer:

$588\pi \text{ in}^3$

Explanation:

To find the volume of a cylinder, we will use the following formula:

$V = \pi \cdot r^2 \cdot h$

where r is the radius, and h is the height of the cylinder.

Now, we know the diameter of the cylinder is 14in. We also know the diameter is two times the radius. Therefore, the radius is 7in.

We know the height of the cylinder is 12in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot (7\text{in})^2 \cdot 12\text{in}$

$V = \pi \cdot 49\text{in}^2 \cdot 12\text{in}$

$V = \pi \cdot 588\text{in}^3$

$V = 588\pi \text{ in}^3$

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Example Question #3 : Cylinders

Your family owns a farm with a silo for storing grain. If the silo is 40 feet tall and 15 feet in diameter, what volume of grain can it hold?

Possible Answers:

$22.50 \pi ft^3$

$2250 \pi ft^3$

$4500 \pi ft^3$

2250 ft^2

Correct answer:

$2250 \pi ft^3$

Explanation:

Your family owns a farm with a silo for storing grain. If the silo is 40 feet tall and 15 feet in diameter, what volume of grain can it hold?

Begin with the formula for volume of a cylinder.

$V=\pi r^2h$

A cylinder is just a circle with height.

So, we know the height is 40 ft, but what is r?

If you said 15, you would be on track to get the problem wrong. That is because the diameter is 15 ft, so our radius is only 7.5 ft.

Plug these in to get our answer:

$V=\pi (7.5ft)^2(40ft)=2250 \pi ft^3$

Our answer should be

$2250 \pi ft^3$

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Example Question #5 : How To Find The Volume Of A Cylinder

Let $\pi = 3.14$

Find the volume of a cylinder with a diameter of 12cm and a height of 7cm.

Possible Answers:

$263.76\text{cm}^3$

$923.16\text{cm}^3$

$527.52\text{cm}^3$

$131.88\text{cm}^3$

$791.28\text{cm}^3$

Correct answer:

$791.28\text{cm}^3$

Explanation:

To find the volume of a cylinder, we will use the following formula:

$V = \pi r^2 h$

where r is the radius and h is the height of the cylinder.

Now, we know the diameter is 12cm. We know that the diameter is two times the radius. Therefore, the radius is 6cm.

We know the height of the cylinder is 7cm.

We know that $\pi = 3.14$ .

Knowing all of this, we can substitute into the formula. We get

$V = 3.14 \cdot (6\text{cm})^2 \cdot 7\text{cm}$

$V = 3.14 \cdot 36\text{cm}^2 \cdot 7\text{cm}$

$V = 3.14 \cdot 252\text{cm}^3$

$V = 791.28\text{cm}^3$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Geometry

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #101 : Solid Geometry

Example Question #101 : Solid Geometry

Example Question #102 : Solid Geometry

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

Example Question #2 : Cylinders

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

Example Question #1 : Cylinders

Example Question #2 : Cylinders

Example Question #3 : Cylinders

Example Question #5 : How To Find The Volume Of A Cylinder

All ISEE Upper Level Math Resources

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