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ISEE Upper Level Math : Solid Geometry

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All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #5 : How To Find The Surface Area Of A Sphere

Find the surface area of a sphere with a diameter of 10in.

Possible Answers:

$100\pi \text{ cm}^2$

$75\pi \text{ cm}^2$

$36\pi \text{ cm}^2$

$50\pi \text{ cm}^2$

$125\pi \text{ cm}^2$

Correct answer:

$100\pi \text{ cm}^2$

Explanation:

To find the surface area of a sphere, we will use the following formula:

$SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 10cm. We also know the diameter is two times the radius. Therefore, the radius is 5cm.

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

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

$SA = 4 \cdot \pi \cdot 25\text{cm}^2$

$SA = 100\text{cm}^2 \cdot \pi$

$SA = 100\pi \text{ cm}^2$

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Example Question #6 : How To Find The Surface Area Of A Sphere

Find the surface area of a sphere with a radius of 10in.

Possible Answers:

$375\pi \text{ in}^2$

$250\pi \text{ in}^2$

$125\pi \text{ in}^2$

$100\pi \text{ in}^2$

$400\pi \text{ in}^2$

Correct answer:

$400\pi \text{ in}^2$

Explanation:

To find the surface area of a sphere, we will use the following formula:

$SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the radius of the sphere is 10in.

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

$SA = 4 \cdot \pi \cdot (10\text{in})^2$

$SA = 4 \cdot \pi \cdot 100\text{in}^2$

$SA = 400\pi \text{ in}^2$

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Example Question #7 : How To Find The Surface Area Of A Sphere

Find the surface area of a sphere with a diameter of 18in.

Possible Answers:

$212\pi \text{ in}^2$

$138\pi \text{ in}^2$

$324\pi \text{ in}^2$

$256\pi \text{ in}^2$

$421\pi \text{ in}^2$

Correct answer:

$324\pi \text{ in}^2$

Explanation:

To find the surface area of a sphere, we will use the following formula:

$SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 18in. We also know the diameter is two times the radius. Therefore, the radius is 9in.

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

$SA = 4 \cdot \pi \cdot (9\text{in})^2$

$SA = 4 \cdot \pi \cdot 81\text{in}^2$

$SA = 324 \pi \text{ in}^2$

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Example Question #8 : How To Find The Surface Area Of A Sphere

Find the surface area of a sphere with a radius of 6in.

Possible Answers:

$121\pi \text{ in}^2$

$48\pi \text{ in}^2$

$72\pi \text{ in}^2$

$144\pi \text{ in}^2$

$36\pi \text{ in}^2$

Correct answer:

$144\pi \text{ in}^2$

Explanation:

To find the surface area of a sphere, we will use the following formula:

$SA = 4\pi r^2$

where r is the radius of the sphere.

Now, we know the radius of the sphere is 6in. Knowing this, we can substitute into the formula. We get

$SA = 4 \cdot \pi \cdot (6\text{in})^2$

$SA = 4 \cdot \pi \cdot 36\text{in}^2$

$SA = 144\pi \text{ in}^2$

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Example Question #2 : How To Find The Surface Area Of A Sphere

Find the surface area of a sphere with a radius of 12in.

Possible Answers:

$144\pi \text{ in}^2$

$576\pi \text{ in}^2$

$216\pi \text{ in}^2$

$432\pi \text{ in}^2$

$864\pi \text{ in}^2$

Correct answer:

$576\pi \text{ in}^2$

Explanation:

To find the surface area of a sphere, we will use the following formula:

$SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the radius of the sphere is 12in.

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

$SA = 4 \cdot \pi \cdot (12\text{in})^2$

$SA = 4 \cdot \pi \cdot 144\text{in}^2$

$SA = 576\pi \text{ in}^2$

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Example Question #3 : How To Find The Surface Area Of A Sphere

Find the surface area of a sphere with a diameter of 20in.

Possible Answers:

$200\pi \text{ in}^2$

$800\pi \text{ in}^2$

$300\pi \text{ in}^2$

$400\pi \text{ in}^2$

$600\pi \text{ in}^2$

Correct answer:

$400\pi \text{ in}^2$

Explanation:

To find the surface area of a sphere, we will use the following formula:

$SA = 4 \cdot \pi \cdot r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 20in. We also know the diameter is two times the radius. Therefore, the radius is 10in.

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

$SA = 4 \cdot \pi \cdot (10\text{in})^2$

$SA = 4 \cdot \pi \cdot 100\text{in}^2$

$SA = 400 \pi \text{ in}^2$

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Example Question #411 : Geometry

Find the surface area of a sphere with a diameter of 12in.

Possible Answers:

$108\pi \text{ in}^2$

$144\pi \text{ in}^2$

$48\pi \text{ in}^2$

$96\pi \text{ in}^2$

$121\pi \text{ in}^2$

Correct answer:

$144\pi \text{ in}^2$

Explanation:

To find the surface area of a sphere, we will use the following formula:

$SA = 4\pi r^2$

where r is the radius of the sphere.

Now, we know the diameter of the sphere is 12in. We also know the diameter is two times the radius. Therefore, the radius is 6in. So, we get

$SA = 4 \cdot \pi \cdot (6\text{in})^2$

$SA = 4 \cdot \pi \cdot 36\text{in}^2$

$SA = 144\pi \text{ in}^2$

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Example Question #21 : Spheres

A spherical buoy has a radius of meters. What is the surface area of the buoy?

Possible Answers:

$50\pi m^2$

75m^2

50 m^2

$100\pi m^2$

100 m^2

Correct answer:

$100\pi m^2$

Explanation:

A spherical buoy has a radius of 5 meters. What is the surface area of the buoy?

To find the surface area of a sphere, use the following:

$SA_{sphere}=4\pi r^2$

Plug in our radius and solve!

$SA_{sphere}=4\pi (5m)^2=100\pi m^2$

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

A cone has height 18 inches; its base has radius 4 inches. Give its volume in cubic feet (leave in terms of $\pi$ )

Possible Answers:

$\frac{1}{3} \pi \textrm{ ft}^{3}$

$\frac{1}{4} \pi \textrm{ ft}^{3}$

$\frac{1}{18} \pi \textrm{ ft}^{3}$

$\frac{1}{6} \pi \textrm{ ft}^{3}$

$\frac{1}{12} \pi \textrm{ ft}^{3}$

Correct answer:

$\frac{1}{18} \pi \textrm{ ft}^{3}$

Explanation:

Convert radius and height from inches to feet by dividing by 12:

Height: 18 inches = $18 \div 12 = \frac{18}{12} = \frac{3}{2}$ feet

Radius: 4 inches = $4 \div 12 = \frac{4}{12} = \frac{1}{3}$

The volume of a cone is given by the formula

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

Substitute $r = \frac{1}{3}, h = \frac{3}{2}$ :

$V = \frac{1}{3}\cdot \left ( \frac{1}{3} \right ) ^{2}\cdot \frac{3}{2} \cdot \pi$

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

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

$V = \frac{1}{18} \pi$

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

Give the volume of a cone whose height is 10 inches and whose base is a circle with circumference $6 \pi$ inches.

Possible Answers:

$30 \pi \textrm{ in}^{3}$

$45 \pi \textrm{ in}^{3}$

$120 \pi \textrm{ in}^{3}$

$360 \pi \textrm{ in}^{3}$

$90 \pi \textrm{ in}^{3}$

Correct answer:

$30 \pi \textrm{ in}^{3}$

Explanation:

A circle with circumference $6 \pi$ inches has as its radius

$r = \frac{C }{2\pi }=\frac{6\pi }{2\pi } = 3$ inches.

The area of the base is therefore

$B = \pi r^{2} = \pi \cdot 3^{2} = 9 \pi$ square inches.

To find the volume of the cone, substitute $B = 9 \pi , h = 10$ in the formula for the volume of a cone:

$V = \frac{1}{3} Bh = \frac{1}{3} \cdot 9 \pi \cdot 10 = 30 \pi$ cubic inches

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Solid Geometry

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #5 : How To Find The Surface Area Of A Sphere

Example Question #6 : How To Find The Surface Area Of A Sphere

Example Question #7 : How To Find The Surface Area Of A Sphere

Example Question #8 : How To Find The Surface Area Of A Sphere

Example Question #2 : How To Find The Surface Area Of A Sphere

Example Question #3 : How To Find The Surface Area Of A Sphere

Example Question #411 : Geometry

Example Question #21 : Spheres

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

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

All ISEE Upper Level Math Resources

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