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ISEE Upper Level Math : Solid Geometry

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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:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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

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

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

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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:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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

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

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Example Question #13 : Spheres

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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

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

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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:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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

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

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Example Question #1 : How To Find The Surface Area Of A Sphere

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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

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

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Example Question #1 : How To Find The Surface Area Of A Sphere

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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

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

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Example Question #411 : Geometry

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

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

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

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

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

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

Possible Answers:

100 m^2 $\displaystyle 100 m^2$

50 m^2 $\displaystyle 50 m^2$

$100\pi m^2$ $\displaystyle 100\pi m^2$

$50\pi m^2$ $\displaystyle 50\pi m^2$

75m^2 $\displaystyle 75m^2$

Correct answer:

$100\pi m^2$ $\displaystyle 100\pi m^2$

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:

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

Plug in our radius and solve!

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

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Example Question #181 : Geometry

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

The volume of a cone is given by the formula

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

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

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

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

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

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

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Example Question #91 : Solid Geometry

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

The area of the base is therefore

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

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

$V = \frac{1}{3} Bh = \frac{1}{3} \cdot 9 \pi \cdot 10 = 30 \pi$ $\displaystyle V = \frac{1}{3} Bh = \frac{1}{3} \cdot 9 \pi \cdot 10 = 30 \pi$ 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 #13 : Spheres

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

Example Question #1 : How To Find The Surface Area Of A Sphere

Example Question #1 : How To Find The Surface Area Of A Sphere

Example Question #411 : Geometry

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

Example Question #181 : Geometry

Example Question #91 : Solid Geometry

All ISEE Upper Level Math Resources

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