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High School Math : Solid Geometry

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Example Questions

Example Question #11 : Solid Geometry

What is the volume of a cone with base radius 4, and height 6?

Possible Answers:

$\frac{24}{3}$

$\frac{96\pi}{3}$

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

$96\pi$

Correct answer:

$\frac{96\pi}{3}$

Explanation:

The volume of a cone is $\frac{\pi*r^2*h}{3}$ , where is the height of the cone and is the base radius.

The volume of this cone is thus:

$\frac{\pi*4^2*6}{3}$

= $\frac{96\pi}{3}$

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Example Question #541 : Geometry

What is the surface area of a cone with a radius of 4 and a height of 3?

Possible Answers:

$25\pi$

$40\pi$

$36\pi$

$16\pi$

$48\pi$

Correct answer:

$36\pi$

Explanation:

Here we simply need to remember the formula for the surface area of a cone and plug in our values for the radius and height.

$\Pi r^{2} + \Pi r\sqrt{r^{2} + h^{2}}= \Pi\ast 4^{2} + \Pi \ast 4\sqrt{4^{2} + 3^{2}} = 16\Pi + 4\Pi \sqrt{25} = 16\Pi + 20\Pi = 36\Pi$

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Example Question #31 : Solid Geometry

The lateral area is twice as big as the base area of a cone. If the height of the cone is 9, what is the entire surface area (base area plus lateral area)?

Possible Answers:

90π

9π

54π

27π

81π

Correct answer:

81π

Explanation:

Lateral Area = LA = π(r)(l) where r = radius of the base and l = slant height

LA = 2B

π(r)(l) = 2π(r²)

rl = 2r²

l = 2r

Cone

From the diagram, we can see that r² + h² = l². Since h = 9 and l = 2r, some substitution yields

r² + 9² = (2r)²

r² + 81 = 4r²

81 = 3r²

27 = r²

B = π(r²) = 27π

LA = 2B = 2(27π) = 54π

SA = B + LA = 81π

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

What is the surface area of a cone with a height of 8 and a base with a radius of 5?

Possible Answers:

$25\pi$

$48.45\pi$

$65\pi$

$62.72\pi$

Correct answer:

$62.72\pi$

Explanation:

To find the surface area of a cone we must plug in the appropriate numbers into the equation

$Surface\ Area=\pi r^{2}+\pi rl$

where is the radius of the base, and is the lateral, or slant height of the cone.

First we must find the area of the circle.

To find the area of the circle we plug in our radius into the equation of a circle which is

This yields $A=25\pi$ .

We then need to know the surface area of the cone shape.

To find this we must use our height and our radius to make a right triangle in order to find the lateral height using Pythagorean’s Theorem.

Pythagorean’s Theorem states

Take the radius and height and plug them into the equation as a and b to yield $5^{2}+8^{2}=c^{2}$

First square the numbers $25+64=c^{2}$

After squaring the numbers add them together $89=c^{2}$

Once you have the sum, square root both sides $\sqrt{89}=\sqrt{c^{2}}$

After calculating we find our length is c=l=9.43

Then plug the length into the second portion of our surface area equation above to get $(4)(\pi)(9.43)=37.72\pi$

Then add the area of the circle with the conical area to find the surface area of the entire figure $25\pi+37.72\pi=62.72\pi$

The answer is $62.72\pi$ .

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Example Question #551 : Sat Subject Test In Math I

What is the surface area of a cone with a radius of 6 in and a height of 8 in?

Possible Answers:

96π in²

36π in²

60π in²

112π in²

66π in²

Correct answer:

96π in²

Explanation:

Find the slant height of the cone using the Pythagorean theorem: r² + h² = s² resulting in 6² + 8² = s² leading to s² = 100 or s = 10 in

SA = πrs + πr² = π(6)(10) + π(6)² = 60π + 36π = 96π in²

60π in² is the area of the cone without the base.

36π in² is the area of the base only.

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

Find the surface area of a cone that has a radius of 12 and a slant height of 15.

Possible Answers:

$180\pi$

$324\pi$

$405\pi$

$144\pi$

$225\pi$

Correct answer:

$324\pi$

Explanation:

The standard equation to find the surface area of a cone is

$SA=\pi rs+\pi r^2$

where denotes the slant height of the cone, and denotes the radius.

Plug in the given values for and to find the answer:

$SA=\pi (12)(15)+\pi (12)^2=180\pi +144\pi =324\pi$

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Example Question #4 : How To Find The Surface Area Of A Cone

Find the surface area of the following cone.

Cone

Possible Answers:

$100 \pi m^2$

$85 \pi m^2$

$90 \pi m^2$

$95 \pi m^2$

$80 \pi m^2$

Correct answer:

$90 \pi m^2$

Explanation:

The formula for the surface area of a cone is:

$SA = A_{base} + A_{lateral}$

$SA = \pi r^2 + \pi r l$

where is the radius of the cone and is the slant height of the cone.

Plugging in our values, we get:

$SA = \pi (5m)^2 + \pi (5m) (13m)$

$SA = 25 \pi m^2 + 65 \pi m^2 = 90 \pi m^2$

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

Find the surface area of the following cone.

Cone

Possible Answers:

$116 \pi m^2$

$86 \pi m^2$

$96 \pi m^2$

$106 \pi m^2$

$76 \pi m^2$

Correct answer:

$96 \pi m^2$

Explanation:

The formula for the surface area of a cone is:

$SA = (base) + \frac{(circumference)(slant height)}{2}$

$SA = (\pi r^2) + \frac{(2 \pi r)(h_s)}{2}$

$SA = (\pi r^2) + ( \pi rh_s)$

Use the Pythagorean Theorem to find the length of the radius:

A^2 + B^2 = C^2

A^2 + (8m)^2 = (10m)^2

A = 6m

Plugging in our values, we get:

$SA = \pi (6m)^2 + \pi (6m)(10m)$

$SA = 96 \pi m^2$

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

Find the surface area of the following half cone.

Half_cone

Possible Answers:

$120 \pi m^2 + 120 m^2$

$88 \pi m^2$

$88 \pi m^2 + 88 m^2$

$120 \pi m^2$

$100 \pi m^2 + 120 m^2$

Correct answer:

$100 \pi m^2 + 120 m^2$

Explanation:

The formula for the surface area of the half cone is:

$SA=(base)+\frac{1}{2}(circumference)(slant\ height)+(triangle)$

$SA=\left(\frac{\pi r^2}{2}\right)+\left(\frac{1}{2}\cdot \frac{2\pi r h_s}{2}\right)+rh$

Where is the radius, h_s is the slant height, and is the height of the cone.

Use the Pythagorean Theorem to find the height of the cone:

A^2+B^2 = C^2

A^2+(8m)^2=(17m)^2

A=15m

Plugging in our values, we get:

$SA=\left(\frac{\pi (8m)^2}{2}\right)+\left( \frac{\pi (8m) (17m)}{2}\right)+(8m)(15m)$

$SA=32 \pi m^2 + 68 \pi m^2 + 120m^2$

$SA = 100 \pi m^2 + 120 m^2$

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

Find the surface area of the following half-cylinder.

Half_cylinder

Possible Answers:

$150+100 \pi m^2$

$75\pi m^2$

$110 \pi m^2$

$90 \pi m^2$

$50+125 \pi m^2$

Correct answer:

$150+100 \pi m^2$

Explanation:

The formula for the surface area of a half-cylinder must include one-half of the surface area of a cylinder, which would be:

$SA = \frac{1}{2} (lateral\ area + 2 \cdot (base))$

We also need to add the area of the new rectangular face that is created by cutting the cylinder in half. The area of this rectangle would be:

$A=l\times w$

where the length of the rectangle is the same as the height of the half-cylinder, and the width of the rectangle is the same as the diameter of the base of the half-cylinder. So we can rewrite the area of the rectangle as:

$A=h\p\times d$

Now we can combine the two area formulas to find the total surface area of the half-cylinder:

$SA = \frac{1}{2} (2 \pi r (h) + 2 (\pi r^2))+h\cdot d$

$SA = \pi r (h) + \pi r^2+hd$

where is the radius of the base and is the length of the height, and is the diameter of the base.

Plugging in our values, we get:

$SA = \pi (5m) (15m) + \pi (5m)^2+15\cdot 10$

$SA = 75 \pi m^2 + 25 \pi m^2+150 = 150+100 \pi m^2$

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High School Math : Solid Geometry

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #11 : Solid Geometry

Example Question #541 : Geometry

Example Question #31 : Solid Geometry

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

Example Question #551 : Sat Subject Test In Math I

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

Example Question #4 : How To Find The Surface Area Of A Cone

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

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

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

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