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ACT Math : Solid Geometry

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Example Questions

Example Question #1 : Cones

The radius of the base of a cone is ; its height is twice of the diameter of that base. Give its surface area in terms of .

Possible Answers:

$\left ( 1+2 \sqrt{5} \right ) \pi r^{2}$

$\left ( 1+ \sqrt{17} \right ) \pi r^{2}$

$5 \pi r^{2}$

$3 \pi r^{2}$

$\left ( 1+ \sqrt{5} \right ) \pi r^{2}$

Correct answer:

$\left ( 1+ \sqrt{17} \right ) \pi r^{2}$

Explanation:

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r l$ .

The base has radius and diameter . The height is twice the diamter, which is . Its slant height can be calculated using the Pythagorean Theorem:

$l = \sqrt{r^{2}+ h^{2}}$

$l = \sqrt{r^{2}+ (4r)^{2}}$

$l = \sqrt{r^{2}+ 16r^{2}}$

$l = \sqrt{17r^{2}}$

$l = r \sqrt{17 }$

Substitute $r \sqrt{17 }$ for in the surface area formula:

$A = \pi r^{2} + \pi r \cdot r \sqrt{17}$

$A = \pi r^{2} + \pi r^{2} \sqrt{17}$

$A = \left ( 1+ \sqrt{17} \right ) \pi r^{2}$

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Example Question #22 : Surface Area

The height of a cone is ; the diameter of its base is twice the height. Give its surface area in terms of .

Possible Answers:

$5 \pi h^{2}$

$\left (1 + \sqrt{2} \right ) \pi h^{2}$

$10 \pi h^{2}$

$4 \pi h^{2}$

$\left (2 +2 \sqrt{2} \right ) \pi h^{2}$

Correct answer:

$\left (1 + \sqrt{2} \right ) \pi h^{2}$

Explanation:

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r l$ .

The diameter of the base is twice the height, which is ; the radius is half this, which is .

The slant height can be calculated using the Pythagorean Theorem:

$l = \sqrt{r^{2}+ h^{2}}$

$l = \sqrt{h^{2}+ h^{2}}$

$l = \sqrt{2h^{2}}$

$l =h \sqrt{2}$

Substitute $h \sqrt{2}$ for and for in the surface area formula:

$A = \pi h^{2} + \pi h \cdot h \sqrt{2}$

$A = \pi h^{2} + \pi h ^{2} \sqrt{2}$

$A = \left (1 + \sqrt{2} \right ) \pi h^{2}$

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Example Question #1 : Cones

The circumference of the base of a cone is 80; the slant height of the cone is equal to twice the diameter of the base. Give the surface area of the cone (nearest whole number).

Possible Answers:

2,485

2,039

4,077

1,975

2,546

Correct answer:

2,546

Explanation:

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r l$ .

The slant height is twice the diameter, or, equivalently, four times the radius, so

l = 4r

and

$A = \pi r^{2} + \pi r \cdot 4r$

$A = \pi r^{2} +4 \pi r^{2}$

$A = 5 \pi r^{2}$

The radius of the base is the circumference divided by $2\pi$ , which is

$80 \div 2 \pi \approx 80 \div (2 \times 3.14 )\approx 12.73$

Substitute:

$A = 5 \pi r^{2}$

$A \approx 5 \cdot 3.14 \cdot (12.73)^{2}$

$A \approx 2,546$

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Example Question #3 : Cones

The circumference of the base of a cone is 100; the height of the cone is equal to the diameter of the base. Give the surface area of the cone (nearest whole number).

Possible Answers:

8,443

1,779

2,575

1,589

2,385

Correct answer:

2,575

Explanation:

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r l$ .

The diameter of the base is the circumference divided by $\pi$ , which is

$d = C \div (2\pi ) \approx 100 \div 3.14 \approx 31.83$

This is also the height .

The radius is half this, or

$r = d \div 2 \approx 31.83 \div 2 \approx 15.92$

The slant height can be found by way of the Pythagorean Theorem:

$l = \sqrt{r^{2}+ h ^{2}}$

$l \approx \sqrt{15.92^{2}+ 31.83 ^{2}}$

$l \approx \sqrt{253.30+ 1,013.21}$

$l \approx \sqrt{1266.51}$

$l \approx 35.60$

Substitute in the surface area formula:

$A = \pi r^{2} + \pi r l$

$A \approx 3.14 \cdot 15.92 ^{2} + 3.14 \cdot 15.92 \cdot 35.60$

$A \approx 795.8+ 1,780.1$

$A \approx 2,575$

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Example Question #3 : Cones

A heat shield on a particular satellite takes the form of a cone. If the surface area of the "face" of the cone (not counting the disk on the bottom) is 2744.36ft^2 , and the stripe of reflective paint from the tip of the cone is down to the base is feet long, what is the diameter of the disk in feet? Round $\pi$ to 3 significant digits. Round your final answer to the nearest foot.

Possible Answers:

12ft

51ft

46ft

38ft

28ft

Correct answer:

46ft

Explanation:

In this problem, we only need to consider the part of the formula for conic surface area that deals with slant height, since that is all the information we have.

We know the formula for the lateral surface area of a cone is $\pi r l$ , and we know that is feet. Plugging in our other values gives us:

$\pi r (38) = 2744.36$

Simplify:

$\pi r (38) = 2744.36$

$r = \frac{2744.36}{(38)(3.14)} \approx 23$

Thus, if our radius is approximately feet, our diameter is approximately feet.

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Example Question #9 : How To Find The Surface Area Of A Cone

What is the surface area in square units of a cone with radius units and height units?

Possible Answers:

$470\pi$

$323\pi$

$677\pi$

$224\pi$

$558\pi$

Correct answer:

$224\pi$

Explanation:

The formula for surface area of a cone is:

$\pi r^2 + \pi r l$ , where is the slant height. Since we know the radius, we can calculate the first part without issue:

$\pi r^2 = \pi (7)^2 = 49\pi$

The second part requires us to calculate slant height. Since all cones have a right angle created by the base and height perpendicular to the base, we can use the Pythagorean theorem to calculate :

a^2 + b^2 = l^2

7^2 + 24^2 = l^2 = 625

l = 25

Now, we can complete our formula. Don't forget to add in the circular base.

$\pi r^2 + \pi r l = 49\pi + \pi (7)(25) = 49\pi + 175\pi = 224\pi$

Thus, our surface area is $224\pi$ square units.

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Example Question #771 : Geometry

Some conical party hats are being decorated with glitter paper. If the base of the hats is inches across and the slant height is inches, how many square inches of glitter paper are needed to decorate one hat?

Possible Answers:

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

$14\pi\textup{ in}^2$

$10\pi\textup{ in}^2$

$57\pi\textup{ in}^2$

$33\pi\textup{ in}^2$

Correct answer:

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

Explanation:

The formula for the surface area of a cone is

$\pi r^2 + \pi rl$ , where is the slant height. Since we're only concerned with the slant height portion of the surface area formula, we can ignore the $\pi r^2$ portion of the equation.

Plug in known values to this part of the equation and solve.

$\pi rl = \pi (6)(8) = 48\pi$

So, each hat requires $48\pi \textup{ in}^2$ of glitter paper.

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Example Question #781 : Geometry

The surface area of a cone equals $24\pi \textup{cm}^2.$ If the radius of the cone equals $3\textup{cm}$ , what is the height of the cone?

Possible Answers:

$4\textup{cm}$

$8\textup{cm}$

$5\textup{cm}$

$3\textup{cm}$

$6\textup{cm}$

Correct answer:

$4\textup{cm}$

Explanation:

To solve this question, you need to know the surface area of a cone equation: $\textup{surface area}=\pi r(r+\sqrt{h^2 +r^2}),$ where is the radius of the cone $(3\textup{cm})$ , and is the height of the cone. We must substitute all of the values that we know, and solve for the height of the cone to get the answer.

$24\textup{cm}^2=\pi *3\textup{cm}(3\textup{cm}+\sqrt{h^2 +(3\textup{cm})^2}),$ Therefore, the height of the cone is $4\textup{cm.}$

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Example Question #4 : Solid Geometry

You have an empty cylinder with a base diameter of 6 and a height of 10 and you have a cone full of water with a base radius of 3 and a height of 10. If you empty the cone of water into the cylinder, how much volume is left empty in the cylinder?

Possible Answers:

$60\pi$

$45\pi$

$30\pi$

$120\pi$

$90\pi$

Correct answer:

$60\pi$

Explanation:

Cylinder Volume = $\pi*r^{2}*h$

Cone Volume = $\frac{1}{3}*\pi*r^{2}*h$

Cylinder Diameter = 6, therefore Cylinder Radius = 3

Cone Radius = 3

Empty Volume = Cylinder Volume – Cone Volume

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

$=\pi*3^{2}*10-\frac{1}{3}*\pi*3^{2}*10$

$=\pi*90-\pi*30$

$=\pi*60$

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

You have two cones, one with a diameter of and a height of and another with a diameter of and a height of . If you fill the smaller cone with sand and dump that sand into the larger cone, how much empty space will be left in the larger cone?

Possible Answers:

$48\pi$

$12\pi$

$24\pi$

$6\pi$

$8\pi$

Correct answer:

$24\pi$

Explanation:

1. Find the volume of each cone:

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

Cone 1:

Since the diameter is , the radius is .

$Volume=\frac{1}{3}\pi (3^{2})(10)=30\pi$

Cone 2:

Since the diameter is , the radius is 1.5 .

$Volume=\frac{1}{3}\pi (1.5^{2})(8)=6\pi$

2. Subtract the smaller volume from the larger volume:

$30\pi-6\pi=24\pi$

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ACT Math : Solid Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Cones

Example Question #22 : Surface Area

Example Question #1 : Cones

Example Question #3 : Cones

Example Question #3 : Cones

Example Question #9 : How To Find The Surface Area Of A Cone

Example Question #771 : Geometry

Example Question #781 : Geometry

Example Question #4 : Solid Geometry

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

All ACT Math Resources

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