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GED Math : Faces and Surface Area

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Example Question #1 : Faces And Surface Area

A circular swimming pool at an apartment complex has diameter 18 meters and depth 2.5 meters throughout.

The apartment manager needs to get the interior of the swimming pool painted. The paint she wants to use covers 40 square meters per can. How many cans of paint will she need to purchase?

You may use 3.14 for $\pi$ .

Possible Answers:

Correct answer:

Explanation:

The pool can be seen as a cylinder with depth (or height) 2.5 meters and a base with diameter 18 meters - and radius half this, or 9 meters.

The bottom of the pool - the base of the cylinder - is a circle with radius 9 meters, so its area is

$B = \pi r ^{2} = 3.14 \cdot 9 \cdot 9 = 254.34$ square meters.

Its side - the lateral face of the cylinder - has area

$LA = 2 \pi r h = 2 \cdot 3.14 \cdot 18 \cdot 2.5 = 141.3$ square meters.

Their sum - the total area to be painted - is 254.34 + 141.3 = 395.64 square feet. Since one can of paint covers 40 square meters, divide:

$395.64 \div 40 \approx 9.891$

Nine cans of paint and part of a tenth will be required, so the correct response is ten.

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Example Question #2 : Faces And Surface Area

A water tank takes the shape of a sphere whose exterior has radius 24 feet; the tank is six inches thick throughout. To the nearest hundred, give the surface area of the interior of the tank in square feet.

Use 3.14 for $\pi$ .

Possible Answers:

$50,900 \textrm{ ft}^{2}$

$6,600 \textrm{ ft}^{2}$

$54,300 \textrm{ ft}^{2}$

$6,900 \textrm{ ft}^{2}$

Correct answer:

$6,900 \textrm{ ft}^{2}$

Explanation:

Six inches is equal to 0.5 feet, so the radius of the interior of the tank is

24-0.5= 23.5 feet.

The surface area of the interior of the tank can be calculated using the formula

$SA = 4 \pi r^{2}$

$SA \approx 4 \cdot 3.14 \cdot 23.5 ^{2} \approx 6,936$ ,

which rounds to 6,900 square feet.

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Example Question #3 : Faces And Surface Area

Cone_1

Give the total surface area of the above cone to the nearest square meter.

Possible Answers:

$126\textrm{ m}^{2}$

$176\textrm{ m}^{2}$

$186\textrm{ m}^{2}$

$135\textrm{ m}^{2}$

Correct answer:

$186\textrm{ m}^{2}$

Explanation:

The base is a circle with radius r = 4 , and its area can be calculated using the area formula for a circle:

$B= \pi r^{2} = \pi \times 4^{2} \approx 3.14 \times 16 = 50.24$ square meters.

To find the lateral area, we need the slant height of the cone. This can be found by way of the Pythagorean Theorem. Treating the height h = 10 and the radius r = 4 as the legs and slant height as the hypotenuse, calculate:

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

$l^{2} = 10^{2} + 4^{2} = 100 + 16 = 116$

$l = \sqrt{116} \approx 10.77$ meters.

The formula for the lateral area can be applied now:

$LA = \pi r l \approx 3.14 \cdot 4 \cdot 10.77 \approx 135.34$

Add the base and the lateral area to obtain the total surface area:

$SA = B + LA \approx 50.24 + 135.34 \approx 185.58$ .

This rounds to 186 square meters.

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Example Question #1 : Faces And Surface Area

A water tank takes the shape of a closed cylinder whose exterior has a height of 40 feet and a base with radius 15 feet; the tank is three inches thick throughout. To the nearest hundred, give the surface area of the interior of the tank in square feet.

Use 3.14 for $\pi$ .

Possible Answers:

$5,000\textrm{ ft}^{2}$

$3,600\textrm{ ft}^{2}$

$3,700\textrm{ ft}^{2}$

$4,900\textrm{ ft}^{2}$

Correct answer:

$5,000\textrm{ ft}^{2}$

Explanation:

Three inches is equal to 0.25 feet, so the height of the interior of the tank is

$40 - 2 \times 0.25 = 39.5$ feet.

The radius of the interior of the tank is

15 - 0.25 = 14.75 feet.

The surface area of the interior of the tank can be determined by using this formula:

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

$SA = 2 \cdot 3.14 \cdot 14.75 (14.75 + 39.5)$

$SA \approx 2 \cdot 3.14 \cdot 14.75 \cdot 54.25\approx 5,025$ ,

which rounds to 5,000 square feet.

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Example Question #1 : Faces And Surface Area

Cone_1

Above is a diagram of a conic tank that holds a city's water supply.

The city wishes to completely repaint the exterior of the tank - sides and base. The paint it wants to use covers 40 square meters per gallon. Also, to save money, the city buys the paint in multiples of 25 gallons.

How many gallons will the city purchase in order to paint the tower?

Possible Answers:

1,450

375

275

975

Correct answer:

375

Explanation:

The surface area of a cone with radius and slant height is calculated using the formula $S = \pi r^{2} + \pi r l$ .

Substitute 35 for and 100 for to find the surface area in square meters:

$S = \pi \cdot 35^{2} + \pi \cdot 35 \cdot 100$

$\approx 3.14 \cdot 1,225 + 3.14\cdot 35 \cdot 100$

$\approx 3,848 + 10,996$

$\approx 14,844$ square meters.

The paint covers 40 square meters per gallon, so the city needs

$14,844 \div 40 =371.1$ gallons of paint.

Since the city buys the paint in multiples of 25 gallons, it will need to buy the next-highest multiple of 25, or 375 gallons.

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Example Question #1 : Faces And Surface Area

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length three inches. Give the total surface area of the octahedron.

Possible Answers:

$18\sqrt{2} \textrm{ in}^{2}$

$18\sqrt{3} \textrm{ in}^{2}$

$9\sqrt{2} \textrm{ in}^{2}$

$9\sqrt{3} \textrm{ in}^{2}$

Correct answer:

$18\sqrt{3} \textrm{ in}^{2}$

Explanation:

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$ .

Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is

$SA = 8 A = 8 \cdot \frac{s^{2}\sqrt{3}}{4} = 2 s^{2}\sqrt{3}$ .

Substitute s = 3 :

$SA =2 \cdot 3^{2}\sqrt{3} = 18 \sqrt{3}$ square inches.

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Example Question #7 : Faces And Surface Area

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

A given regular icosahedron has edges of length two inches. Give the total surface area of the icosahedron.

Possible Answers:

$10 \sqrt{3} \textrm{ in}^{2}$

$20 \sqrt{3} \textrm{ in}^{2}$

$10 \sqrt{2} \textrm{ in}^{2}$

$20 \sqrt{2} \textrm{ in}^{2}$

Correct answer:

$20 \sqrt{3} \textrm{ in}^{2}$

Explanation:

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$ .

Since there are twenty equilateral triangles that comprise the surface of the icosahedron, the total surface area is

$SA = 20 A = 20 \cdot \frac{s^{2}\sqrt{3}}{4} = 5 s^{2}\sqrt{3}$ .

Substitute s = 2 :

$SA = 5 \cdot 2^{2}\sqrt{3} = 5 \cdot 4 \sqrt{3} = 20 \sqrt{3}$ square inches.

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Example Question #8 : Faces And Surface Area

A regular tetrahedron has four congruent faces, each of which is an equilateral triangle.

A given tetrahedron has edges of length five inches. Give the total surface area of the tetrahedron.

Possible Answers:

$25 \sqrt{2}\textrm{ in}^{2}$

$25 \sqrt{3}\textrm{ in}^{2}$

$\frac{25\sqrt{3}}{2} \textrm{ in}^{2}$

$\frac{25\sqrt{2}}{2} \textrm{ in}^{2}$

Correct answer:

$25 \sqrt{3}\textrm{ in}^{2}$

Explanation:

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$ .

Since there are four equilateral triangles that comprise the surface of the tetrahedron, the total surface area is

$SA = 4 A = 4 \cdot \frac{s^{2}\sqrt{3}}{4} = s^{2}\sqrt{3}$ .

Substitute s = 5 :

$SA = 5^{2}\sqrt{3} = 25 \sqrt{3}$ square inches.

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Example Question #9 : Faces And Surface Area

A cube has a height of 9cm. Find the surface area.

Possible Answers:

$324\text{cm}^2$

$486\text{cm}^2$

$108\text{cm}^2$

$405\text{cm}^2$

$729\text{cm}^2$

Correct answer:

$486\text{cm}^2$

Explanation:

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

$SA = 6 \cdot l \cdot w$

where l is the length, and w is the width of the cube.

Now, we know the height of the cube is 9cm. Because it is a cube, all lengths, widths, and heights are the same. Therefore, the length and the width are also 9cm.

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

$SA = 6 \cdot 9\text{cm} \cdot 9\text{cm}$

$SA = 6 \cdot 81\text{cm}^2$

$SA = 486\text{cm}^2$

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Example Question #1 : Faces And Surface Area

A sphere has a radius of 7in. Find the surface area.

Possible Answers:

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

$56\pi \text{ in}^2$

$196\pi \text{ in}^2$

$294\pi \text{ in}^2$

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

Correct answer:

$196\pi \text{ in}^2$

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 7in.

So, we can substitute into the formula. We get

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

$SA = 4 \cdot \pi \cdot 49\text{in}^2$

$SA = 196\text{in}^2 \cdot \pi$

$SA = 196\pi \text{ in}^2$

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GED Math : Faces and Surface Area

Study concepts, example questions & explanations for GED Math

All GED Math Resources

Example Questions

Example Question #1 : Faces And Surface Area

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