3-Dimensional Geometry - GED Math

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Question

How many edges and vertices are found on a square pyramid?

Answer

The base of a square pyramid is, as the name suggests, a square which has edges and vertices. The vertices of the square each have edges that meet at a single point, adding an additional vertex and additional edges. Together, a square pyramid has edges and vertices.

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Question

How many vertices does an octagonal pyramid have?

Answer

An octagonal pyramid has a base with eight vertices, each of which is a vertex of the pyramid. There is one more vertex, or the apex, which is connected to each of the vertices of the base by an edge. Nine is the correct choice.

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Question

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 $\pi$ .

Answer

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 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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Question

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.

Answer

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 :

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

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Question

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.

Answer

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 :

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

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Question

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.

Answer

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 :

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

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Question

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 $\pi$ .

Answer

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

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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Question

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 $\pi$ .

Answer

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

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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Question

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

Answer

The base is a circle with radius , 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 and the radius 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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Question

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?

Answer

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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Question

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

Answer

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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Question

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

Answer

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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Question

Find the surface area of a cube with a length of 12in.

Answer

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 length of the cube is 12in. Because it is a cube, all sides are equal. Therefore, the width is also 12in. So, we can substitute. We get

$SA = 6 \cdot 12\text{in} \cdot 12\text{in}$

$SA = 6 \cdot 144\text{in}^2$

$SA = 864\text{in}^2$

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Question

A cube has a height of 8cm. Find the surface area.

Answer

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 8cm. Because it is a cube, all lengths, widths, and heights are the same. Therefore, the length and the width are also 8cm.

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

$SA = 6 \cdot 8\text{cm} \cdot 8\text{cm}$

$SA = 6 \cdot 64\text{cm}^2$

$SA = 384\text{cm}^2$

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Question

A sphere has a radius of 5in. Find the surface area.

Answer

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 5in.

So, we can substitute into the formula. We get

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

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

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

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

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Question

Find the surface area of a cube with a height of 13in.

Answer

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 length of the cube is 13in. Because it is a cube, all sides are equal. Therefore, the width is also 13in. So, we can substitute. We get

$SA = 6 \cdot 13\text{in} \cdot 13\text{in}$

$SA = 6 \cdot 169\text{in}^2$

$SA = 1014\text{in}^2$

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Question

Find the surface area of a sphere with a radius of 8in.

Answer

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 8in.

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

$SA = 4 \cdot \pi \cdot (8\text{in})^2$

$SA = 4 \cdot \pi \cdot 64\text{in}^2$

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

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Question

Give the surface area of the above sphere.

Answer

Given the radius of a sphere, the surface area can be calculated using the formula

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

Set $r = \sqrt{7}$ in the formula and evaluate:

$A = 4 \pi ( \sqrt{7})^{2}$

$A = 4 \pi \cdot 7$

$A = 28 \pi$

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Question

A cube has a width of 8in. Find the surface area.

Answer

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 width of the cube is 8in. Because it is a cube, all sides/lengths are equal. Therefore, the length is also 8in. So, we can substitute. We get

$SAA = 6 \cdot 8\text{in} \cdot 8\text{in}$

$SA = 6 \cdot 64\text{in}^2$

$SA = 384\text{in}^2$

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Question

Let $\pi = 3.14$

Find the surface area of a sphere with a radius of 5in.

Answer

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 $\pi = 3.14$ . We know the radius of the sphere is 5in. So, we can substitute. We get

$SA = 4 \cdot 3.14 \cdot (5\text{in})^2$

$SA = 4 \cdot 3.14 \cdot 25\text{in}^2$

$SA = 4 \cdot 25\text{in}^2 \cdot 3.14$

$SA = 100\text{in}^2 \cdot 3.14$

$SA = 314\text{in}^2$

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