GED Math - 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?

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.

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.

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

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

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

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

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 :

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

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

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

$512\pi \text{ in}^2$

$384\pi \text{ in}^2$

$1024\pi \text{ in}^2$

$64\pi \text{ in}^2$

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 8in.

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

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

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

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

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.

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

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

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

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

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 :

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

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

$384\text{in}^2$

$512\text{in}^2$

$200\text{in}^2$

$96\text{in}^2$

$128\text{in}^2$

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 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

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

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

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

Find the surface area of a sphere with a radius of 10in.

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

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

$40\pi \text{ in}^2$

$25\pi \text{ in}^2$

$75\pi \text{ in}^2$

Explanation

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

$SA = 4\pi r^2$

where r is the radius of the sphere.

Now, we know the radius of the sphere is 10in. So, we will substitute. We get

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

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

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

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

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

Find the surface area of a cube with a length of 9in.

$486\text{in}^2$

$729\text{in}^2$

$324\text{in}^2$

$108\text{in}^2$

$384\text{in}^2$

Explanation

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

where a is the length of any side of the cube.

Now, we know the length of the cube is 9in. Because it is a cube, all sides/lengths are equal. Therefore, we can use any side of the cube in the formula. So, we get

$SA = 6 \cdot (9\text{in})^2$

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

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

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.

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

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

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

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

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 :

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

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

$384\text{cm}^2$

$512\text{cm}^2$

$448\text{cm}^2$

$576\text{cm}^2$

$256\text{cm}^2$

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 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

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