# SAT II Math I : Surface Area

## Example Questions

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### Example Question #1 : Surface Area

A circular swimming pool has diameter 32 meters and depth  meters throughout. Which of the following expressions gives the total area of the inside of the pool, in square meters?

None of the other responses is correct.

Explanation:

The bottom of the pool is a circle with diameter 32, and, subsequently, radius half this, or 16; its area is

The side of the pool is the lateral surface of a cylinder with radius 16 and height ; the area of this is

The area of the inside of the pool is the sum of these two, or

### Example Question #2 : Surface Area

A circular swimming pool at an apartment complex has diameter 50 feet and depth six feet throughout.

The apartment manager needs to get the interior of the swimming pool painted. The paint she wants to use covers 350 square feet per gallon. How many one-gallon cans of paint will she need to purchase?

The correct answer is not given among the other responses.

Explanation:

The pool can be seen as a cylinder with depth (or height) six feet and a base with diameter 50 feet - and radius half this, or 25 feet.

The bottom of the pool - the base of the cylinder - is a circle with radius 25 feet, so its area is

square feet.

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

square feet.

Their sum - the total area to be painted - is  square feet. Since one gallon of paint covers 350 square feet, divide:

Eight cans of paint and part of a ninth will be required, so the correct response is nine.

### Example Question #3 : Surface Area

Figure not drawn to scale.

Find the surface area of the cylinder above.

73.44 in2

56.55 in2

94.25 in2

87.25 in2

122.14 in2

94.25 in2

Explanation:

In order to find the surface area of a cylinder, you need to find the surface areas of the circles that are the top and bottom of the cylinder (2 x pi x radius2) and add it to the surface are of the rectangle that is the side of the cylinder (diameter x height).

### Example Question #4 : Surface Area

Figure not drawn to scale

# What is the surface area of the sphere above?

512.63 yd2

712.12 yd2

815.44 yd2

615.75 yd2

615.75 yd3

615.75 yd2

Explanation:

In order to find the surface area of a sphere, you must use the equation below:

### Example Question #5 : Surface Area

What is the surface area of the following cone?

Explanation:

The formula for the surface area of a cone is:

,

where represents the radius of the cone base and represents the slant height of the cone.

Plugging in our values, we get:

### Example Question #2 : Solid Geometry

The surface area of cone  is . If the radius of the base of the cone is , what is the height of the cone?

Explanation:

To figure out , we must use the equation for the surface area of a cone, , where  is the radius of the base of the cone and  is the length of the diagonal from the tip of the cone to any point on the base's circumference. We therefore first need to solve for  by plugging what we know into the equation:

This equation can be reduced to:

For a normal right angle cone,  represents the line from the tip of the cone running along the outside of the cone to a point on the base's circumference. This line represents the hypotenuse of the right triangle formed by the radius and height of the cone. We can therefore solve for  using the Pythagorean theorem:

so

Our  is therefore:

The height of cone  is therefore

### Example Question #6 : Surface Area

A circle of radius five is cut into two pieces,  and . The larger section is thrown away. The smaller section is curled until the two straight edges meet, and a bottom is made for the cone.

What is the area of the bottom?

Explanation:

When the smaller portion of the circle is curled in, it will make the top of a cone. The circumfrence of the circle on the bottom is  (where r is the radius of the circle on the bottom). The circumference of the bottom is also  of the circumfrence of the original larger circle, which is  (where R is the radius of the original, larger circle)

Therefore we use the circumference formula to solve for our new r:

Substituting this value into the area formula, the area of the small circle becomes:

### Example Question #7 : Surface Area

A cone has a bottom area of  and a height of , what is the surface area of the cone?

Explanation:

The area of the bottom of the cone yields the radius,

The height of the cone is , so the Pythagorean Theorem will give the slant height,

The area of the side of the cone is  and adding that to the  given as the area of the circle, the surface area comes to

### Example Question #8 : Surface Area

If the surface area of a right angle cone  is , and the distance from the tip of the cone to a point on the edge of the cone's base is , what is the cone's radius?

Explanation:

Solving this problem is going to take knowledge of Algebra, Geometry, and the equation for the surface area of a cone: , where  is the radius of the cone's base and  is the distance from the tip of the cone to a point along the edge of the cone's base. First, let's substitute what we know in this equation:

We can divide out  from every term in the equation to obtain:

We see this equation has taken the form of a quadratic expression, so to solve for  we need to find the zeroes by factoring. We therefore need to find factors of  that when added equal . In this case,  and :

This gives us solutions of  and . Since  represents the radius of the cone and the radius must be positive, we know that  is our only possible answer, and therefore the radius of the cone is .

### Example Question #9 : Surface Area

For a right circular cone , the radius is  and the height of the cone is . What is the surface area of the cone in terms of ?