SAT Math : Plane Geometry

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #331 : Plane Geometry

Find the area of a circle given radius of 7.

Possible Answers:

Correct answer:

Explanation:

To solve, simply use the formula for the area of a circle. Thus,

To remember this formula, think about area and how it is in two dimensions, not one or three. So, one of our variables will have to be squared. By process of elimination, it must be r because pi doesn't change so it only makes sense to square the variable that changes between all circles.

Example Question #332 : Sat Mathematics

The circumference of a circle is . What is its area?

Possible Answers:

None of the given answers. 

Correct answer:

Explanation:

The circumference C of a circle with radius r is expressed by:

Our circle's circumference is . Therefore,

Plug this r-value into the formula for the area of a circle.

 

Example Question #33 : How To Find The Area Of A Circle

Target

In the above figure, .

What percent of the figure is white?

Possible Answers:

Correct answer:

Explanation:

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for  in the formula :

 - this is the area of the white circle in the center.

 - this is the total area of the figure.

The area of the white ring is the difference between those at of the second-largest and third-largest circles:

The total area of the white portion of the figure is 

which is 

of the figure.

Example Question #334 : Geometry

A circle with radius  units fits inside of a square such as the circle is tangent to the sides of the square. What is the area inside of the square but outside of the circle? 

Possible Answers:

 square units

 square units

 square units

 square units

 square units

Correct answer:

 square units

Explanation:

If the radius of the circle is  units, that means that the diameter is  units. Because the circle is tangent to the sides of the square, we also know that each side of the square is also  units. 

The area of the square is given by its length times its width, or:

The area of the circle is given by:

The area that we're looking for lies outside of the circle but inside of the square. To get this area, we subtract the area of the circle from the area of the square. This gives us our answer of  square units. 

Example Question #335 : Geometry

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The area of the square in this figure is . What is the area of the entire figure given that the semi-circle atop the square is exactly half of a full circle.

Possible Answers:

None of the given answers. 

Correct answer:

Explanation:

If the area of the square is , then that means that each side is  since the area of a square is given by 

In the figure, the side of the square is equal to the diameter of the semicircle. Therefore, the semicircle's radius is 

The area of the semicircle is given by:

The area of the entire figure is the sum of the two component areas. 

Our total area is 

Example Question #91 : Circles

A circle is circumscribed about a regular hexagon of perimeter 120, Give its area. 

Possible Answers:

Correct answer:

Explanation:

Below is the figure referenced; note that the hexagon is divided by its diameters.

Hexagon 3

The hexagon is divided into six equilateral triangles; the length of each side is equal to the radius of the hexagon, and, consequently, the radius of the circle. The regular hexagon has perimeter 120, so the length of each congruent side is . The radius of the hexagon, and of the circle, is 20 also. 

The area of the circle can be found using the area formula, setting :

 

Example Question #337 : Geometry

A circle is inscribed in a regular hexagon of perimeter 120, Give its area. 

Possible Answers:

Correct answer:

Explanation:

Below is the figure referenced; note that the hexagon is divided by its diameters, and that an apothem - a perpendicular segment from the center to one side - has been drawn.

Hexagon 3

The hexagon is divided into six equilateral triangles. The regular hexagon has perimeter 120, so the length of each congruent side is .

Specifically,  is a 30-60-90 triangle.  is the midpoint of , so 

By the 30-60-90 Theorem, since  and  are the short and long legs of a right triangle, 

.

This is the apothem of the hexagon; this is also the radius of the circle. 

The area of the circle can be found using the area formula, setting :

 

Example Question #332 : Plane Geometry

What is  in radians?

Possible Answers:

Correct answer:

Explanation:

To convert degrees to radians, we need to remember the following formula.

.

Now lets substitute for degrees.

Example Question #335 : Sat Mathematics

A circle is circumscribed by a square, and that square is the base of a cube. If the volume of the cube is 216, what is the area of the circle?

Possible Answers:

Correct answer:

Explanation:

The equation for the volume of a cube is 

If V= 216, then s = 6. 

When a circle is circumscribed by a square the diameter of the circle is equal to the length of one side of the square. In this case, one side of the square is equal to 6; therefore the diameter of the circle is also 6. We can find the area of a circle with the equation 

and since the radius is equal to half the diameter: 

Example Question #1 : Circumference Of A Circle

If a circle has an area of , what is the circumference of the circle?

Possible Answers:

Correct answer:

Explanation:

The formula for  the area of a circle is πr2. For this particular circle, the area is 81π, so 81π = πr2. Divide both sides by π and we are left with r2=81. Take the square root of both sides to find r=9. The formula for the circumference of the circle is 2πr = 2π(9) = 18π. The correct answer is 18π.

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