Basic Geometry : Plane Geometry

Example Questions

Example Question #31 : Basic Geometry

The diameter of a circle is 16 centimeters.  What is the area of the circle in centimeters?

Explanation:

The area of a circle is found by squaring the circle's radius and multiplying the result by 3.14.  The radius is half of the diameter (8 in this case).

Example Question #72 : Circles

A circle is inscribed in a square whose side is 6 in. What is the difference in area between the square and the circle, rounded to the nearest square inch?

Explanation:

The circle is inscribed in a square when it is drawn within the square so as to touch in as many places as possible. This means that the side of the square is the same as the diameter of the circle.

Let

So the approximate difference is in area

Example Question #21 : How To Find The Area Of A Circle

Two equal circles are cut out of a rectangular sheet of paper with the dimensions 10 by 20. The circles were made to have the greatest possible diameter. What is the approximate area of the paper after the two circles have been cut out?

23

16

56

43

43

Explanation:

The length of 20 represents the diameters of both circles. Each circle has a diameter of 10 and since radius is half of the diameter, each circle has a radius of 5. The area of a circle is A = πr2 . The area of one circle is 25π. The area of both circles is 50π. The area of the rectangle is (10)(20) = 200. 200 - 50π gives you the area of the paper after the two circles have been cut out. π is about 3.14, so 200 – 50(3.14) = 43.

Example Question #564 : Plane Geometry

Kate has a ring-shaped lawn which has an inner radius of 10 feet and an outer radius 25 feet. What is the area of her lawn?

525π ft2

175π ft2

125π ft2

325π ft2

275π ft2

525π ft2

Explanation:

The area of an annulus is

where  is the radius of the larger circle, and  is the radius of the smaller circle.

Example Question #22 : How To Find The Area Of A Circle

A 6 by 8 rectangle is inscribed in a circle. What is the area of the circle?

Explanation:

The image below shows the rectangle inscribed in the circle. Dividing the rectangle into two triangles allows us to find the diameter of the circle, which is equal to the length of the line we drew. Using a2+b2= c2 we get 6+ 82 = c2. c2 = 100, so c = 10. The area of a circle is  . Radius is half of the diameter of the circle (which we know is 10), so r = 5.

Example Question #31 : Basic Geometry

What is the ratio of the area of a circle to the area of the square it circumscribes?

Explanation:

Since each angle of a square is a right angle, each intercepts a semicircle; as a result, a diagonal of the square doubles as a diameter of the circle. We will call that common measure , as it will make our ratio easier to calculate.

The area of the circle is .

The area of the square is  .  This formula is important, even though it is not the more commonly known formula of Area = side2.

Therefore the ratio of the area of the circle to that of the square is  to , which simplifies to  to 2.

Example Question #31 : Circles

What is the area of a circle with a circumference of 12 inches?

Explanation:

The relationship of a circumference of a circle to its radius is .

Equivalently, .

Set :

To find the area, use the formula .

Plug in our new value of r:

square inches

Example Question #23 : How To Find The Area Of A Circle

A park wants to build a circular fountain with a walkway around it.  The fountain will have a radius of 40 feet, and the walkway is to be 4 feet wide.  If the walkway is to be poured at a depth of 1.5 feet, how many cubic feet of concrete must be mixed to make the walkway?

None of the other answers are correct.

Explanation:

The following diagram will help to explain the solution:

We are searching for the surface area of the shaded region.  We can multiply this by the depth (1.5 feet) to find the total volume of this area.

The radius of the outer circle is 44 feet.  Therefore its area is 442π = 1936π.  The area of the inner circle is 402π = 1600π.  Therefore the area of the shaded area is 1936π – 1600π = 336π.  The volume is 1.5 times this, or 504π.

Example Question #567 : Plane Geometry

How many times greater is the area of a circle with a radius of 4in., compared to a circle with a radius of 2in.?

Explanation:

The area of a circle can be solved using the equation

The area of a circle with radius 4 is while the area of a circle with radius 2 is .

Example Question #21 : How To Find The Area Of A Circle

What is the area of a circle whose diameter is 8?

32π

8π

16π

64π

12π