Question

Find the area of the circles.

Answer 1

See Explanation!

Explanation:

Brief History:

While a circle, symbolically, represents many different things to many different groups of people including concepts such as eternity, timelessness, and totality, a circle by definition is a simple closed shape. It is a set of all points in a plane that are equidistant from a given point, called the center. It can also be defined as a curve traced by a point where the distance from a given point remains constant as the point moves. The distance between any point of a circle and the center of a circle is called its radius, while the diameter of a circle is defined as the largest distance between any two points on a circle. Essentially, the diameter is twice the radius, as the largest distance between two points on a circle has to be a line segment through the center of a circle. The circumference of a circle can be defined as the distance around the circle, or the length of a circuit along the circle. All of these values are related through the mathematical constant π, or pi, which is the ratio of a circle's circumference to its diameter, and is approximately 3.14159. π is an irrational number meaning that it cannot be expressed exactly as a fraction (though it is often approximated as 22/7) and its decimal representation never ends or has a permanent repeating pattern. It is also a transcendental number, meaning that it is not the root of any non-zero, polynomial that has rational coefficients. Interestingly, the proof by Ferdinand von Lindemann in 1880 that π is transcendental finally put an end to the millennia-old quest that began with ancient geometers of "squaring the circle." This involved attempting to construct a square with the same area as a given circle within a finite number of steps, only using a compass and straightedge. While it is now known that this is impossible, and imagining the ardent efforts of flustered ancient geometers attempting the impossible by candlelight might evoke a ludicrous image, it is important to remember that it is thanks to people like these that so many mathematical concepts are well defined today.

Solution: (Circle 1)

⇒ Given :

• Radius is 5
• Diameter is 10 (Can be known by adding the radius together.)

Using the formulas

`A=\pi r^(2) \\d=2r`

Solving for
`A`

`A=(1)/(4) \pi d^(2) =(1)/(4) *\pi *10^(2)` ≈
`78.53982`

`A` ≈
`78.54...rounded`

Solution: (Circle 2)

⇒ Given :

• Diameter is 16
• Radius is 8

Using the formulas

`A=\pi r^(2) \\\d=2r`

Solving for
`A`

`A=(1)/(4) \pi d^(2) *\pi *16^(2)` ≈
`201.06193`

`A` ≈
`201.06...rounded`

Answer 2

200.96 square units

Explanation:

d = 16

r = diameter ÷ 2 = 16 ÷ 2

r = 8

`\boxed{Area \ of \ circle = \pi r^2}`

= 3.14 * 8* 8

= 200.96 square units