7th Grade - Area of Circles

Introduction

• A circle is a two-dimensional closed figure formed by joining a set of all the points equidistant from a fixed point.
• The fixed point is called the circle's center, and the fixed distance is called the circle's radius.

• The most extended chord that passes through the center of a circle joining the two points on the circumference of a circle, is called its diameter.

$d=2r$, where $d=$diameter of the circle & $r=$radius of the circle.

• The circumference of a circle is the length of the boundary of a circle, and can be calculated using the formula $C=2\mathrm{\pi r}$, where $r=$radius of the circle.

Area of Circles

• The area of a circle is the measure of the region enclosed inside it.
• The area of a circle depends on the length of its radius.
• If we know the radius of a circle, we can calculate its area using the formula mentioned below.

$Are{a}_{circle}=\mathrm{\pi }·{\mathrm{r}}^2$, where $\mathrm{\pi }=\frac{22}{7}\approx 3.14$, and $r=$the radius of the circle.

• Likewise, if the diameter of a circle is known, its area can be calculated using the formula mentioned below.

$Are{a}_{circle}=\frac{\mathrm{\pi }}{4}·{\mathrm{d}}^2$, where $d=$the diameter of the circle $\left(d=2r\right)$

Solved Examples

Example 1: What is the area of a circle with a radius of 7 cm?

Solution: $Are{a}_{circle}= \pi ·r^2$$=\frac{22}{7}\times {\left(7\right)}^2$$=22\times 7$$=154 c{m}^2$

 

Example 2: What is the area of a circle with a diameter of 2 m? (Use $\mathrm{\pi }=3.14$)

Solution: $Are{a}_{circle}=\frac{\mathrm{\pi }}{4}·d^2$$=\frac{3.14}{4}\times {\left(2\right)}^2$$=3.14 m^2$

 

Example 3: Find the circumference of a circle with a radius of $3\frac{1}{2} in$.

Solution: $C=2\mathrm{\pi r}$$=2\times \frac{22}{7}\times \left(3\frac{1}{2}\right)$$=2\times \frac{22}{7}\times \frac{7}{2}$$=22 in$

Cheat Sheet

• Relation between the diameter & radius of a circle: $d=2r$
• Circumference of a circle: $C=2\mathrm{\pi r}$
• Area of a circle when the radius is known: $Are{a}_{circle}=\mathrm{\pi }·{\mathrm{r}}^2$
• Area of a circle when the diameter is known: $Are{a}_{circle}=\frac{\mathrm{\pi }}{4}·{\mathrm{d}}^2$

Blunder Areas

• The radius is half of the diameter, not the vice-versa.
• The area of a semi-circle is half the area of a circle.
• The perimeter of a semi-circle is not half the perimeter of a circle.