Lesson 5: Solve Geometry Applications: Circles and Irregular Figures

New Concept

This lesson introduces circle properties, letting you calculate circumference ($C=2\pi r$) and area ($A=\pi r^2$). You'll also learn to find the area of irregular figures by breaking them down into familiar shapes like rectangles and triangles.

What’s next

Next, you'll apply these formulas in a series of interactive examples and practice problems to master calculating the area and perimeter of any shape.

Property

• $r$ is the length of the radius
• $d$ is the length of the diameter
• $d = 2r$

Remember that we approximate $\pi$ with $3.14$ or $\frac{22}{7}$ depending on whether the radius of the circle is given as a decimal or a fraction.

Examples

• A circle has a radius ($r$) of 5 cm. Its diameter ($d$) is calculated as $d = 2r = 2(5) = 10$ cm.

• If a circle's diameter is 14 inches, its radius is half of that. We find the radius by solving $14 = 2r$, so $r = 7$ inches.

Property

Circumference is the perimeter of a circle. The formula for circumference is:

Since the diameter of a circle is two times the radius, we could write the formula for the circumference in terms of $d$:

Examples

• An extra-large pizza has a radius of 8 inches. Its circumference is $C = 2\pi r \approx 2(3.14)(8) = 50.24$ inches.

• A circular table has a diameter of 4 feet. Using the formula $C = \pi d$, its circumference is approximately $C \approx (3.14)(4) = 12.56$ feet.

Property

The formula for area of a circle is:

Examples

• A circular sandbox has a radius of 2.5 feet. Its area is $A = \pi r^2 \approx 3.14(2.5)^2 = 3.14(6.25) = 19.625$ square feet.

• The lid of a paint bucket has a radius of 7 inches. Using $\pi \approx \frac{22}{7}$, the area is $A \approx \frac{22}{7}(7)^2 = \frac{22}{7}(49) = 154$ square inches.

Property

An irregular figure is a figure that is not a standard geometric shape. To find the area of one of these irregular figures, we can split it into figures whose formulas we know and then add the areas of the figures.

Examples

• An L-shaped figure can be split into two rectangles. If one is 12 by 4 units ($A=48$) and the other is 2 by 6 units ($A=12$), the total area is $48 + 12 = 60$ square units.

• A shape is formed by a rectangle and a triangle. The rectangle is 8 by 4 units ($A=32$). The triangle on top has a base of 3 and height of 3 ($A=4.5$). The total area is $32 + 4.5 = 36.5$ square units.