9.5 Solve Geometry Applications: Circles and Irregular Figures

Circles

Circumference and Area of Circles

Every circle calculation starts with two key measurements: the radius and the diameter. The radius ($r$) is the distance from the center of the circle to any point on its edge. The diameter ($d$) is a line segment that passes through the center with both endpoints on the circle. The diameter is always twice the radius, so a circle with a radius of 5 cm has a diameter of 10 cm.

One more term worth knowing: a chord is any line segment with both endpoints on the circle. The diameter is actually the longest possible chord.

Circumference is the distance around a circle (think of it as the circle's perimeter). You can calculate it two ways:

• $C = 2\pi r$ if you know the radius
• $C = \pi d$ if you know the diameter

Both formulas give the same answer since $d = 2r$. The symbol $\pi$ (pi) is a constant approximately equal to 3.14159. For most problems in this course, using 3.14 works fine unless your teacher says otherwise.

Area is the space inside the circle:

• $A = \pi r^2$

Notice that the radius gets squared here. A common mistake is to square the diameter instead of the radius, so always double-check which measurement you're working with.

Circle Components

These terms come up less often in calculations but are good to recognize:

• Central angle: an angle formed by two radii, with its vertex at the center of the circle
• Arc: a portion of the circle's circumference (like a curved segment between two points)
• Sector: the "pie slice" region bounded by two radii and an arc
• Tangent: a line that touches the circle at exactly one point without crossing through it

Irregular Figures

Area of Irregular Figures

Irregular figures are shapes that don't match a single standard formula. The strategy is to break them apart into familiar shapes, find each area separately, then add the areas together.

The shapes you'll typically break them into:

• Rectangles: $A = lw$
• Triangles: $A = \frac{1}{2}bh$
• Circles (or semicircles): $A = \pi r^2$ (use half this for a semicircle)

Example: An irregular figure is made up of a rectangle (4 cm by 6 cm) with a semicircle attached to one of the 4 cm sides. The semicircle has a radius of 2 cm (half the 4 cm side).

1. Rectangle area: $A = 4 \times 6 = 24 \text{ cm}^2$
2. Semicircle area: $A = \frac{1}{2}\pi r^2 = \frac{1}{2}(3.14159)(2^2) = 6.28 \text{ cm}^2$
3. Total area: $24 + 6.28 = 30.28 \text{ cm}^2$

Applications of Geometric Formulas

Word problems ask you to recognize which shapes and formulas to use. Here's a reliable approach:

1. Read the problem and identify the shape(s) involved.
2. Write down the measurements given (radius, diameter, length, width, etc.).
3. Pick the right formula for each shape.
4. Plug in values and calculate.
5. Check your units and ask yourself whether the answer makes sense.

Example 1: A circular garden has a diameter of 10 ft. How much fencing do you need to go around it?

You need the circumference: $C = \pi d = 3.14159 \times 10 = 31.42 \text{ ft}$

Example 2: A room is shaped like a 12 ft by 16 ft rectangle with a triangular alcove on one side. The triangle has a base of 12 ft and a height of 8 ft. What's the total floor area?

1. Rectangle area: $A = 12 \times 16 = 192 \text{ ft}^2$
2. Triangle area: $A = \frac{1}{2}(12)(8) = 48 \text{ ft}^2$
3. Total area: $192 + 48 = 240 \text{ ft}^2$

That 240 ft² is how much carpet you'd need to cover the whole floor.