Geometry Reference
2D Area and Perimeter
Area measures the space inside a flat shape. Perimeter measures distance around the outside. For composite figures, split the shape into familiar pieces.
Solved Examples
Jump directly to a worked example by shape or measurement type.
Fact Table
| Shape | Area | Perimeter or Circumference |
|---|---|---|
| Rectangle | $A=lw$ | $P=2l+2w$ |
| Triangle | $A=\frac12 bh$ | Add the side lengths. |
| Parallelogram | $A=bh$ | Add the side lengths. |
| Trapezoid | $A=\frac12(b_1+b_2)h$ | Add the side lengths. |
| Circle | $A=\pi r^2$ | $C=2\pi r=\pi d$ |
| Sector | $A=\frac{\theta}{360^\circ}\pi r^2$ | Arc length $=\frac{\theta}{360^\circ}2\pi r$ |
Content Formulas
Triangle
$$A=\frac12 bh$$
Trapezoid
$$A=\frac12(b_1+b_2)h$$
Circle
$$A=\pi r^2\quad C=2\pi r$$
Sector
$$A=\frac{\theta}{360^\circ}\pi r^2$$
Height means perpendicular height, not necessarily a slanted side.
Classic Examples
Trapezoid Area
A trapezoid has bases $8$ and $14$ and height $5$. Find its area.
Solution Steps
- Use the trapezoid area formula.
- Substitute the two bases and height.
- State the area: $55$ square units.
$$A=\frac12(b_1+b_2)h$$
$$=\frac12(8+14)(5)$$
$$=55$$
Circle Area and Circumference
A circle has radius $6$. Find its area and circumference.
Solution Steps
- Use the radius in the area formula.
- State the area.
- Use the radius in the circumference formula.
- State the circumference.
$$A=\pi(6)^2$$
$$=36\pi$$
$$C=2\pi(6)$$
$$=12\pi$$
Composite Area
A rectangle is $10$ by $6$. A semicircle of radius $3$ is attached to one side. Find the total area.
Solution Steps
- Add the rectangle area and the attached semicircle area.
- State the total area.
$$A=10(6)+\frac12\pi(3)^2$$
$$=60+\frac92\pi$$