8.3 Perimeter, Circumference & Area of 2D Figures
Key Takeaways
- Perimeter measures the total outer boundary distance of a polygon (linear units), while area measures the 2D surface enclosed inside a shape (square units).
- Circle circumference is $C = 2\pi r = \pi d$, and circle area is $A = \pi r^2$. On test day, use $\pi \approx 3.14$, $\frac{22}{7}$, or exact $\pi$ notation as indicated.
- Standard area formulas: Rectangle ($A = lw$), Square ($A = s^2$), Triangle ($A = \frac{1}{2}bh$), Parallelogram ($A = bh$), and Trapezoid ($A = \frac{1}{2}(b_1 + b_2)h$).
- Altitude/height ($h$) in triangles, parallelograms, and trapezoids must always be measured perpendicular ($90^\circ$) to the base, never along a slanted side.
- Composite figures are solved by decomposing into simpler geometric shapes (additive method) or by subtracting inner unshaded regions from an outer bounding area (subtractive method).
Perimeter, Circumference & Area of 2D Figures
Quick Summary: Two-dimensional measurement questions on the HiSET focus on perimeter (the linear distance around the outside edge of a figure) and area (the number of square units enclosed within a boundary). Circles require specialized formulas for circumference ($C = 2\pi r$) and area ($A = \pi r^2$). When working with triangles, parallelograms, and trapezoids, the height ($h$) must always be perpendicular to the base. Complex shapes are solved by decomposing them into standard sub-shapes or subtracting unshaded areas.
Familiarity with the official HiSET formula sheet and mastery of composite shape decomposition ensure quick, accurate solutions on test day.
Perimeter & Circumference Fundamentals
- Perimeter ($P$): The total distance around the exterior boundary of a polygon. Calculated by summing the lengths of all outer sides.
- Circumference ($C$): The perimeter of a circle. Where $r$ is the radius (center to edge) and $d$ is the diameter ($d = 2r$, passing through the center).
Circumference and Diameter of a Circle
┌───────────────┐
•──┼───────► r │ Radius (r) = Half-distance across
/ │ \ │ Diameter (d) = Full distance across (2r)
│ ◄──┴─────────► │ Circumference (C) = 2πr = πd
\ d / │ Area (A) = πr²
•──────────• └───────────────┘
Comprehensive 2D Area Reference Guide
Below is the complete inventory of 2D area formulas provided on or tested by the HiSET Mathematics subtest:
Standard 2D Geometry Area Formulas Visualized
Rectangle Triangle Parallelogram Trapezoid
┌─────────────┐ /\ /─────────────/ ┌─────────┐ b₁
│ │ w / \ h /│ / /│ │\
│ │ / \ / │ h / / │ h │ \
└─────────────┘ /______\ /__│__________/ /__│_________│__\
l b b b₂
A = l · w A = 1/2 · b · h A = b · h A = 1/2 · (b₁ + b₂) · h
Formula Reference Table
| Geometric Figure | Perimeter / Circumference Formula | Area Formula | Key Measurement Rules |
|---|---|---|---|
| Square | $P = 4s$ | $A = s^2$ | All 4 sides $s$ are congruent and meet at $90^\circ$. |
| Rectangle | $P = 2l + 2w = 2(l + w)$ | $A = l \cdot w$ | Opposite sides are equal; angles are $90^\circ$. |
| Triangle | $P = s_1 + s_2 + s_3$ | $A = \frac{1}{2} b h$ | Height $h$ is perpendicular (at $90^\circ$) to base $b$. |
| Parallelogram | $P = 2a + 2b$ | $A = b \cdot h$ | Never use the slant side as height; use perpendicular altitude $h$. |
| Trapezoid | $P = s_1 + s_2 + b_1 + b_2$ | $A = \frac{1}{2}(b_1 + b_2)h$ | $b_1$ and $b_2$ are parallel bases; $h$ is perpendicular distance between them. |
| Circle | $C = 2\pi r = \pi d$ | $A = \pi r^2$ | If given diameter $d$, divide by $2$ to get radius $r$ before squaring! |
| Semicircle | Boundary $= \pi r + 2r$ | $A = \frac{1}{2}\pi r^2$ | Half the area of a full circle. |
Composite and Irregular Figures
A composite figure is a geometric shape composed of two or more basic shapes (such as rectangles, triangles, and semicircles). There are two fundamental strategies for finding the area of composite figures:
Two Methods for Composite Figures
1. Additive Method (Decomposition) 2. Subtractive Method (Hollow / Shaded)
┌───────┐ ┌────────────────────────┐
│ A₁ │ (Triangle) │ │ (Outer Rect)
├───────┴────────┐ │ ┌──────────┐ │
│ │ │ │ Unshaded │ │
│ A₂ │ (Rectangle) │ │ A₂ │ │
│ │ │ └──────────┘ │
└────────────────┘ └────────────────────────┘
Total Area = A₁ + A₂ Shaded Area = A_outer - A_inner
Method 1: The Additive Method (Summing Sub-Shapes)
- Draw auxiliary lines to divide the irregular shape into familiar polygons.
- Determine the missing dimensions of each sub-region using parallel side relationships.
- Compute the individual area of each component.
- Add the component areas together.
Method 2: The Subtractive Method (Shaded Regions)
- Calculate the total area of the entire outer bounding figure.
- Calculate the area of the unshaded or hollow inner shape(s).
- Subtract the unshaded area from the total area:
Worked Examples
Worked Example 1: Area of a Trapezoid
A concrete patio is shaped like a trapezoid with parallel bases measuring $14\text{ feet}$ and $22\text{ feet}$, and a perpendicular distance (height) between them of $9\text{ feet}$. What is the total surface area of the patio?
- Identify given dimensions: $b_1 = 14\text{ ft}, b_2 = 22\text{ ft}, h = 9\text{ ft}$.
- Apply the trapezoid area formula:
- Conclusion: The patio surface area is $162\text{ square feet}$.
Worked Example 2: Shaded Deck Area with Circular Cutout
A circular pool with a diameter of $12\text{ feet}$ is centered inside a square wooden deck measuring $20\text{ feet}$ on each side. What is the area of the exposed wooden deck? (Use $\pi \approx 3.14$).
- Area of square deck:
- Area of circular pool:
- Diameter $d = 12\text{ ft} \implies \text{radius } r = \frac{12}{2} = 6\text{ ft}$.
- Subtract inner circle from outer square:
- Conclusion: The exposed deck area is $286.96\text{ square feet}$.
High-Frequency HiSET Traps & Exam Tips
- Diameter vs. Radius Confusion: In circle area ($A = \pi r^2$), using the diameter instead of the radius produces an answer that is $4\times$ too large! Always double-check if the problem states "diameter" or "radius".
- Perimeter of Composite Figures: When calculating the perimeter of a composite figure, only sum the outer exposed edges. Never include internal dividing seams where two shapes touch.
- Slant Height Trap: In triangles and parallelograms, never multiply the base by the slanted side length. Always use the perpendicular altitude ($h$).
A concrete patio is shaped like a trapezoid with parallel bases measuring 14 feet and 22 feet, and a perpendicular height of 9 feet. What is the total surface area of the patio?
A circular pool with a diameter of 12 feet is installed in the center of a square wooden deck measuring 20 feet on each side. What is the area of the remaining exposed wooden deck in square feet? (Use pi = 3.14)
A decorative garden plot is in the shape of a rectangle surmounted by a semicircle on one of its shorter ends. The rectangle is 12 meters long and 6 meters wide. The diameter of the attached semicircle equals the 6-meter width of the rectangle. What is the total perimeter (outer boundary length) of the garden plot? (Use pi = 3.14)
A triangle has a base that is 4 inches longer than its perpendicular height. If the total area of the triangle is 48 square inches, what is the length of the base?