Overview
Area and perimeter of plane figures is a foundational SAT Geometry and Trigonometry topic tested in approximately 5–7 geometry questions per exam. Questions require applying standard area formulas for triangles, rectangles, parallelograms, trapezoids, and circles, often to composite figures where multiple shapes combine. The SAT reference sheet provides area and perimeter formulas for the most common shapes; however, students must know how to apply them under algebraic manipulation and understand scaling laws.
Key Points
1. Core Area and Perimeter Formulas
The SAT reference sheet includes the following; verify you can use each without hesitation:
| Shape | Area | Perimeter / Circumference |
|---|---|---|
| Rectangle | A = lw | P = 2l + 2w |
| Triangle | A = ½bh | P = a + b + c |
| Parallelogram | A = bh | P = 2a + 2b |
| Trapezoid | A = ½(b₁ + b₂)h | P = sum of all sides |
| Circle | A = πr² | C = 2πr |
For the trapezoid formula, note that h is the perpendicular height, not a slanted side.
2. Polygon Interior Angle Sums
For any polygon with n sides:
Sum of interior angles = (n − 2) × 180°
Each interior angle of a regular n-gon:
Interior angle = (n − 2) × 180° / n
Common values:
- Triangle (n=3): 180°
- Quadrilateral (n=4): 360°
- Pentagon (n=5): 540°
- Hexagon (n=6): 720°
3. Composite Figures
Many SAT questions show an irregular shape that can be split into rectangles, triangles, and semicircles. Strategy:
- Identify each sub-shape within the composite figure.
- Calculate each sub-shape’s area or perimeter.
- Add areas (or subtract for cutouts).
- For perimeter, trace only the outer boundary — do not include any shared interior sides.
4. Scaling Laws
When all linear dimensions of a figure are multiplied by scale factor k:
New perimeter = k × original perimeter
New area = k² × original area
Example: A rectangle with sides 3 and 4 (area = 12) is scaled by k = 2. New sides: 6 and 8. New area = 48 = 12 × 4 = 12 × 2².
The SAT tests this by asking: “If the dimensions of a figure are tripled, how does the area change?” Answer: area is multiplied by 9 (= 3²), not 3.
5. Step-by-Step: Solving Area/Perimeter Problems
- Read the question carefully — is it asking for area or perimeter?
- Identify the shape(s) involved.
- Confirm which measurements are given (base? height? side lengths?).
- Apply the correct formula.
- If a dimension is unknown, set up an algebraic equation and solve.
Pitfalls and Common Mistakes
Mistake 1: Using slant height instead of perpendicular height Parallelograms and trapezoids have a slanted side that is NOT the height. The height h is always the perpendicular distance between the parallel bases. Fix: Only use a measurement as height if it meets the base at a 90° angle (marked with a right-angle symbol).
Mistake 2: Solving for area when the question asks for perimeter (or vice versa) The SAT frequently asks for perimeter after presenting information that makes area easy to compute first. Fix: Underline the word “area” or “perimeter” in the problem before starting any calculation.
Mistake 3: Double-counting shared sides in composite figures When two rectangles share a side in a composite figure, that shared side is an interior edge and must NOT be counted in the perimeter. Fix: Physically trace the outer boundary of the figure with your finger (or cursor), counting only edges that form the exterior.
Mistake 4: Forgetting that area scales by k², not k If a student sees “dimensions doubled” and writes “area doubled,” they will lose this point. Fix: Memorize: perimeter scales by k, area scales by k², volume scales by k³.
Mistake 5: Assuming a diagram is to scale The SAT explicitly warns that figures may not be drawn to scale. A shape that looks like a square may be a rectangle. Fix: Trust only labeled measurements and marked relationships.
Related Entries
- Right_Triangles_Pythagorean
- Similar_Triangles
- Properties_Circles
- Volume_Surface_Area
- Lines_Angles_Parallel
Quick Reference Card
| Formula | Expression |
|---|---|
| Rectangle area | A = lw |
| Triangle area | A = ½bh |
| Parallelogram area | A = bh |
| Trapezoid area | A = ½(b₁+b₂)h |
| Circle area | A = πr² |
| Polygon angle sum | (n−2) × 180° |
| Regular polygon angle | (n−2) × 180° / n |
| Scaling — perimeter | × k |
| Scaling — area | × k² |