📏 Perimeter & Area – SAT Mathematics Study Notes
Goal: Master quick calculations of perimeter and area for common shapes (rectangles, squares, triangles, circles, and composite figures) and solve related SAT word problems in 2–3 minutes.
1. Simple Explanation
| Concept | What It Is | How It Appears on the SAT |
|---|---|---|
| Perimeter | Total length around a shape. | “How many inches long is the border?” |
| Area | Amount of space inside a shape. | “How many square inches fill the floor?” |
Why It Matters
- SAT geometry problems often hide perimeter/area calculations in “find the missing side length” or “total area of a composite figure” questions.
- Knowing the formulas and being able to manipulate them quickly saves time.
2. Key Points & Formulas
| Shape | Perimeter Formula | Area Formula | Notes |
|---|---|---|---|
| Rectangle / Square | (P = 2(l+w)) | (A = l \times w) | For a square, (l = w). |
| Triangle | (P = a + b + c) | (A = \frac{1}{2} \times \text{base} \times \text{height}) | Use base & height; or Heron’s formula for all sides. |
| Circle | (P = 2\pi r) (circumference) | (A = \pi r^2) | Use (\pi \approx 3.14) unless given exact. |
| Composite Figures | Sum of individual perimeters except for shared edges (count once). | Sum of individual areas. | Identify overlapping parts. |
| Right Triangle | Same as triangle | (A = \frac{1}{2}ab) where (a) & (b) are legs | Useful shortcut for right triangles. |
| Trapezoid | (P = a + b + c + d) | (A = \frac{1}{2}(a+b)\times h) | (a, b) are bases; (h) is height. |
Quick Tips
- Perimeter always adds sides; no need to multiply by 2 unless shape is symmetrical.
- Area is always length × width (or a variation), never just a sum.
- If a figure is described as “composite” (e.g., a rectangle plus a semicircle), split into parts.
- For unknown side problems, use given perimeter/area to set up an equation.
3. Worked Example (SAT‑Style)
Problem:
A rectangular garden is 12 ft long and 8 ft wide. A circular fountain with a radius of 3 ft sits in the center.
Questions:
- What is the perimeter of the garden?
- What is the total area inside the garden (including the fountain)?
Step 1: Identify the shapes
- Rectangle: length = 12 ft, width = 8 ft.
- Circle: radius = 3 ft.
Step 2: Compute perimeter of the rectangle
[ P_{\text{rect}} = 2(l + w) = 2(12 + 8) = 2(20) = 40 \text{ ft} ]
Answer 1: 40 ft
Step 3: Compute area of the rectangle
[ A_{\text{rect}} = l \times w = 12 \times 8 = 96 \text{ ft}^2 ]
Step 4: Compute area of the fountain
[ A_{\text{circ}} = \pi r^2 = 3.14 \times 3^2 = 3.14 \times 9 \approx 28.26 \text{ ft}^2 ]
Step 5: Total area inside the garden
Because the fountain is inside the garden, we add the fountain’s area to the garden’s area: [ A_{\text{total}} = A_{\text{rect}} + A_{\text{circ}} \approx 96 + 28.26 = 124.26 \text{ ft}^2 ]
Answer 2: Approximately 124.3 ft²
Note: If the fountain were outside the garden, we would subtract its area.
4. Quick Revision Summary
| Concept | Formula | Common Mistake | Quick Fix |
|---|---|---|---|
| Perimeter of Rectangle | (2(l+w)) | Forgetting the factor 2 | Double-check both sides |
| Area of Rectangle | (l\times w) | Mixing up length & width | Label axes clearly |
| Perimeter of Circle | (2\pi r) | Using diameter instead of radius | Remember (r = d/2) |
| Area of Circle | (\pi r^2) | Using (r^2) incorrectly | Write out (r) then square |
| Area of Triangle | (\frac{1}{2}bh) | Using base × height directly | Multiply first, then halve |
| Composite Figures | Sum individual | Double‑count shared edges | Count shared edges once |
Quick Practice Drill (2 min):
| Question | Answer |
|---|---|
| A square has a perimeter of 48 cm. What is its side length? | 12 cm |
| A right triangle has legs 6 cm and 8 cm. Find its area. | 24 cm² |
| A circle’s circumference is 31.4 cm. Find its radius. | 5 cm |
| A trapezoid has bases 10 cm and 6 cm, height 4 cm. Find its area. | 28 cm² |
| A rectangle 5 m × 3 m has a semicircle cut out from one corner with radius 1.5 m. Find remaining area. | (15 - \tfrac{1}{2}\pi(1.5)^2 \approx 12.43 \text{ m}^2) |
Final Tip:
- Read the question carefully → identify shape(s), determine if you’re adding or subtracting areas, and note any shared edges.
- Write down known values → avoid misreading.
- Use estimation → quickly check if your answer is in the right ballpark.
Good luck mastering perimeter & area on the SAT!