Area of Plane Figures
Program: NMMS – Mathematics (SAT)
Topic: Area of Plane Figures
1️⃣ Simple Explanation
Area tells you how many unit squares fit inside a shape.
Think of it as the “size” of the shape’s surface.
In the SAT you’ll be asked to compute areas of:
- Basic shapes: rectangle, square, triangle, circle, trapezoid, parallelogram.
- Composite figures: shapes made by adding/subtracting the above shapes.
All formulas use the same units (e.g., square inches, square centimeters).
The SAT usually gives you the measurements in the same units, so no need to convert.
2️⃣ Key Points & Formulas
| Shape | Formula | Notes |
|---|---|---|
| Rectangle / Square | Area = length × width |
For a square, l = w. |
| Triangle | Area = ½ × base × height |
Height is perpendicular to the base. |
| Parallelogram | Area = base × height |
Height is the perpendicular distance between the parallel sides. |
| Trapezoid | Area = ½ × (base₁ + base₂) × height |
base₁ and base₂ are the two parallel sides. |
| Circle | Area = π × r² |
Use π ≈ 3.14 or π = 22/7 on the SAT. |
| Composite Figures | Add or subtract the areas of constituent shapes | Always work in the same unit. |
Quick Tips for the SAT
- Identify the shape – look for right angles, parallel sides, or a curved boundary.
- Choose the right formula – e.g., use the triangle formula even if the triangle is not right‑angled.
- Use the common base – when a rectangle shares a side with a triangle, the base of the triangle may be the same as the rectangle’s side.
- Keep units consistent – if the problem gives inches and centimeters, convert one to the other before computing.
- Approximate π carefully – most SAT problems accept
π = 3.14orπ = 22/7. - Check for “whole shape” vs. “partial shape” – sometimes a figure is a whole square minus a triangle; compute both and subtract.
3️⃣ Worked Example
Problem
A composite figure consists of:
- a rectangle 8 in. long and 6 in. wide,
- a right triangle on top of the rectangle that shares the 8 in. side as its base and has a height of 4 in.,
- a semicircle on the right side of the rectangle that uses the 6 in. side as its diameter.
Find the total area of the figure.
Step‑by‑step Solution
| Part | Calculation | Result |
|---|---|---|
| Rectangle | A_rect = 8 × 6 = 48 |
48 in² |
| Triangle | A_tri = ½ × 8 × 4 = 16 |
16 in² |
| Semicircle | Diameter = 6 → radius = 3. | |
A_semi = ½ × π × r² = ½ × 3.14 × 3² = ½ × 3.14 × 9 = 14.13 |
≈14.13 in² |
|
| Total Area | 48 + 16 + 14.13 = 78.13 |
≈78.13 in² |
Answer: The composite figure has an area of ≈ 78.13 square inches.
4️⃣ Quick Revision Summary
- Area of a rectangle:
l × w. - Area of a triangle:
½ × base × height. - Area of a parallelogram:
base × height. - Area of a trapezoid:
½ × (b₁ + b₂) × height. - Area of a circle:
π r²(useπ ≈ 3.14or22/7). - Composite figures: compute each part’s area, then add or subtract.
- Always keep units consistent.
- Key SAT trick: look for shared bases or heights to simplify calculations.
- Remember π: 3.14 or 22/7; the SAT usually accepts either.
Keep practicing with mixed‑shape problems—knowing how to break a figure into familiar pieces is the fastest way to ace area questions!