Area and Perimeter of Composite Shapes — ACT Math Guide

Q: What You Need to Know

• **Composite shapes** = figures made from combining basic geometric shapes • **Area strategy**: Add areas of individual shapes (subtract if there's overlap or cutouts) • **Perimeter strategy**: Add up all outer edges (don't count internal boundaries) • **Basic shape formulas**: Rectangle (A = lw),

Area and perimeter of composite shapes ACT questions test your ability to break down complex figures into simpler geometric pieces. These problems combine multiple basic shapes like rectangles, triangles, circles, and semicircles into one larger figure. The ACT math section includes 3-5 composite shape problems out of 60 questions in 60 minutes, making this a key skill for boosting your ACT math score. With the right approach, these questions become manageable puzzles that you can solve systematically.

What You Need to Know

Composite shapes = figures made from combining basic geometric shapes

Area strategy: Add areas of individual shapes (subtract if there's overlap or cutouts)

Perimeter strategy: Add up all outer edges (don't count internal boundaries)

Basic shape formulas: Rectangle (A = lw), Triangle (A = ½bh), Circle (A = πr²)

Common combinations: Rectangle + triangle, circle + rectangle, semicircle + square

Units matter: Keep area units squared (ft², cm²) and perimeter units linear (ft, cm)

📐 KEY FORMULA: Total Area = Area₁ + Area₂ + Area₃... (subtract overlapping regions)

⏱️ ACT TIME TIP: Sketch and label each basic shape separately — with 1 minute per question, clear organization prevents costly errors.

How to Solve Area and Perimeter of Composite Shapes on the ACT

Example Question 1 — Easy/Medium Difficulty

A composite figure consists of a rectangle with dimensions 8 ft by 6 ft, with a right triangle attached to one of the 8 ft sides. The triangle has a base of 8 ft and a height of 4 ft. What is the total area of the composite figure?

A) 32 ft²

B) 48 ft²

C) 64 ft²

D) 80 ft²

E) 96 ft²

Solution:

Step 1: Calculate the rectangle's area: A₁ = length × width = 8 × 6 = 48 ft²

Step 2: Calculate the triangle's area: A₂ = ½ × base × height = ½ × 8 × 4 = 16 ft²

Step 3: Add the areas together: Total area = 48 + 16 = 64 ft²

✅Answer: C — The composite shape's area equals the sum of its component parts.

Example Question 2 — Hard Difficulty

A semicircle with radius 5 inches is attached to a square with side length 10 inches. The diameter of the semicircle lies along one side of the square. What is the perimeter of the composite figure? (Use π ≈ 3.14)

A) 25.7 inches

B) 35.7 inches

C) 40.0 inches

D) 45.7 inches

E) 50.0 inches

Solution:

Step 1: Identify the outer edges: 3 sides of the square + curved edge of semicircle

Step 2: Calculate square's contribution: 3 × 10 = 30 inches (don't count the side where semicircle attaches)

Step 3: Calculate semicircle's curved edge: ½ × 2πr = ½ × 2π(5) = 5π ≈ 15.7 inches

Step 4: Add the perimeter components: 30 + 15.7 = 45.7 inches

✅Answer: D — Remember to exclude internal boundaries when calculating composite perimeters.

Common ACT Math Mistakes to Avoid

❌Mistake: Counting internal boundaries as part of the perimeter

✅Fix: Only measure the outer edge that you could trace with your finger

❌Mistake: Forgetting to subtract overlapping areas when shapes intersect

✅Fix: Carefully identify if shapes overlap or if one is cut out from another

❌Mistake: Using the wrong formula for curved sections (like using diameter instead of circumference)

✅Fix: Review your circle formulas — circumference is 2πr, area is πr²

❌Mistake: Mixing up units or forgetting to square area units

✅Fix: Always check that area answers have squared units (ft², cm²) and perimeter has linear units

Practice Question — Try It Yourself

A rectangular swimming pool measuring 20 ft by 12 ft has a semicircular hot tub with radius 4 ft attached to one of its shorter sides. What is the total area of the water surface? (Use π ≈ 3.14)

A) 240 ft²

B) 265.1 ft²

C) 290.2 ft²

D) 315.3 ft²

E) 340.4 ft²

Show Answer

Answer: B — Rectangle area: 20 × 12 = 240 ft². Semicircle area: ½π(4²) = ½(3.14)(16) = 25.1 ft². Total: 240 + 25.1 = 265.1 ft².

Key Takeaways for the ACT

Break composite shapes into familiar basic shapes before calculating

For area: add component areas together (subtract if there are cutouts)

For perimeter: trace only the outer boundary of the composite figure

ACT math allows calculators throughout — use them for π calculations and decimal arithmetic

Practice identifying which edges count toward perimeter in composite figures