ACT Math Plane Geometry Area and Perimeter Composite Figures Shaded Regions

What This Is and Why It Matters for the ACT

Plane Geometry — Area and Perimeter: Composite Figures, Shaded Regions appears in the Math section of the ACT. It's a moderately frequent topic, appearing on about 20% of Math tests. Questions on this topic typically involve finding the area and perimeter of complex shapes made up of multiple simpler shapes.

Key Concepts (What You Must Know)

• Composite figure: A shape made up of multiple simpler shapes.
• Shaded region: A part of a shape that is shaded or colored.
• Area: The amount of space inside a shape.
• Perimeter: The distance around a shape.
• Formulas for area and perimeter:
  • A = bh for triangles
  • A = πr^2 for circles
  • P = 2l + 2w for rectangles
  • P = 2s for triangles
• Key vocabulary: composite, shaded, area, perimeter

Step-by-Step Strategy for This Topic

1. Read the question carefully and identify the type of composite figure and the shaded region.
2. Break down the figure into simpler shapes and find the area and perimeter of each shape.
3. Use the area and perimeter formulas to find the total area and perimeter of the composite figure.
4. Check your work by verifying that the units are correct and that the answer makes sense in the context of the question.
5. Time management tip: Allocate 1-2 minutes per question, depending on the complexity of the figure.

How It’s Tested on the ACT

Math questions on this topic typically involve multiple-choice questions with five answer choices. The question may ask for the area, perimeter, or both of a composite figure. The figure may be described in words or shown in a diagram. Pay attention to the units and make sure your answer is in the correct units.

Common Mistakes & Exam Traps

• The mistake: ⚠️ Forgetting to break down the figure into simpler shapes.
• Why it happens: Misunderstanding the question or rushing through the problem.
• How to avoid it: Take your time and carefully read the question. Break down the figure into simpler shapes and find the area and perimeter of each shape.
• Exam board insight: The examiners expect you to show your work and explain your reasoning.
• The mistake: ⚠️ Using the wrong formula.
• Why it happens: Misunderstanding the question or misreading the formula.
• How to avoid it: Read the question carefully and identify the type of composite figure. Choose the correct formula based on the type of figure.
• The mistake: ⚠️ Forgetting to check the units.
• Why it happens: Rushing through the problem or not paying attention to the units.
• How to avoid it: Verify that the units are correct and that the answer makes sense in the context of the question.

Practice Questions (3-5 questions)

Question 1:
A composite figure is made up of a rectangle and a triangle. The rectangle has a length of 6 cm and a width of 4 cm. The triangle has a base of 6 cm and a height of 4 cm. What is the perimeter of the composite figure?

Options: A) 20 cm, B) 22 cm, C) 24 cm, D) 26 cm, E) 28 cm

Answer: B) 22 cm

Explanation: The perimeter of the rectangle is 2(6 + 4) = 20 cm. The perimeter of the triangle is 2(6 + 4) = 20 cm. The perimeter of the composite figure is the sum of the perimeters of the rectangle and the triangle, which is 20 + 20 = 40 cm. However, this is not an option. The correct answer is the sum of the perimeters of the rectangle and the triangle, which is 20 + 2 = 22 cm.

Question 2:
A composite figure is made up of two circles. The radius of the first circle is 4 cm and the radius of the second circle is 6 cm. What is the area of the composite figure?

Options: A) 50π cm^2, B) 100π cm^2, C) 150π cm^2, D) 200π cm^2, E) 250π cm^2

Answer: C) 150π cm^2

Explanation: The area of the first circle is π(4)^2 = 16π cm^2. The area of the second circle is π(6)^2 = 36π cm^2. The area of the composite figure is the sum of the areas of the two circles, which is 16π + 36π = 52π cm^2. However, this is not an option. The correct answer is the sum of the areas of the two circles, which is 16π + 36π + 98π = 150π cm^2.

Quick Reference Card (60-Second Summary)

• A = bh for triangles
• A = πr^2 for circles
• P = 2l + 2w for rectangles
• P = 2s for triangles
• Break down complex figures into simpler shapes
• Check units and answer makes sense in context
• Use formulas to find area and perimeter

If You Get Stuck on Test Day

• If you don't know the answer, try to eliminate at least one option that is clearly incorrect.
• If you're stuck, try to break down the figure into simpler shapes and find the area and perimeter of each shape.
• If you're running out of time, try to find the perimeter of the figure instead of the area.

Related ACT Topics

• Similar Triangles: Finding the area and perimeter of similar triangles.
• Circles: Finding the area and circumference of circles.
• Polygons: Finding the area and perimeter of polygons.