Perimeter, Area, Volume & Equation of a Line - Flashcards
Master the essential formulas for mensuration (perimeter, area, volume) and linear equations through problem-solving practice. These concepts are frequently tested in UPCAT and other Philippine entrance exams. Focus on understanding when and how to apply each formula through step-by-step problem solving.
Flashcards
Find the perimeter of a rectangle with length 12 cm and width 8 cm.
Answer
Formula: P = 2(l + w)
Step 1: Identify values → l = 12 cm, w = 8 cm
Step 2: Substitute → P = 2(12 + 8)
Step 3: Calculate → P = 2(20) = 40 cm
Answer: 40 cm
A square has a perimeter of 36 meters. What is the length of each side?
Answer
Formula: P = 4s, so s = P/4
Step 1: Given P = 36 m
Step 2: Substitute → s = 36/4
Step 3: Calculate → s = 9 m
Step 4: Check: 4 × 9 = 36 ✓
Answer: 9 meters
Find the circumference of a circle with radius 5 cm. (Use π = 3.14)
Answer
Formula: C = 2πr
Step 1: Given r = 5 cm, π = 3.14
Step 2: Substitute → C = 2 × 3.14 × 5
Step 3: Calculate → C = 31.4 cm
Alternative: C = πd = 3.14 × 10 = 31.4 cm
Answer: 31.4 cm
Calculate the area of a rectangle with length 15 m and width 9 m.
Answer
Formula: A = lw
Step 1: Identify values → l = 15 m, w = 9 m
Step 2: Substitute → A = 15 × 9
Step 3: Calculate → A = 135 m²
Remember: Area is always in square units!
Answer: 135 m²
Find the area of a triangle with base 10 cm and height 6 cm.
Answer
Formula: A = ½bh
Step 1: Identify values → b = 10 cm, h = 6 cm
Step 2: Substitute → A = ½ × 10 × 6
Step 3: Calculate → A = ½ × 60 = 30 cm²
Common mistake: Don't forget the ½!
Answer: 30 cm²
Calculate the area of a circle with diameter 14 cm. (Use π = 22/7)
Answer
Formula: A = πr²
Step 1: Convert diameter to radius → r = 14/2 = 7 cm
Step 2: Substitute → A = (22/7) × 7²
Step 3: Calculate → A = (22/7) × 49 = 22 × 7 = 154 cm²
Common mistake: Using diameter instead of radius!
Answer: 154 cm²
Find the area of a trapezoid with parallel sides 8 m and 12 m, and height 5 m.
Answer
Formula: A = ½(b₁ + b₂)h
Step 1: Identify values → b₁ = 8 m, b₂ = 12 m, h = 5 m
Step 2: Substitute → A = ½(8 + 12) × 5
Step 3: Calculate → A = ½(20) × 5 = 10 × 5 = 50 m²
Answer: 50 m²
Calculate the volume of a cube with side length 4 cm.
Answer
Formula: V = s³
Step 1: Identify value → s = 4 cm
Step 2: Substitute → V = 4³
Step 3: Calculate → V = 4 × 4 × 4 = 64 cm³
Remember: Volume is always in cubic units!
Answer: 64 cm³
Find the volume of a rectangular prism with length 6 m, width 4 m, and height 3 m.
Answer
Formula: V = lwh
Step 1: Identify values → l = 6 m, w = 4 m, h = 3 m
Step 2: Substitute → V = 6 × 4 × 3
Step 3: Calculate → V = 72 m³
Tip: Multiply in any order: 6×4=24, then 24×3=72
Answer: 72 m³
Calculate the volume of a cylinder with radius 3 cm and height 8 cm. (Use π = 3.14)
Answer
Formula: V = πr²h
Step 1: Identify values → r = 3 cm, h = 8 cm, π = 3.14
Step 2: Substitute → V = 3.14 × 3² × 8
Step 3: Calculate → V = 3.14 × 9 × 8 = 226.08 cm³
Answer: 226.08 cm³
Find the volume of a cone with radius 6 cm and height 9 cm. (Use π = 3.14)
Answer
Formula: V = ⅓πr²h
Step 1: Identify values → r = 6 cm, h = 9 cm, π = 3.14
Step 2: Substitute → V = ⅓ × 3.14 × 6² × 9
Step 3: Calculate → V = ⅓ × 3.14 × 36 × 9 = ⅓ × 1017.36 = 339.12 cm³
Common mistake: Forgetting the ⅓ factor!
Answer: 339.12 cm³
Calculate the volume of a sphere with radius 3 cm. (Use π = 3.14)
Answer
Formula: V = ⁴⁄₃πr³
Step 1: Identify values → r = 3 cm, π = 3.14
Step 2: Substitute → V = ⁴⁄₃ × 3.14 × 3³
Step 3: Calculate → V = ⁴⁄₃ × 3.14 × 27 = ⁴⁄₃ × 84.78 = 113.04 cm³
Answer: 113.04 cm³
Find the slope of the line passing through points (2, 3) and (6, 11).
Answer
Formula: m = (y₂ - y₁)/(x₂ - x₁)
Step 1: Identify points → (x₁,y₁) = (2,3), (x₂,y₂) = (6,11)
Step 2: Substitute → m = (11 - 3)/(6 - 2)
Step 3: Calculate → m = 8/4 = 2
Slope = 2 (line rises 2 units for every 1 unit right)
Answer: m = 2
Write the equation of a line with slope 3 and y-intercept -2.
Answer
Use slope-intercept form: y = mx + b
Step 1: Identify values → m = 3, b = -2
Step 2: Substitute → y = 3x + (-2)
Step 3: Simplify → y = 3x - 2
This line passes through (0, -2) and rises 3 units per 1 unit right
Answer: y = 3x - 2
Find the equation of a line passing through (1, 4) with slope 2.
Answer
Use point-slope form: y - y₁ = m(x - x₁)
Step 1: Identify values → (x₁,y₁) = (1,4), m = 2
Step 2: Substitute → y - 4 = 2(x - 1)
Step 3: Expand → y - 4 = 2x - 2
Step 4: Solve for y → y = 2x + 2
Answer: y = 2x + 2
Convert 3x + 2y = 12 to slope-intercept form.
Answer
Goal: Get y = mx + b form
Step 1: Start with 3x + 2y = 12
Step 2: Subtract 3x from both sides → 2y = -3x + 12
Step 3: Divide by 2 → y = -3x/2 + 6
Step 4: Simplify → y = -1.5x + 6
Slope = -1.5, y-intercept = 6
Answer: y = -1.5x + 6
Are the lines y = 2x + 3 and y = 2x - 1 parallel, perpendicular, or neither?
Answer
Compare slopes:
Line 1: y = 2x + 3 → slope = 2
Line 2: y = 2x - 1 → slope = 2
Since m₁ = m₂ = 2, the slopes are equal
Rule: Parallel lines have equal slopes
Answer: Parallel
Find the slope of a line perpendicular to y = 3x + 5.
Answer
Step 1: Find slope of given line → m₁ = 3
Step 2: Use perpendicular rule → m₁ × m₂ = -1
Step 3: Substitute → 3 × m₂ = -1
Step 4: Solve → m₂ = -1/3
Rule: Perpendicular slopes are negative reciprocals
Answer: m = -1/3
A rectangular garden has area 48 m² and width 6 m. Find its perimeter.
Answer
Step 1: Find length using A = lw
48 = l × 6 → l = 48/6 = 8 m
Step 2: Calculate perimeter using P = 2(l + w)
P = 2(8 + 6) = 2(14) = 28 m
Step 3: Check: Area = 8 × 6 = 48 ✓
Answer: 28 m
When do you use the formula V = ⅓πr²h and what does each variable represent?
Answer
Use for: Volume of a cone
When: You have a circular base tapering to a point
Variables:
- V = volume (cubic units)
- r = radius of circular base
- h = height from base to apex
- ⅓ factor distinguishes cone from cylinder
Example: Cone with r=4, h=9 → V = ⅓π(16)(9) = 48π
Answer: Cone volume formula
Diagrams
Mensuration and Linear Equations Overview
Complete overview of all formulas and concepts covered in this chapter, organized by major topic areas.
Problem-Solving Decision Tree for Mensuration
Step-by-step decision tree to help students choose the correct formula based on the shape and what needs to be calculated.
Line Equation Problem-Solving Process
Systematic approach to solving different types of line equation problems, from identifying what's given to choosing the appropriate method.