USTET Mathematics — Perimeter, Area, Volume & Equation of a LineFlash Cards

Find the perimeter of a rectangle with length 12 cm and width 8 cm.

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?

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)

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.

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.

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)

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.

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.

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.

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)

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)

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)

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).

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.

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.

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.

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?

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.

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.

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?

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