This content introduces the geometric concept of cylinders, focusing on their surface area calculation by relating cylinders to prisms and using nets. It explains formulas, common mistakes, real-life applications, and provides examples and practice problems to deepen understanding.
Cylinders: The Soup Can Glow-Up Cylinders are like prisms who went to art school: still about that straight-edge life, but now with curves. You already unboxed prisms and flattened them into neat rectangles. You saw how lateral area was just perimeter times height. Now we level up: cylinders. Sa...
What Even Is a Cylinder? (The Recap Remix) A right cylinder has: Two congruent circular bases (top and bottom) A curved surface that wraps around A height h measured straight from base to base (like a prism) We used nets for prisms. For cylinders, if you slice the curved side straight down...
From Prisms to Cylinders: Same Script, Different Shape Remember: For prisms, lateral area = perimeter of base × height. For cylinders, the base perimeter is the circle's circumference. So we copy-paste the idea with extra seasoning: Lateral area of a cylinder = circumference × height = 2πr × h...
Quick Symbol Guide r = radius of the circular base h = height of the cylinder π ≈ 3.14 or use the π button for accuracy Area units are square units (cm², m², etc.) Part Formula When to use Lateral area 2πrh Labels, side wrapping, no top/bottom Area of one base πr² Top or ...
Worked Example 1: Classic Soup Can A right cylinder has radius r = 3 cm and height h = 10 cm. Find the total surface area. Lateral area: 2πrh = 2 × π × 3 × 10 = 60π cm² ≈ 188.5 cm² Two bases: 2πr² = 2 × π × 3² = 18π cm² ≈ 56.5 cm² Total: 60π + 18π = 78π cm² ≈ 244.9 cm² Answer: about 245 cm...
Worked Example 2: The Label Detective (Pythagorean Cameo) You measure a can's label as a rectangle when unwrapped. The height is 10 cm and the diagonal of the rectangle is 26 cm. What is the radius of the can, and what is the label area? The unwrapped label is a rectangle with sides: height h = ...
How To Do It Every Time (Mini Algorithm) Given r and h: 1) Compute lateral area: LA = 2πrh 2) Compute caps: Caps = 2πr² (if closed) or πr² (if one end) or 0 (if open both ends) 3) Total = LA + Caps 4) Slap on units squared If you do not have r directly: Given diameter d: use r = d/2 Given c...
Common Mistakes (AKA Math Plot Twists) Mixing radius and diameter. If they give diameter and you use it like radius, everything doubles and your teacher cries softly. Forgetting there are two circular bases on a closed cylinder. Two pizzas, not one. Using area of a circle for the side. The sid...
Speed Summary Net of a cylinder = two circles + one rectangle Lateral area = 2πrh Two bases = 2πr² Total surface area = 2πr² + 2πrh = 2πr(r + h) Use Pythagoras on the label rectangle if you know its diagonal and height Tiny Practice Prompts A cylinder has diameter 8 cm and height 12 cm...
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