Summary of Geometric Measurement and Unit Conversion

## Introduction Surface area is the total area of all the outer faces of a three-dimensional object. In this material you will learn how to identify which faces to include, how to compute surface area for common solids, and how to apply those calculations to practical packing and wrapping problems. > **Definition:** Surface area is the sum of the areas of all the outer surfaces of a 3D shape. ## Key ideas — breaking the concept down ### 1. What counts as "surface" - Surface area counts only the outside faces of a 3D object, not the inside volume or any internal parts. - If a face is missing (for example, a box with no lid), do not include that face in the total. ### 2. Common solids and their surface pieces - **Cube:** 6 congruent square faces. - **Rectangular prism (box):** 6 rectangles — opposite faces are equal in area. - **Cylinder:** 2 circular faces (top and bottom) plus one curved surface. The curved surface can be treated as a rectangle when unwrapped: its length equals the circle's circumference and its height equals the cylinder height. > **Definition:** Circumference of a circle is $2\pi r$, where $r$ is the radius. ### 3. Formulas (pieces to use) - Area of a square: $\text{side}^2$. - Area of a rectangle: $\text{length} \times \text{width}$. - Area of a circle: $\pi r^2$. - Circumference of a circle: $2\pi r$. ## Surface area formulas (assembly from faces) - Cube with side $s$: $$\text{SA} = 6s^2$$ - Rectangular prism with edges $l$, $w$, $h$: $$\text{SA} = 2(lw + lh + wh)$$ - Cylinder with radius $r$ and height $h$: full surface area (including top and bottom) is $$\text{SA} = 2\pi r^2 + 2\pi r h$$ The curved (side) area alone is $$\text{Side area} = 2\pi r h$$ > **Definition:** When you unwrap the curved surface of a cylinder you get a rectangle of width $2\pi r$ and height $h$. ## Step-by-step approach for solving surface area questions 1. Identify the 3D shape. 2. List each distinct face and decide whether to include it (e.g., is there a lid?). 3. Write area for one face of each type, then multiply by how many such faces exist. 4. Add all face areas; keep units as squared units (e.g., cm$^2$). 5. Check units and whether the problem asks for a partial surface (for example, only the curved side of a cylinder for wrapping). ## Worked examples and practical applications ### Example 1: Cube with side $2\text{ cm}$ - Each face area: $2^2 = 4\text{ cm}^2$. - Six faces: $6\times4 = 24\text{ cm}^2$. - If the cube has no lid (one face removed): surface area is $5\times4 = 20\text{ cm}^2$. ### Example 2: Rectangular prism with $l=10\text{ cm}$, $w=5\text{ cm}$, $h=2\text{ cm}$ - Faces: two $10\times5$, two $10\times2$, two $5\times2$. - Surface area: $$\text{SA} = 2(10\times5) + 2(10\times2) + 2(5\times2) = 100 + 40 + 20 = 160\text{ cm}^2.$$ ### Example 3: Cylinder with diameter $2\text{ cm}$ (so radius $r=1\text{ cm}$) and height $18\text{ cm}$ - Two circular faces: $2\pi r^2 = 2\pi(1^2) = 2\pi$. - Curved side (unwrapped rectangle): width $2\pi r = 2\pi(1)=2\pi$, height $=18$. Side area $=2\pi\times18 = 36\pi$. - Full surface area: $$\text{SA} = 2\pi + 36\pi = 38\pi\text{ cm}^2.$$ - If the top is missing (no lid), omit one circle: $$\text{SA}_{\text{no lid}} = \pi r^2 + 2\pi r h = \pi + 36\pi = 37\pi\text{ cm}^2.$$ ### Real-world application: Packaging many cylinders in one box (manufacturer example) - Suppose cylinders each have diameter $5\text{ cm}$, height $20\text{ cm}$, and are packed 6 across in one direction and 3 across in the other (a single layer). - Box interior length = $6\times5 = 30\text{ cm}$. - Box interior width = $3\times5 = 15\text{ cm}$. - Box interior height = $20\text{ cm}$ (single layer). - If the box will be wrapped with cling wrap but has no lid, compute area of bottom, two long sides, and two short sides (omit top): - Bottom: $30\times15 = 450\text{ cm}^2$. - Two short sides (left and right): $2\times(20\times15) = 600\text{ cm}^2$