This lesson advances your skills in calculating surface area of 3-D shapes by combining formulas, visualization through nets, and application of the Pythagorean theorem to find missing dimensions. It includes strategies, key formulas, worked examples, practice problems, and tips to effectively solve surface area problems involving various 3-D solids and their combinations.
"Surface area is just geometry's way of asking: how much paint do we need for this drama?" — your slightly dramatic math TA You already know how to visualize surface area with nets and how to calculate cylinders (we did that detective work last time). Now we level up: we solve real problems that mi...
Quick reminder (so we don't reinvent the wheel) A net helps you lay a 3‑D shape flat so you can see all faces at once — we used nets earlier to visualize what needs painting. For a cylinder , you know the total surface area formula: 2πr(r + h). If you forgot, go peek at Position 3 in the lesson ch...
Strategy: How to solve surface area problems (a step‑by‑step detective checklist) Draw the shape and, if helpful, draw the net. Label all given dimensions. Identify each face that contributes to surface area (including bases!). Ask: does the shape have hidden triangles, rectangles, or curved surfa...
Key formulas (your crime‑scene toolkit) Code block style so it looks official and intimidating: Cube: SA = 6a^2 Rectangular prism: SA = 2(lw + lh + wh) Cylinder: SA = 2πr(r + h) (2πr^2 for the bases + 2πrh for the curved part) Right circular cone: SA = πr(r + l) (l = slant height) Square pyramid: ...
Worked example 1 — A cone with secrets (slant height via Pythagoras) Problem: A right circular cone has a base radius r = 6 cm and a vertical height h = 8 cm. Find the total surface area. Step 1: We need the slant height l. Right triangle with legs 6 and 8, hypotenuse l. l^2 = r^2 + h^2 = 6^2 + 8...
Worked example 2 — Compound solid: cylinder with a cone on top Problem: A water tank is a vertical cylinder (radius 5 m, height 10 m) topped by a right cone (same base radius 5 m) whose vertical height is 3 m. Find the total surface area exposed to air (so include the cone, the curved part of the c...
Quick table: Common faces and what to compute Shape Faces to include (total SA) Rectangular prism 6 rectangles (2 of each pair) Cylinder 2 circles + 1 rectangle (curved surface as 2πrh) Cone 1 circle + 1 curved triangular face (πr l) Square pyramid 1 square + 4 triangular faces (use sl...
Practice time — try these (answers below so you can self‑grade) A rectangular prism is 4 cm by 3 cm by 6 cm. Find SA. A right pyramid has a square base with side 8 cm and slant height 5 cm. Find SA. A cone has radius 7 cm and slant height 25 cm. Find SA. A composite solid: a hemisphere (radius 6...
Final tips (the stuff that saves marks on tests) Always list which faces you are including — graders love clues that you know what’s happening. Keep units consistent and square them at the end (m², cm²). If slant heights are missing, ask if the problem implies a right cone/pyramid — then use Pyt...
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