Surface Area and Volume Unit | 3D Geometry | Grades 7–10 With Solutions

Teach the complete 3D Geometry unit in one ready-to-print resource — from guided notes through quizzes and a comprehensive test, all with fully worked solutions.

This Surface Area and Volume Complete Unit covers every solid your Grades 7–10 students need to master: prisms, cylinders, pyramids, cones, spheres, hemispheres, composite figures, and real-world density applications. Every section includes 3D diagrams placed directly beside the worked solutions so students can see the shape and the maths at the same time — no flipping between pages.

Built for teachers who want to save planning time without sacrificing rigor or visual quality.

─── WHAT'S INCLUDED ───

6 Guided Note Sections (with 3D diagrams, vocabulary tables, formula reference tables, and multiple worked examples per section)

Section 1 — Prisms: Rectangular, triangular (right & equilateral), trapezoidal, and hexagonal prisms. Lateral surface area, total surface area, nets, volume, and composite prism problems. Includes 8 fully worked examples. Section 2 — Cylinders: Lateral & total surface area, volume, and hollow cylinder problems. Includes a cylinder net diagram and 3 fully worked examples. Section 3 — Pyramids: Square and triangular pyramids. Lateral SA using slant height, nets, volume, and frustum of a pyramid. Includes 4 fully worked examples. Section 4 — Cones: Lateral and total surface area, slant height relationships, volume, frustum of a cone, and a cone-vs-cylinder comparison. Includes 3 fully worked examples. Section 5 — Spheres & Hemispheres: Surface area, volume, hemisphere SA and volume, Archimedes' result. Includes 2 fully worked examples. Section 6 — Composite Figures: Decomposing strategy (add/subtract), cylinder-plus-hemisphere composite, hollowed solid problems. Includes 2 fully worked examples. 2 Enrichment Activities

Cavalieri's Principle & Cross-Sections: Explanation of the principle with the formula, its application to why cones and pyramids have a 1/3 factor, and a cross-section identification table for all 5 main solids. 3D Net Identification & Folding Activity: 4 labelled nets (rectangular prism, cylinder, square pyramid, cone) with identification blanks for students to name the solid and record key measurements. 80 Practice Problems — Full Solutions Included

40 Surface Area Problems covering all 8 solid types (prisms ×8, cylinders ×8, pyramids ×8, cones ×8, spheres/hemispheres ×8). Every problem includes the full worked solution with formula, substitution, and final answer in a side-by-side diagram layout. 40 Volume Problems covering all solid types in the same format with fully worked solutions. 10 Real-World Application Problems with diagrams and solutions — scenarios include: painting a room, grain silo cost, ice cream cone capacity, tennis ball canister air space, trapezoidal cross-section pool volume, sphere balloon surface area and cost, pyramid roof area, rectangular metal box material, cylindrical candle wrapping, hollow water tank material volume.

10 Density & Mass Application Problems with solutions — including gold bar, steel ball bearing, aluminum can, wooden block (find density), copper pipe, concrete pillar, mercury flask, comparing gold vs silver spheres, lead shot, and water tank mass.

Cumulative Unit Review

Complete formula quick-reference table for all 7 solid types (LSA, TSA, Volume) Key relationships & theorems (Archimedes, Cavalieri, cone:cylinder ratio, slant height formulas) 3 open-ended concept check questions 2 Unit Quizzes + Full Solutions

Quiz 1: Prisms & Cylinders (rectangular prism, triangular prism, cylinder, hollow cylinder, word problem) — questions page + separate solutions page Quiz 2: Pyramids, Cones & Spheres (square pyramid, cone, conical frustum, sphere/hemisphere combo, cone vs cylinder) — questions page + separate solutions page Comprehensive Unit Test + Full Solutions

10 questions across 4 parts: Prisms & Cylinders, Pyramids & Cones, Spheres & Hemispheres, Composite & Applications Separate solutions section with all diagrams and full worked steps ─── SKILLS COVERED ───

Finding the lateral surface area and total surface area of all prism types Finding the volume of all prism types (rectangular, triangular, trapezoidal, hexagonal) Calculating cylinder lateral SA, total SA, and volume Solving hollow cylinder (annulus cross-section) surface area and volume problems Applying slant height relationships in pyramids and cones (l = √(h² + (b/2)²) and l = √(r² + h²)) Surface area and volume of square and triangular pyramids Volume of a frustum of a pyramid — V = (1/3)h(B₁ + B₂ + √(B₁B₂)) Surface area and volume of cones Volume of a frustum of a cone — V = (1/3)πh(R² + Rr + r²) Surface area and volume of spheres (SA = 4πr², V = (4/3)πr³) Surface area and volume of hemispheres Comparing cone and cylinder volumes (1:3 ratio) Composite solid volume and surface area (add/subtract strategy) Hollowed solid volume (sphere minus cylindrical hole) Density, mass, and volume applications: Mass = Density × Volume Reading and interpreting 3D nets and identifying the solid formed Cavalieri's Principle and cross-section identification ─── HOW TO USE ───

As a Full Unit: Teach sections in order — each builds on prior knowledge. Use the guided notes as direct instruction, assign practice problems in class or as homework, and use the quizzes and test for formal assessment.

As a Targeted Supplement: Assign specific sections (e.g., Cones only, or just the density problems) to plug gaps in existing instruction.

For Differentiation: Use the worked-solution practice sets as guided practice for students who need more scaffolding, or as a self-checking independent work tool.

For Test Prep: The cumulative review and unit test make excellent end-of-unit or exam-preparation tools.

Print Format: Pages are formatted for US Letter (8.5" × 11"). Print single-sided for student workbooks or double-sided to save paper.

─── GRADE SUITABILITY ───GradeBest UseGrade 7Prisms, cylinders — sections 1 & 2, SA/Volume practice setsGrade 8All prisms, cylinders, pyramids, cones — sections 1–4Grade 9Full unit including spheres, composite figures, densityGrade 10Full unit including frustums, Cavalieri's principle, all applications

─── TEACHER NOTES ───

All answers are included — no separate answer key needed. All symbols render in standard print fonts — no special software needed to open or print. Solutions are placed on separate pages (quizzes, test) so student question.