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 the solids listed for Grades 7–10: **prisms, cylinders, pyramids, cones, spheres, hemispheres, composite figures, and real-world density applications**. Every section includes 3D diagrams placed directly beside worked solutions so students can see the shape and the mathematics together—without 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**

Each section includes **3D diagrams, vocabulary tables, formula reference tables, and multiple worked examples**.

**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 surface area 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 surface area and volume, Archimedes' result. Includes **2 fully worked examples**.

**Section 6 — Composite Figures:** Decomposing strategy (add/subtract), cylinder-plus-hemisphere composite, and 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 the main solids.

**3D Net Identification & Folding Activity:** 4 labeled nets—rectangular prism, cylinder, square pyramid, and 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 the listed solid types, with every problem including a full worked solution using formula, substitution, and final answer in a side-by-side diagram layout.

**40 Volume Problems:** Covering the solid types in the same format with fully worked solutions.

### **10 Real-World Application Problems**

Includes diagrams and solutions for scenarios involving:

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**

Includes solutions involving:

Gold bar Steel ball bearing Aluminum can Wooden block (find density) Copper pipe Concrete pillar Mercury flask Comparing gold vs. silver spheres Lead shot Water tank mass

### **Cumulative Unit Review**

Complete formula quick-reference table Key relationships & theorems, including Archimedes, Cavalieri, cone:cylinder ratio, and 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, and a word problem. Includes a questions page and separate solutions page.

**Quiz 2:** Pyramids, Cones & Spheres — square pyramid, cone, conical frustum, sphere/hemisphere combination, and cone-vs-cylinder comparison. Includes a questions page and separate solutions page.

### **Comprehensive Unit Test + Full Solutions**

**10 questions across 4 parts:**

Prisms & Cylinders Pyramids & Cones Spheres & Hemispheres Composite & Applications

A separate solutions section includes diagrams and full worked steps.

### ─── SKILLS COVERED ───

Finding lateral surface area and total surface area of prism types Finding volume of rectangular, triangular, trapezoidal, and hexagonal prisms Calculating cylinder lateral surface area, total surface area, and volume Solving hollow-cylinder surface area and volume problems Applying slant-height relationships in pyramids and cones 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 using the stated 1:3 ratio Composite-solid volume and surface area using add/subtract strategies Hollowed-solid volume 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, using the guided notes for direct instruction, practice problems for classwork or homework, and the quizzes and test for assessment.

**As a Targeted Supplement:** Assign specific sections, such as Cones or density problems, to address particular instructional needs.

**For Differentiation:** Use worked-solution practice sets as guided practice for students who need more support or as self-checking independent work.

**For Test Prep:** Use the cumulative review and unit test for end-of-unit or exam preparation.

**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 ───

**Grade 7:** Prisms and cylinders—Sections 1 & 2 and related surface-area/volume practice **Grade 8:** All prisms, cylinders, pyramids, and cones—Sections 1–4 **Grade 9:** Full unit including spheres, composite figures, and density **Grade 10:** Full unit including frustums, Cavalieri's Principle, and 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 for quizzes and the test so student questions remain separate from solutions.