Geometry Fundamentals II Practice Quiz with a Step-by-Step Interactive Lesson
Use the question set below to practice Geometry Fundamentals II: regular polygons and angle relationships, counting faces, edges, and vertices, and mastering surface area, lateral area, and volume formulas for prisms, pyramids, cylinders, cones, and spheres. If you want a refresher, click Start lesson to open a step-by-step guide with worked examples and quick checks.
Answer the question set and review your mistakes at the end.
How this geometry practice works
1. Take the practice set: answer the geometry questions below.
2. Open the lesson (optional): review key geometry facts and formulas with worked examples and quick checks.
3. Retry: return to the question set and apply the geometry formulas immediately.
What you will learn in the Geometry Fundamentals II lesson
Polygons & angle measures
Adjacent angles and common angle vocabulary
Interior angle sum: \((n-2)\cdot 180^\circ\)
Regular polygons: each interior angle \(\dfrac{(n-2)\cdot 180^\circ}{n}\)
3D solids: faces, edges, vertices
Polyhedra (prisms, pyramids) vs curved solids (cylinder, cone, sphere)
Count faces, edges, vertices reliably (and avoid common traps)
Dihedral angle (angle between faces/planes) as a key solid-geometry idea
Surface area & lateral area
Nets and "add the areas of all faces"
Rectangular prism: \(SA=2(lw+lh+wh)\)
Cylinder & cone formulas (lateral vs total surface area)
Volume formulas & real-world geometry
Prisms and cylinders: \(V=Bh\)
Pyramids and cones: \(V=\dfrac{1}{3}Bh\)
Cubes: \(V=s^3\) and combining volumes without overlap
Purpose: Build a strong foundation in Geometry Fundamentals II—regular polygons, angle relationships, and solid geometry formulas for surface area, lateral area, and volume.
Success criteria
Use accurate angle vocabulary (including adjacent angles).
Find interior-angle measures for regular polygons using \(\dfrac{(n-2)\cdot 180^\circ}{n}\).
Identify and count faces, edges, and vertices of common solids (prisms, pyramids, cylinder, cone, sphere).
Compute surface area of prisms and cubes using correct formulas (and interpret nets).
Compute lateral surface area (side area only) vs total surface area (including bases).
Compute volume of prisms/cylinders (\(V=Bh\)) and cones/pyramids (\(V=\tfrac13Bh\)).
Recognize a dihedral angle as the angle between two faces (two planes).
Key vocabulary
Adjacent angles: share a common vertex and a common side (and do not overlap).
Regular polygon: all sides and all angles equal.
Face / edge / vertex: a flat surface / where faces meet / a corner point.
Prism: two parallel, congruent bases connected by rectangles.
Pyramid: one base with triangular faces meeting at an apex.
Slant height \(l\): the length along the side of a cone (used for surface area).
Dihedral angle: the angle between two planes (e.g., two faces of a polyhedron).
Quick pre-check
Pre-check 1: Angles that share a common side and vertex are called what?
Hint: They touch at the same vertex and share one side.
Pre-check 2: How many faces does a rectangular prism have?
Hint: Think “top, bottom, and four sides.”
Polygons & Angles
Regular polygons and interior angles
Learning goal: Find interior-angle measures for regular polygons and use a reliable formula every time.
Key idea
A polygon with \(n\) sides has an interior-angle sum:\[\text{Sum of interior angles}=(n-2)\cdot 180^\circ.\]If the polygon is regular (all angles equal), then each interior angle is:\[\text{Each interior angle}=\frac{(n-2)\cdot 180^\circ}{n}.\]
Worked example
Example: Find the measure of each interior angle of a regular enneagon (9-sided polygon).
Use \(n=9\). First find the sum:\[(9-2)\cdot 180^\circ = 7\cdot 180^\circ = 1260^\circ.\]Then divide by 9 (because it’s regular):\[\frac{1260^\circ}{9}=140^\circ.\]
Try it
Try it 1: What is the measure of each interior angle of a regular enneagon (9-sided polygon)?
Hint: Use \(\dfrac{(n-2)\cdot 180^\circ}{n}\) with \(n=9\).
Try it 2: What is the area of a regular hexagon with side length \(s\)? (in terms of \(s\))
Hint: Split the hexagon into 6 equilateral triangles of side \(s\).
Summary
Interior-angle sum: \((n-2)\cdot 180^\circ\).
Regular polygon: each interior angle \(\dfrac{(n-2)\cdot 180^\circ}{n}\).
Learning goal: Identify faces, edges, and vertices and count them correctly for prisms, pyramids, and common polyhedra.
Key idea
For polyhedra (solids with flat faces): faces are flat surfaces, edges are where faces meet, and vertices are corner points. A handy check for many convex polyhedra is Euler’s formula:\[V - E + F = 2,\]where \(V\) is vertices, \(E\) edges, and \(F\) faces.
Worked example
Example: How many vertices and edges does a square pyramid have?
A square pyramid has a square base (\(n=4\)) and one apex. Vertices: \(n+1=4+1=5\). Edges: base edges \(4\) plus lateral edges \(4\), so \(8\) total.
Try it
Try it 1: How many vertices does a square pyramid have?
Hint: A pyramid has all base vertices plus one apex.
Try it 2: How many edges does a pentagonal prism have?
Hint: A prism with \(n\)-gon bases has \(3n\) edges (top \(n\), bottom \(n\), vertical \(n\)).
Try it 3: How many vertices does a tetrahedron have?
Hint: A tetrahedron is a pyramid with a triangular base (\(n=3\)), so vertices \(=n+1=4\).
Summary
Square pyramid: \(5\) vertices and \(8\) edges.
Pentagonal prism: \(15\) edges.
Tetrahedron: \(4\) vertices.
Surface Area
Surface area and lateral surface area
Learning goal: Compute total and lateral surface area using correct formulas and a “net” mindset.
Key idea
Surface area is the sum of the areas of all outside faces. A net is a “flattened” layout of those faces—useful for checking what to include.
For a rectangular prism with dimensions \(l\times w\times h\):\[SA = 2(lw + lh + wh).\]For a cube with side length \(s\):\[SA_{\text{total}} = 6s^2, \quad SA_{\text{lateral}} = 4s^2.\]
Worked example
Example: What is the surface area of a rectangular prism with dimensions \(2 \times 3 \times 4\)?
Compute the three face areas: \(lw=2\cdot 3=6\), \(lh=2\cdot 4=8\), \(wh=3\cdot 4=12\). Add and multiply by 2:\[SA=2(6+8+12)=2(26)=52.\]
Try it
Try it 1: What is the surface area of a rectangular prism with dimensions \(2 \times 3 \times 4\)?
Hint: Use \(SA=2(lw+lh+wh)\) with \(l=2,w=3,h=4\).
Try it 2: What is the lateral surface area of a cube with side length \(2\)?
Hint: Lateral area is 4 side faces. Each face has area \(s^2\).
Summary
Rectangular prism: \(SA=2(lw+lh+wh)\).
Cube: total \(6s^2\), lateral \(4s^2\).
Volume
Volume of prisms and cubes
Learning goal: Compute volume using base area times height, and combine volumes when shapes don’t overlap.
Key idea
For prisms (including rectangular prisms) and cylinders, volume is:\[V = Bh,\]where \(B\) is the area of the base and \(h\) is the height (perpendicular distance). For a rectangular prism, this becomes \(V=lwh\). For a cube, \(V=s^3\).
Worked example
Example: What is the volume of a rectangular prism with dimensions \(2 \times 3 \times 4\)?
Multiply the three dimensions:\[V = 2\cdot 3\cdot 4 = 24.\]The units are cubic units.
Try it
Try it 1: What is the volume of a rectangular prism with dimensions \(2 \times 3 \times 4\)?
Hint: Multiply \(2\cdot 3\cdot 4\).
Try it 2: What is the combined volume of two cubes with side lengths \(2\) and \(3\)?
Hint: Cube volume is \(s^3\). Add \(2^3\) and \(3^3\).
Summary
Prisms: \(V=Bh\). Rectangular prism: \(V=lwh\).
Cubes: \(V=s^3\). Add volumes if solids don’t overlap.
Cones & Pyramids
Cones: volume and total surface area
Learning goal: Use \(\tfrac13Bh\) for cone volume and distinguish lateral vs total surface area using slant height.
Key idea
A cone (and any pyramid) has volume:\[V=\frac{1}{3}Bh.\]For a right circular cone, the base area is \(B=\pi r^2\), so:\[V=\frac{1}{3}\pi r^2 h.\]The cone’s lateral surface area uses slant height \(l\):\[SA_{\text{lateral}}=\pi r l,\]and total surface area adds the base:\[SA_{\text{total}}=\pi r^2+\pi r l.\]
Worked example
Example: A right circular cone has radius \(4\) and height \(3\). What is its volume?
Try it 2: What is the total surface area of a right circular cone with radius \(1\) and slant height \(2\)?
Hint: \(SA_{\text{total}}=\pi r^2+\pi r l\). With \(r=1,l=2\), that is \(\pi+2\pi\).
Summary
Cone volume: \(V=\dfrac13\pi r^2 h\).
Cone total surface area: \(SA=\pi r^2+\pi r l\).
Cylinders & Spheres
Cylinders and spheres: area, volume, and “parts”
Learning goal: Use cylinder surface-area formulas correctly and understand edges/vertices for curved solids.
Key idea
A cylinder’s lateral surface area (side only) is:\[SA_{\text{lateral}} = 2\pi r h,\]and its total surface area adds the two circular bases:\[SA_{\text{total}} = 2\pi r^2 + 2\pi r h.\]A sphere has no edges and no vertices. Its surface area is:\[SA_{\text{sphere}} = 4\pi r^2.\]
Worked example
Example: What is the lateral surface area of a cylinder with radius \(2\) and height \(3\)?
\[SA_{\text{lateral}}=2\pi r h = 2\pi(2)(3)=12\pi.\]
Try it
Try it 1: What is the lateral surface area of a cylinder with radius \(2\) and height \(3\)?
Hint: Use \(SA_{\text{lateral}}=2\pi r h\).
Try it 2: How many edges does a cylinder have?
Hint: Count the circular boundaries where the bases meet the curved surface.
Try it 3: How many edges does a sphere have?
Hint: A sphere has a smooth surface with no line segments where faces meet.
Learning goal: Combine definitions and formulas, and build the habit of checking answers for reasonableness.
Key idea: dihedral angle
A dihedral angle is the angle between two planes. In solid geometry, it’s the angle between two faces of a polyhedron. For a cube, any two adjacent faces meet at a right angle, so the dihedral angle is \(90^\circ\).
Real-world connections
Packaging and shipping: surface area helps estimate wrapping material; volume helps estimate capacity.
Architecture and engineering: angles and face intersections matter for joints and corners.
Design and 3D printing: knowing edges/vertices helps visualize models and nets.
Try it
Try it 1: What is the dihedral angle between two faces of a cube?
Hint: Cube faces meet like two walls meeting at a corner.
Try it 2: What is the total number of edges on a cube and a tetrahedron combined?
Hint: Cube has 12 edges. Tetrahedron has 6 edges.
Try it 3: How many edges does a square pyramid have?
Hint: 4 base edges + 4 edges from base vertices to the apex.
Fun facts (a little geometry history)
Euclid: Many foundational geometry results were organized in Euclid’s Elements.
Archimedes: Early work on areas and volumes paved the way for modern measurement formulas.
Big idea: The same surface-area and volume reasoning shows up in science, engineering, and computer graphics.
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the geometry skill you need.
Practice set
Geometry fundamentals II practice questions with instant score
Answer all 10 questions below, then get your final score and a mistake review at the end so you know exactly what to improve.
0/10answered
Question 1Not answered
How many faces does a cube have?
Correct answer: B. 6
Explanation: A cube has 6 faces.
Question 2Not answered
What is the volume of a cube with side length 3?
Correct answer: D. 27
Explanation: Volume = side³ = \(3^3 = 27\).
Question 3Not answered
How many edges does a cube have?
Correct answer: D. 12
Explanation: A cube has \(12\) edges.
Question 4Not answered
How many vertices does a rectangular prism have?
Correct answer: A. 8
Explanation: A rectangular prism has \(8\) vertices.
Question 5Not answered
What is the surface area of a cube with side length \(2\)?
