Study guide: Area and Volume Geometry

📋 Course Outline

1. Tiling and Area Reasoning
2. Area Strategies and Units
3. Parallelogram Area
4. Triangle Area
5. Polygons and Their Areas
6. Surface Area and Nets
7. Polyhedra, Prisms, and Pyramids
8. Length, Area, and Volume
9. Powers and Geometric Measures

📖 1. Tiling and Area Reasoning

🔑 Key Concepts & Definitions

• Tiling: Covering a two-dimensional region with copies of the same shape or shapes without gaps or overlaps.

📝 Essential Points

• Comparing which shapes or patterns cover more of the plane is a way to begin reasoning about area.

📖 2. Area Strategies and Units

★ Must-know

📌 Figures that can be placed exactly on top of one another have the same area.

• A figure's area can be found by decomposing it into pieces, rearranging the pieces into known shapes, finding their areas, and adding them.

• A figure's area can be found by calculating the area of a containing shape and subtracting the area of the missing piece.

Further detail

• Area is measured in square units, and a rectangle whose side lengths are measured in centimeters has area in square centimeters.

• Two squares each with area 1 square unit can be decomposed into four equal triangles, each with area one-half square unit, and rearranged to form a larger triangle of area 2 square units.

💡 Memory Hook

Decompose, rearrange, or subtract the missing piece.

📖 3. Parallelogram Area

🔑 Key Concepts & Definitions

• Parallelogram: A quadrilateral whose opposite sides are parallel and equal in length and whose opposite angles have equal measures.
• Base of a parallelogram: Any side of a parallelogram can be chosen as its base, and the term base refers both to the side and its length.
• Height of a parallelogram: The length of a perpendicular segment from a point on the base to the opposite side; all such segments have the same length.

★ Must-know

• A parallelogram's area can be found by decomposing and rearranging it into a rectangle, or by enclosing it in a rectangle and subtracting the corner triangles.

📐 Formula - The area of a parallelogram with base $b$ and corresponding height $h$ is $b \cdot h$.

Further detail

• A parallelogram with base 6 units and height 4 units and the same parallelogram with base 5 units and height 4.8 units both have area 24 square units.

📌 Parallelograms with the same base and corresponding height have the same area, even if their shapes differ.

📖 4. Triangle Area

🔑 Key Concepts & Definitions

• Base of a triangle: Any of the three sides of a triangle can be chosen as its base, and base refers both to the side and its length.
• Height of a triangle: The length of a perpendicular segment connecting the base, or its extended line, to the opposite vertex or a parallel line through that vertex.

★ Must-know

• A segment joining opposite vertices decomposes any parallelogram into two identical triangles, and two identical copies of any triangle can be joined along matching sides to form a parallelogram.

📐 Formula - The area of a triangle with base $b$ and corresponding height $h$ is $A = \frac{1}{2} \cdot b \cdot h$.

Further detail

• Joining two identical triangles along any one of their three matching sides can produce different parallelograms.

• A triangle with base 5 units and height 6 units has area 15 square units, and one with base 3 units and height 3 units has area 4.5 square units.

• In a right triangle, the two perpendicular sides can serve as a base and its corresponding height.

📖 5. Polygons and Their Areas

🔑 Key Concepts & Definitions

• Polygon: A two-dimensional figure made of straight line segments whose sides do not cross and whose vertices are the points where the segments connect.
• Area of a polygon: The area of the region enclosed by the polygon.

★ Must-know

• A polygon's area can be found by decomposing its interior into triangles and rectangles and adding their areas, or by subtracting the areas of corner triangles from an enclosing rectangle.

Further detail

• A polygon has an equal number of vertices and sides.

📖 6. Surface Area and Nets

🔑 Key Concepts & Definitions

• Surface area: The number of unit squares needed to cover an entire surface without gaps or overlaps.
• Net: A two-dimensional arrangement of polygons that form the faces of a polyhedron and can be folded into a model of it.
• Face: A flat side of a three-dimensional figure.

★ Must-know

📌 The surface area of a three-dimensional figure with flat faces is the sum of the areas of all its faces.

• A polyhedron's surface area can be found from its net by calculating the area of every face and adding the face areas.

Further detail

• A rectangular prism with edge lengths 2 cm, 3 cm, and 4 cm has surface area 52 square centimeters.

📖 7. Polyhedra, Prisms, and Pyramids

🔑 Key Concepts & Definitions

• Polyhedron: A three-dimensional figure made of polygonal faces that meet along complete edges and enclose a three-dimensional region.
• Prism: A polyhedron with two identical parallel faces called bases, connected by rectangles or sometimes non-rectangular parallelograms.
• Pyramid: A polyhedron with one base and triangular remaining faces that all meet at a single vertex.

📝 Essential Points

• A prism or pyramid is named according to the shape of its base.

• A cube has six square faces, and in a folded cube net each square's edges join edges of other squares even though some edges are open in the flat net.

💡 Memory Hook

A prism has two parallel bases; a pyramid has one base and triangular faces meeting at a vertex.

📖 8. Length, Area, and Volume

🔑 Key Concepts & Definitions

• Length: A one-dimensional attribute measured in units such as millimeters, centimeters, meters, kilometers, inches, feet, yards, and miles.
• Area: A two-dimensional attribute measured in square units.
• Volume: A three-dimensional attribute measured in cubic units.

★ Must-know

📌 Two figures can have the same volume but different surface areas, or the same surface area but different volumes.

Further detail

• Rectangular prisms measuring 1 cm by 2 cm by 2 cm and 1 cm by 1 cm by 4 cm both have volume 4 cubic centimeters, but their surface areas are 16 and 18 square centimeters, respectively.

💡 Memory Hook

Length is one-dimensional, area two-dimensional, and volume three-dimensional.

📖 9. Powers and Geometric Measures

🔑 Key Concepts & Definitions

• Exponent: The raised number that indicates how many times a number is multiplied by itself.

★ Must-know

• Squaring a number means multiplying it by itself, and cubing a number means multiplying it by itself three times.

📐 Formula - The surface area of a cube with edge length $s$ is $6 \cdot s^2$ because its six identical square faces each have area $s^2$.

Further detail

• A square with side length 5 inches has area 25 square inches, and a cube with edge length 4 cm has volume 64 cubic centimeters.