Volume of Three-Dimensional Solids Notes

Volume of Three-Dimensional Solids

Calculate the volume of prisms, pyramids, cones, cylinders, and spheres.
Calculate the volume of composite three-dimensional figures.
Determine the area of composite two-dimensional figures.
Use appropriate units of measure when calculating volume.
Calculate volume for problems with real-world context.
Explain the reasonableness of the solution to volume problems with real-world context.
Calculate the changes in surface area and volume when the dimensional change is proportional.
Calculate the changes in surface area and volume when the dimensional change is non-proportional.
Sentence stems for calculating volume:
- To calculate the volume of a , first I _ Next I ____ After that I ____ Finally I _
- To calculate the volume of a , I…

G.11D: Apply the formulas for the volume of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure.
G.10B: Determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional change.

Parallelogram: A = bh
Rectangle: A = lw
Regular Polygon: A = \frac{1}{2}Pa
Prism and Cylinder:
- Lateral Area: LA = Ph
- Surface Area: SA = LA + 2B
- Volume: V = Bh
Pyramid and Cone:
- Lateral Area: LA = \frac{1}{2}Pl
- Surface Area: SA = LA + B
- Volume: V = \frac{1}{3}Bh
Triangle: A = \frac{1}{2}bh
Equilateral Triangle: A = \frac{s^2\sqrt{3}}{4}
Regular Hexagon: A = 6 \cdot \frac{s^2\sqrt{3}}{4}
Trapezoid: A = \frac{1}{2}h(b1 + b2)
Circle:
- Circumference: C = 2\pi r
- Area: A = \pi r^2
Square: A = s^2
Rhombus, Kite, and Square: A = \frac{1}{2}d1d2
Sphere:
- Surface Area: SA = 4\pi r^2
- Volume: V = \frac{4}{3}\pi r^3

Volume: The space that a three-dimensional figure occupies. Measured in cubic units.
Cavalieri’s Principle: If two 3-D figures have the same height and the same cross-sectional area at every level, then they have the same volume.
The volume of a prism or cylinder is the product of the base area and the height: V = Bh

The volume of a pyramid is one-third the product of the area of the base and the height of the pyramid: V = \frac{1}{3}Bh
Cavalieri’s Principle applies to pyramids, including oblique pyramids.
The volume of a cone is one-third the product of the area of the base and the height of the cone: V = \frac{1}{3}Bh or V = \frac{1}{3}\pi r^2 h
This formula applies to all cones, including oblique cones.

Rotating a region around an axis creates a solid of revolution. The volume can be determined based on the shape and axis of rotation (refer to the tables for specific formulas based on the graph of the region).

Sphere: The set of all points in space equidistant from a given point called the center.
Radius: A segment that has one endpoint at the center and the other endpoint on the sphere.
Diameter: A segment passing through the center with endpoints on the sphere.
When a plane and a sphere intersect in more than one point, the intersection is a circle.
Great Circle: If the center of the circle is also the center of the sphere.
The circumference of a great circle is the circumference of the sphere.
A great circle divides a sphere into two hemispheres.
Surface Area of a Sphere: S = 4\pi r^2
Volume of a Sphere: V = \frac{4}{3}\pi r^3

Solids that have the same shape but different sizes are said to be similar.
Similarity can be determined by comparing the ratios of corresponding linear measurements.
Relationships between ratios of similar solids:
- If the dimensions of a figure are multiplied by k, the surface area will be k^2 times larger, and the volume will be k^3 times larger.

Comprehensive review covering volumes of various 3D shapes and their surface areas, including composite figures and solids of revolution, as well as practical problems involving rates and dimensional changes.