Geometry 3D Shapes Unit Summary

Polyhedra and Basic Concepts

A polyhedron is a solid shape bounded by polygons called faces that enclose a region of space. Think of shapes like cubes, pyramids, and prisms.

A prism is a polyhedron with two congruent faces called bases that lie in parallel planes. The bases stay the same regardless of how you position the prism.

When working with pyramids or cones, remember that height and slant height are different measurements. The height is the perpendicular distance from the apex to the base, while slant height measures from the apex to the edge of the base.

Helpful Tip: Cavalieri's Principle states that if two solids have the same height and the same cross-sectional area at every level, they have the same volume—even if they look different!

Density and Surface Area Formulas

Density is calculated as mass divided by volume. Similarly, population density equals population divided by area. These concepts help us understand how tightly packed matter or people are in a space.

For rectangular prisms, the surface area formula is SA = 2(lw + lh + wh), where l is length, w is width, and h is height.

For all other prisms and cylinders, the surface area formula is SA = hp + 2B, where h is the height between bases, p is the perimeter of the base, and B is the area of the base.

For pyramids, use SA = $lp/2$ + B, where l is slant height, p is perimeter of base, and B is base area.

Remember: Surface area calculations always include ALL faces of the solid!

Surface Area of Curved Shapes

The surface area of a cone can be calculated using the formula SA = πr$r + l$, where r is the radius of the base and l is the slant height. Think of this as the circular base plus the curved surface.

For a sphere, the surface area formula is SA = 4πr². You can visualize this as four circles with the same radius as the sphere.

A hemisphere (half a sphere) has a surface area of SA = 3πr². This includes both the curved surface and the circular base.

Visualization Tip: The surface area of a sphere (4πr²) is exactly four times the area of its great circle (πr²), which is any circle that divides the sphere into two equal hemispheres.

Volume of Prisms and Pyramids

For rectangular prisms, the volume formula is V = lwh, where l is length, w is width, and h is height. This is simply the area of the base times the height.

The same principle applies to all prisms and cylinders: V = Bh, where B is the area of the base and h is the height.

For pyramids, the volume formula is V = (Bh)/3, where B is the area of the base and h is the height of the pyramid. Notice pyramids have exactly 1/3 the volume of prisms with the same base and height.

For cylinders, use V = πr²h, where r is the radius of the circular base and h is the height.

Fun Fact: If you fill a pyramid and a prism that share the same base and height with water, you'd need to empty the pyramid exactly three times to fill the prism!

Volume of Curved Solids

The volume of a cone is V = (πr²h)/3, where r is the radius of the base and h is the height. Notice this is exactly 1/3 the volume of a cylinder with the same base and height.

For a sphere, the volume formula is V = $4/3$πr³, where r is the radius. This formula helps calculate the space inside a ball or globe.

A hemisphere (half sphere) has a volume of V = $2/3$πr³, which is exactly half the volume of a complete sphere.

Application Note: These volume formulas are essential for real-world problems—like calculating how much water a tank can hold, how much air is in a balloon, or how much material is needed to make a solid object.