Chapter 12 Final Review

In chapter 12, we learned about surface area and volume.

Section 12.1: Surface Areas of Prisms

In every prism there are two bases and n number of lateral faces, where n is the number of the sides of the face. The lateral faces are always parallelograms. The lateral area of a prism is the surface area of all of the lateral faces and total surface area of a prism is the sum of the lateral areas and the surface area of the two bases.

The formula: S=2B+L, where L is the lateral area and B is the base of the prism. Also, the perimeter * height equals the lateral area, so a more useful formula is S=2B+ph, where p is the perimeter of the base and h is the height of the prism.

Section 12.2: Surface Areas of Pyramids

In every pyramid, there is one base and n lateral faces, when n is the number of sides on the base. In a regular pyramid, the base is a regular polygon and all of the lateral edges are congruent.

The formula: S=B+L, when B is the area of the base and L is the lateral area of the pyramid. One half of l, the slant height, and p, the perimiter of the base, equals the lateral area, so a more useful formula is S=B+1/2 pl, where p is the perimiter of the base and h is the height of the pyramid.

Section 12.3: Surface Area of Circular Solids

A cylinder is an infinitely sided prism, so the formula for its surface area is S=2B+ph. Because the cylinder's base is a circle, the area of the base is πr2(pi * radius squared, superscript doesn't work), and the perimiter is 2πr, so the formula is S=πr2+2πrh, where h is the height and r is the radius of the cylinder.

A cone is an infinitely sided pyramid, so the formula for its surface area is S=B+1/2 pl. Because the cone's base is a circle, the area of the base is πr2 (again, radius squared), and the perimiter is 2πr, so the formula is S=πr2+πrl, where l is the slant height and r is the radius of the cylinder.

A sphere's Surface area is S=4πr2, and a hemisphere's formula for surface area is 3πr2, not 2πr2, because you are adding another circle when you cut the sphere in half.

We also learned about a frustum during this section.

A frustum is a pyramid with a pyramid similar to the original one sliced off, leaving the bases parallel. In a conical frustum, the Surface area is S=πR2-πr2+πRL-πrl, where R is the radius of the big cone, r is the radius of the small cone, L is the slant height of the big cone, and l is the slant height of the small cone.

Section 12.4: Volumes of Prisms and Cylinders

Volume is the measure of the space enclosed by a solid. in prisms, the formula for volume is V=Bh, where B is the area of the base and h is the height of the prism. A cylinder's base is a circle, so B=πr2, and the formula is V=πr2h, where h is the height of the cylinder and r is the radius of the cylinder.

Section 12.5: Volumes of Pyramids and Cones

The formula for volume in a pyramid is 1/3 Bh, where B is the area of the base and h is the height of the pyramid. A cone's base is a circle, so its area is πr2, and the formula for volume is V=1/3 πr2h. Also, the volume for a frustum is 1/3 πR2H-1/3 πr2h, where R and H are the radius and height of the large cone and r and h are the radius and height of the small cone.

Section 12.6: Volumes of spheres

The volume of a sphere is 4/3 πr3 (pi * radius cubed), and the volume of a hemisphere is half that, or 2/3 πr3.

I hope that you found this helpful and good luck on the final!

-Jacob