1. Volume and Surface Area
Geometry Day 80
Volume and Surface Area

2. Review of Terms
Polyhedron – Closed three-dimensional solid made of flat polygonal regions called faces. Pairs of faces intersect in line segments called edges; points where three or more edges intersect are vertices.
Prism – polyhedron with the following characteristics:
Two faces (called bases) are made of congruent, parallel polgons.
The other faces (called lateral faces) are formed by parallelograms.
The intersection of two adjacent lateral faces (called lateral edges) are congruent, parallel segments. The intersection of a lateral face with a base is a base edge.

3. Review of Terms
Cylinder – A 3D solid with bases that are congruent, parallel circles. The axis of a cylinder is the segment that connects the centers of their bases.
Pyramid – A polyhedron with a polygonal base, and the lateral faces are triangles that meet at a single point (the vertex).
Cone – A solid with a circular base, a vertex not in the same plane as the base, and a curved lateral area composed of all the points in the segments that connect the vertex to the base edge.

4. Area of the Base
To find the surface area or volume of any figure, you usually need to first find the area of that figure’s base. The area of the base is represented by B.

5. Right vs. Oblique
The altitude of a figure is the perpendicular segment that joins the two bases (prisms, cylinders) or the base to the opposite vertex (pyramid, cone).
A right prism or cylinder is one where the sides are perpendicular to the bases. In other words, the altitude is a lateral edge (prism), or the axis (cylinder).
An oblique prism or cylinder has slanted sides.
Note: unless told otherwise, figures are assumed to be right.

6. Surface Area and Lateral Area
There are various formulas for finding surface area, but all you need to remember is this:
Surface area is found by calculating the area of each face, and adding them together. The surface area of a figure is the area of the net of that figure.
Lateral area is like the above, only you don’t count the base(s).

7. Surface Area and Lateral Area
The lateral area of a right prism is the perimeter of the base (P) times the height (h).
L = Ph
It’s the same for a right cylinder, except since the base is always a circle, we use the formula for circumference:
L = 2πrh
The surface area adds in the area of each base (B).
Prism: SA = Ph + 2B
Cylinder: SA = 2πrh + 2πr2

8. Surface Area – Cylinders
If you were to ‘unwrap’ a cylinder, what would you get?
Two circles & a rectangle.

9. Surface Area – Cylinders
So we have to add the areas of the circles (each πr2) and the rectangle (bh).
The base of the rectangle is the same as the circumference of the circle.
Doing a little substitution:

10. Regular Pyramids
A regular pyramid has a regular polygon for a base, and the lateral faces are congruent isosceles triangles. The altitude has one end at the center of the base and the other at the vertex.
The slant height (l) of the pyramid is the height of each lateral face.
L = ½Pl
SA = ½Pl + B

11. Slant height – Cones
The distance from the top of a cone to its base, moving along the surface, is called the slant height.
Note: you may see slant height represented as s or l, depending on the text.

12. Pyramids and Cones L = πrl SA = πrl + πr2
Pyramids and cones can be right or oblique.
The lateral area of a right cone is a sector of a circle with a radius of the slant height (l) of the cone.
L = πrl
SA = πrl + πr2

13. Volume
For a prism or cylinder, the volume is the area of the base (B) times the height (h) of the figure.
V = Bh
Prism Cylinder

14. Cylinders
The base of a cylinder is a circle. To find the volume of the cylinder, find the area of the circle and multiply it by the height of the cylinder.
V = πr2h

15. Cavalleri’s Principle
If two solids have the same height (h) and the same cross-section area (B) at every level, then they have the same volume.

16. Pyramids and Cones
The volume of pyramids and cones are found by the following formula:
Pyramid Cone
V = ⅓Bh V = ⅓πr2h
The volumes of pyramids and cones are one-third of the volumes of prisms and cylinders with the same dimensions.

17. Spheres
A sphere is the locus of all points in space that are the same distance away from a center point.
A radius (r) connects the center to any point on the sphere.
A chord connects any two points on the sphere.
A diameter is a chord that contains the center.
A tangent to the sphere is a line that intersects it in only one point.
A cross section of the sphere that contains its center is a great circle. It is the largest circle that can be drawn on a sphere, and separates it into two hemispheres.

18. Spheres – Volume and Surface Area

19. Note
The formulas given for lateral and surface area are dependent on the figures being right prisms, cylinders, cones, and in the case of pyramids, regular.
If the figure is oblique, or a non-regular pyramid, then you need to find the area of each face separately, and add them together.
The volume formulas work whether the figure is right or oblique, thanks to Cavalleri’s Principle.

20. Assignments Homework 49 Homework 50 Workbook, pp. 152, 154