Surface Areas & Volumes of Solids
Altitude: of a prism or cylinder--the segment to both bases with endpoints in each base.
of a pyramid or cone--the segment from the vertex to the base.
Apothem (a): the perpendicular distance from the center of a polygon to one of the sides.
Axis: of a cylinder--the segment whose endpoints are centers of the bases.
of a cone--the segment whose endpoints are the vertex and the center of the base.
Base: of a prism or cylinder--the two congruent, parallel faces.
of a pyramid--the face that does not meet at the vertex.
of a cone--the circle.
Base area (B): area of one base (see next section for formulas)
Base length (b): length of one side of a polygon.
Circumference (C): distance around the circular base.
Diagonal (d): length of a diagonal in a polygon.
Edge: line segments where faces of a polyhedron intersect.
Face: any flat surface of a polyhedron.
Height (h): of a prism or cylinder-- the distance between the bases (the length of the altitude).
of a polygon--the length of the altitude.
Lateral area (LA): area of all lateral faces (does not include the bases).
# of sides (n): the number of sides of a regular polygon.
Perimeter (P): distance around the base.
Polyhedron: solid with all flat surfaces that enclose a single region of space.
Side length (s): the length of a side of a regular polygon.
Slant height (l): of a pyramid--the height of a lateral face.
of a right cone--the measure of any segment joining the vertex to the edge of the base.
Surface area (SA): area of all surfaces.
Volume (V): the measure of the amount of space that a figure encloses.
Area formulas: Use one formula to find the area of a base of a polyhedron.
Parallelograms: A = bh
1
2
Triangles: A =
bh
1
Rhombi: A = 2 d1d2
Trapezoids:
1
1
Regular Polygons: A = 2 san or A = 2 Pa
Circles: A = r 2
Prism: a solid with the following characteristics
1. Two faces, called bases, are formed by congruent polygons that lie in parallel planes.
2. The faces that are not bases, called lateral faces, are formed by parallelograms.
3. The intersections of two adjacent lateral faces are called lateral edges and are parallel segments.
Regular prism: bases are regular polygons.
Right prism: lateral edges are also altitudes (bases
Oblique prism: not a right prism.
Cube: prism in which all faces are squares.
For right prisms:
faces).
LA = Ph
SA = LA + 2B or SA = Ph + 2B
V = Bh
h
Cylinder: a solid with bases formed by congruent circles in parallel planes. It’s not a polyhedron.
Right cylinder: the axis is also an altitude.
Oblique cylinder: not a right cylinder.
axis
For right cylinders:
h
LA = Ch or LA = 2 rh
SA = LA + 2B or SA = 2 rh + 2 r 2
V = Bh or V = r 2h
r
Pyramid: a solid with the following characteristics:
1. All the faces except one intersect at a point called the vertex.
2. The base is the face that does not intersect the other faces at the vertex. The base is always a polygon.
3. The faces that intersect at the vertex are called lateral faces and form triangles. The edges of the lateral
faces that have the vertex as an endpoint are called lateral edges.
vertex
4. The altitude is the segment from the vertex perpendicular to the base.
h
l
Regular pyramid:
1. The base is a regular polygon.
2. The altitude’s endpoints are the vertex and the center of the base.
3. All lateral faces are congruent triangles.
The height of each lateral face is called the slant height, l.
For regular pyramids:
LA =
1
2
Pl
SA = LA + B or SA =
V=
1
3
1
2
Pl + B
Bh
Cone: a circular base with a vertex.
For a right cone: LA =
1
2
vertex
l
h
Cl or L = rl
SA = LA + B or SA = rl + r 2
1
V = 3 Bh
r
Sphere: set of points in space that are a given distance from a given point, the center.
Objects related to spheres:
radius: a segment whose endpoints are the center of the sphere and a point on the sphere.
chord: a segment whose endpoints are points on the sphere.
diameter: a chord that contains the sphere’s center.
tangent: a line that intersects the sphere in exactly one point.
great circle: the intersection of a sphere and a plane that contains the center of the sphere.
hemisphere: two congruent halves of a sphere separated by a great circle.
For a sphere:
SA = 4 r 2
4
V = 3 r3
r