A pyramid is a solid with one base and lateral faces that meet at a common vertex. The edges between the lateral faces are lateral edges. The edges between the base and the lateral faces are base edges.
A regular pyramid is a pyramid where the base is a regular polygon. All regular pyramids also have a slant height, which is the height of a lateral face. A non-regular pyramid does not have a slant height.
The net shows the surface area of a pyramid. If you ever forget the formula, use the net.
To find the volume of any solid you must figure out how much space it occupies. The basic unit of volume is the cubic unit.
What if you were given a solid three-dimensional figure with one base and lateral faces that meet at a common vertex? How could you determine how much two-dimensional and three-dimensional space that figure occupies?
Find the slant height of the square pyramid.
The slant height is the hypotenuse of the right triangle formed by the height and half the base length. Use the Pythagorean Theorem.
Find the surface area of the regular triangular pyramid.
“Regular” tells us the base is an equilateral triangle. Let’s draw it and find its area.
The surface area is:
Find the height and then volume of the pyramid.
In this example, we are given the slant height. Use the Pythagorean Theorem.
Find the volume of the pyramid with a right triangle as its base.
Use the formula for volume and plug in the information we were given. Then solve for the height.
Fill in the blanks about the diagram to the left.
xis the ___________.
- The slant height is ________.
yis the ___________.
- The height is ________.
- The base is _______.
- The base edge is ________.
For questions 7-8, sketch each of the following solids and answer the question. Your drawings should be to scale, but not one-to-one. Leave your answer in simplest radical form.
- Draw a square pyramid with an edge length of 9 in and a 12 in height. Find the slant height.
- Draw an equilateral triangle pyramid with an edge length of 6 cm and a height of 6 cm. What is the height of the base?
Find the surface area and volume of the regular pyramid. Round your answers to the nearest hundredth.
- A regular tetrahedronhas four equilateral triangles as its faces.
- Find the height of one of the faces if the edge length is 6 units.
- Find the area of one face.
- Find the total surface area of the regular tetrahedron.
- If the surface area of a square pyramid is
40 ft2and the base edge is 4 ft, what is the slant height?
- If the lateral area of a square pyramid is
800 in2and the slant height is 16 in, what is the length of the base edge?
- If the lateral area of a regular triangle pyramid is
252 in2and the base edge is 8 in, what is the slant height?
- The volume of a square pyramid is 72 square inches and the base edge is 4 inches. What is the height?
- The volume of a triangle pyramid is
170 in3and the base area is 34 in2. What is the height of the pyramid?
To see the Review answers, open this PDF file and look for section 11.5.