- Find the surface area of pyramids and cones.
- A rectangular prism has sides of 5 cm, 6 cm, and 7 cm. What is the surface area?
- A cylinder has a diameter of 10 in and a height of 25 in. What is the surface area?
- A cylinder has a circumference of . and a height of 24 ft. What is the surface area?
- Draw the net of a square pyramid.
Know What? A typical waffle cone is 6 inches tall and has a diameter of 2 inches. What is the surface area of the waffle cone? (You may assume that the cone is straight across at the top)
Parts of a Pyramid
Pyramid: A solid with one base and the lateral faces 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.
Regular Pyramid: 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.
Example 1: Find the slant height of the square pyramid.
Solution: The slant height is the hypotenuse of the right triangle formed by the height and half the base length. Use the Pythagorean Theorem.
Surface Area of a Regular Pyramid
Using the slant height, which is labeled , the area of each triangular face is .
Example 2: Find the surface area of the pyramid from Example 1.
Solution: The four triangular faces are . To find the total surface area, we also need the area of the base, which is . The total surface area is .
From this example, we see that the formula for a square pyramid is:
is the area of the base and is the number of triangles.
Surface Area of a Regular Pyramid: If is the area of the base, then .
The net shows the surface area of a pyramid. If you ever forget the formula, use the net.
Example 3: Find the area of the regular triangular pyramid.
Solution: “Regular” tells us the base is an equilateral triangle. Let’s draw it and find its area.
The surface area is:
Example 4: If the lateral surface area of a square pyramid is and the base edge is equal to the slant height. What is the length of the base edge?
Solution: In the formula for surface area, the lateral surface area is . We know that and . Let’s solve for .
Surface Area of a Cone
Cone: A solid with a circular base and sides taper up towards a vertex.
A cone has a slant height, just like a pyramid.
A cone is generated from rotating a right triangle, around one leg, in a circle.
Surface Area of a Right Cone: .
Area of the base:
Area of the sides:
Example 5: What is the surface area of the cone?
Solution: First, we need to find the slant height. Use the Pythagorean Theorem.
The surface area would be .
Example 6: The surface area of a cone is and the radius is 4 units. What is the slant height?
Solution: Plug in what you know into the formula for the surface area of a cone and solve for .
Know What? Revisited The standard cone has a surface area of .
- Questions 1-10 use the definitions of pyramids and cones.
- Questions 11-19 are similar to Example 1.
- Questions 20-26 are similar to Examples 2, 3, and 5.
- Questions 27-31 are similar to Examples 4 and 6.
- Questions 32-25 are similar to Example 5.
Fill in the blanks about the diagram to the left.
is the ___________.
- The slant height is ________.
is the ___________.
- The height is ________.
- The base is _______.
- The base edge is ________.
Use the cone to fill in the blanks.
is the ___________.
- The height of the cone is ______.
is a __________ and it is the ___________ of the cone.
is the _____________ ____________.
For questions 11-13, 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 right cone with a radius of 5 cm and a height of 15 cm. What is the slant height?
- 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 slant height, , of one lateral face in each pyramid or of the cone. Round your answer to the nearest hundredth.
Find the area of a lateral face of the regular pyramid. Round your answers to the nearest hundredth.
Find the surface area of the regular pyramids and right cones. Round your answers to 2 decimal places.
- A regular tetrahedron has 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 lateral surface area of a cone is and the radius is 5 cm, what is the slant height?
- If the surface area of a cone is and the slant height is 8 cm, what is the radius?
- If the surface area of a square pyramid is and the base edge is 4 ft, what is the slant height?
- If the lateral area of a square pyramid is and the slant height is 16 in, what is the length of the base edge?
- If the lateral area of a regular triangle pyramid is and the base edge is 8 in, what is the slant height?
The traffic cone is cut off at the top and the base is a square with 24 in sides. Round answers to the nearest hundredth.
- Find the area of the entire square. Then, subtract the area of the base of the cone.
- Find the lateral area of the cone portion (include the 4 inch cut off top of the cone).
- Subtract the cut-off top of the cone, to only have the lateral area of the cone portion of the traffic cone.
- Combine your answers from #27 and #30 to find the entire surface area of the traffic cone.
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