# 11.3: Surface Area of Pyramids and Cones

## Learning Objectives

- Find the surface area of pyramids and cones.

## Review Queue

- 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

**meet at a common**

*lateral faces*

*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 .

## Review Questions

- 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
has four equilateral triangles as its faces.*regular tetrahedron*- 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.