1. 3-D Shapes Topic 14: Lesson 7
3-D Shapes Topic 14: Lesson 7 Surface Area of Pyramids Holt Geometry Texas ©2007

2. Objectives and Student Expectations
TEKS: G2B, G3B, G6B, G8D, G11D
The student will make conjectures about 3-D figures and determine the validity using a variety of approaches.
The student will construct and justify statements about geometric figures and their properties.
The student will use nets to represent and construct 3-D figures.
The student will find surface area and volume of prisms, cylinders, cones, pyramids, spheres, and composite figures.
The student will describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed.

3. The vertex of a pyramid is the point opposite the base of the pyramid
The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base.

5. Example: 1 Lateral area of a regular pyramid P = 4(14) = 56 cm
Find the lateral area and surface area of a regular square pyramid with base edge length 14 cm and slant height 25 cm. Round to the nearest tenth, if necessary.
Lateral area of a regular pyramid
P = 4(14) = 56 cm
Surface area of a regular pyramid
B = 142 = 196 cm2

6. Example: 2
Find the lateral area and surface area of the regular pyramid.
Step 1 Find the base perimeter and apothem.
The base perimeter is 6(10) = 60 in.
The apothem is , so the base area is
Or also remember that you can find the area of a
regular hexagon by using six equilateral triangles!

7. Example: 2 continued Step 2 Find the lateral area.
Find the lateral area and surface area of the regular pyramid.
Step 2 Find the lateral area.
Lateral area of a regular pyramid
Substitute 60 for P and 16 for ℓ.
Step 3 Find the surface area.
Surface area of a regular pyramid
Substitute for B.

8. Example: 3
Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft.
Step 1 Find the base perimeter and apothem.
The base perimeter is 3(6) = 18 ft.
The apothem is so the base area is
Step 2 Find the lateral area.
Lateral area of a regular pyramid
Substitute 18 for P and 10 for ℓ.

9. Example: 3 continued
Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft.
Step 3 Find the surface area.
Surface area of a regular pyramid
Substitute for B.

10. There are only 5 kinds of regular polyhedrons, which are named the Platonic solids:
A regular tetrahedron has 4 faces that are equilateral triangles.
A cube has 6 faces that are squares.
A regular octahedron has 8 faces that are equilateral triangles.
A regular dodecahedron has 12 faces that are regular pentagons.
A regular icosahedron has 20 faces that are equilateral triangles.

11. Example: 4
Which Platonic solid does this net represent? Draw the 3-dimensional solid and find the surface area.

12. Example: 5
The base edge length and slant height of the regular hexagonal pyramid are both divided by 5. Describe the effect on the surface area.

13. Example: 5 continued base edge length and slant height divided by 5:
original dimensions:
S = Pℓ + B 12
S = Pℓ + B 12

14. Example: 5 continued base edge length and slant height divided by 5:
original dimensions:
Notice that
If the base edge length and slant height are divided by 5, the surface area is divided by 52, or 25.

15. Example: 6
The base edge length and slant height of the regular square pyramid are both multiplied by Describe the effect on the surface area.

16. Example: 6 continued original dimensions: multiplied by two-thirds:
8 ft
10 ft
8 ft
original dimensions:
multiplied by two-thirds:
S = Pℓ + B 12
S = Pℓ + B 12
= 585 cm2
= 260 cm2
By multiplying the dimensions by two-thirds, the surface area was multiplied by

17. Example: 7 Find the surface area of the composite figure.
Surface Area of Cube without the top side:
S = 4bh + B
S = 4(2)(2) + (2)(2) = 20 yd2
Surface Area of Pyramid without base:
Surface Area of Composite:
Surface of Composite = SA of Cube + SA of Pyramid