## Pyramid

##### Pyramid = Three-Dimensional Shape with Polygonal Base Connecting to a Point Basics

h = height
b = base = distance along a side of pyramid’s base
s = slant height= distance along pyramid’s side
Apex = point or peak to which a pyramid connects
Base = foundation of pyramid
p = perimeter of base
B = area of base
SA = Surface Area = area covering surface, including base, of pyramid (measured in square units)
LA = Lateral Area = area covering surface, excluding base, of pyramid (measured in square units)
V = Volume = space occupied by pyramid (measured in cubic units)

The pyramid pictured above, with triangular sides and a square base, is the type of pyramid probably featured most frequently on the GED.  However, pyramids with bases of other polygonal shapes (triangle, pentagon, etc.) are possible.

Formulas (provided by Mathematics Formula Sheet)

SA = $\bf\displaystyle\frac{1}{2}$ps + B

V = $\bf\displaystyle\frac{1}{3}$Bh

Formulas (NOT provided by Mathematics Formula Sheet)

p = 4b (square Base)
B = b² (square Base)

LA = $\bf\displaystyle\frac{1}{2}$ps

h = $\bf\displaystyle\frac{3V}{B}$

s = $\bf\displaystyle\frac{2(SA-B)}{p}$
Parentheses protect order of operations.

Question
An ancient, square-based pyramid has a base of 600 meters, height 800 meters, and slant height 1000 meters.
What is its surface area?

1560000 square meters

p = 4b
p = 4 × 600 = 2400

B = b²
B = 600² = 360000

SA = $\bf\displaystyle\frac{1}{2}$ps + B

SA = $\bf\displaystyle\frac{1}{2}$*2400*1000+360000 = 1560000

Calculator Click What You See Comment blinker clears screen
4 × 600 4*600 p = 4b 2400 p
600 600² B = b²
enter 360000 B 1 2 × 2400 × 1000 + 360000 $\bf\displaystyle\frac{1}{2}$*2400*1000+360000 SA = $\bf\displaystyle\frac{1}{2}$ps + B
###### Pyramid

Question
An ancient, square-based pyramid has a base of 600 meters, height 800 meters, and slant height 1000 meters.  What is its volume?

96000000 cubic meters

B = b²
B = 600² = 360000

V = $\bf\displaystyle\frac{1}{3}$Bh

V = $\bf\displaystyle\frac{1}{3}$*360000*800 = 96000000

Calculator Click What You See Comment blinker clears screen
600 600² B = b² 360000 B 1 3 × 360000 × 800 $\bf\displaystyle\frac{1}{3}$*360000*800 V = $\bf\displaystyle\frac{1}{3}$Bh 96000000 Answer
###### Pyramid

Question
An ancient, square-based pyramid has a base of 600 meters, height 800 meters, and slant height 1000 meters.
What is its lateral area?

1200000 square meters

p = 4b
p = 4 × 600 = 2400

LA = $\bf\displaystyle\frac{1}{2}$ps

LA = $\bf\displaystyle\frac{1}{2}$*2400*1000 = 1200000

Calculator Click What You See Comment blinker clears screen
4 × 600 4*600 p = 4b 2400 p 1 2 × 2400 × 1000 $\bf\displaystyle\frac{1}{2}$*2400*1000 LA = $\bf\displaystyle\frac{1}{2}$ps 1200000 Answer
###### Pyramid

Question
An ancient, square-based pyramid has a base of 300 meters and a surface area of 309000 square meters.  What is its slant height?

500 meters

p = 4b
p = 4 × 300 = 1200

B = b²
B = 300² = 90000

s = $\bf\displaystyle\frac{2(SA-B)}{p}$
s = $\bf\displaystyle\frac{2(309000-9000)}{1200}$ = 500

Calculator Click What You See Comment blinker clears screen
4 × 300 4*300 p = 4b 1200 p
300 300² B = b² 90000 B 2 309000 – 9000  1200  $\bf\displaystyle\frac{2(309000-9000)}{1200}$ s = $\bf\displaystyle\frac{2(SA-B)}{p}$ 500 Answer
###### Pyramid

Question
An ancient, square-based pyramid has a base of 300 meters and a volume of 1200000 cubic meters.  What is its height?

40 meters

B = b²
B = 300² = 90000

h = $\bf\displaystyle\frac{3V}{B}$

h = $\bf\displaystyle\frac{3*1200000}{90000}$ = 40

Calculator Click What You See Comment blinker clears screen
300 300² B = b² 90000 B 3 × 1200000 90000  $\bf\displaystyle\frac{3*1200000}{90000}$ h = $\bf\displaystyle\frac{3V}{B}$ 40 Answer
###### Pyramid

Practice – Questions
1.  An ancient, square-based pyramid has a base of 300 meters, height 400 meters, and slant height 500 meters.  What is its surface area?

2.  An ancient, square-based pyramid has a base of 300 meters, height 400 meters, and slant height 500 meters.  What is its volume?

3.  An ancient, square-based pyramid has a base of 300 meters, height 400 meters, and slant height 500 meters.  What is its lateral area?

4.  An ancient, square-based pyramid has a base of 600 meters and a surface area of 1560000 square meters.  What is its slant height?

5.  An ancient, square-based pyramid has a base of 600 meters and a volume of 96000000 cubic meters.  What is its height?