Pyramid

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

pyramid


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?

Answer
1560000 square meters

Answer Process
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
clear blinker clears screen
4 × 600 4*600 p = 4b
enter 2400 p
600 square 600² B = b²
enter 360000 B
fraction 1 down 2 right
× 2400 × 1000 + 360000
\bf\displaystyle\frac{1}{2}*2400*1000+360000 SA = \bf\displaystyle\frac{1}{2}ps + B
enter 1560000 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 volume?

Answer
96000000 cubic meters

Answer Process
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
clear blinker clears screen
600 square 600² B = b²
enter 360000 B
fraction 1 down 3 right
× 360000 × 800
\bf\displaystyle\frac{1}{3}*360000*800 V = \bf\displaystyle\frac{1}{3}Bh
enter 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?

Answer
1200000 square meters

Answer Process
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
clear blinker clears screen
4 × 600 4*600 p = 4b
enter 2400 p
fraction 1 down 2 right
× 2400 × 1000
\bf\displaystyle\frac{1}{2}*2400*1000 LA = \bf\displaystyle\frac{1}{2}ps
enter 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?

Answer
500 meters

Answer Process
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
clear blinker clears screen
4 × 300 4*300 p = 4b
enter 1200 p
300 square 300² B = b²
enter 90000 B
fraction 2 ( 309000 – 9000 )
down 1200 right
\bf\displaystyle\frac{2(309000-9000)}{1200} s = \bf\displaystyle\frac{2(SA-B)}{p}
enter 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?

Answer
40 meters

Answer Process
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
clear blinker clears screen
300 square 300² B = b²
enter 90000 B
fraction 3 × 1200000 down 90000 right \bf\displaystyle\frac{3*1200000}{90000} h = \bf\displaystyle\frac{3V}{B}
enter 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?


Practice – Answers
1.  390000 square meters

2.  12000000 cubic meters

3.  300000 square meters

4.  1000 meters

5.  800 meters

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