Volume and Surface Area of Pyramids - Geometry in Three Dimensions (Pre-Algebra)

Begin by considering the given diagram of the trellises.

Each trellis is shaped like a pyramid and has a surface area of

117 + 93

square meters. The base of the trellis is an equilateral triangle with sides of

6

meters each. The surface area

S A

of a pyramid can be found using the following formula.

S A = \frac{1}{2} p ℓ + B

In this formula,

B

is the base area,

p

is the base perimeter, and

ℓ

is the slant height of the pyramid. The perimeter and area of the base must be found first to calculate the slant height of the trellis.

Perimeter of the Base

Since the base is an equilateral triangle with a side length of

6

meters, multiply this side length by

3

to get the perimeter of the base.

Perimeter of the Base 6 \cdot 3 = 18 meters

Area of the Base

The area of a triangle is calculated as half the product of its base and height.

A = \frac{1}{2} b h

This means that the height of the triangle must be determined first. The height of an equilateral triangle is a perpendicular segment that extends from one vertex to the opposite side, effectively bisecting that side.

When drawing the height of this triangle, two right triangles are formed. Each of these right triangles has a base of

3

meters, which is half the length of the equilateral triangle's side. Substituting

c = 6

and

b = 3

into the Pythagorean Theorem, the missing side

a

of these right triangles can be determined.

c^{2} = a^{2} + b^{2}

SubstituteII

$c = 6$ , $b = 3$

6^{2} = a^{2} + 3^{2}

▼

Solve for

a

CalcPow

Calculate power

36 = a^{2} + 9

SubEqn

$LHS - 9 = RHS - 9$

27 = a^{2}

RearrangeEqn

Rearrange equation

a^{2} = 27

SqrtEqn

$LHS = RHS$

a = \pm 27

SplitIntoFactors

Split into factors

a = \pm 9 \cdot 3

SqrtProd

$a \cdot b = a \cdot b$

a = \pm 9 \cdot 3

CalcRoot

Calculate root

a = \pm 33

$a > 0$

a = 33

The value of

a

corresponds to the height of the equilateral triangle. Then, height

h

of the equilateral triangle is

33

meters. Next, substitute the base and height of the equilateral triangle into the formula for the area of a triangle to determine its area.

A = \frac{1}{2} b \cdot h

SubstituteII

$b = 6$ , $h = 33$

A = \frac{1}{2} \cdot 6 \cdot 33

Multiply

A = \frac{1}{2} \cdot 183

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

A = \frac{1 8 3}{2}

CalcQuot

Calculate quotient

A = 93

The area of this triangle is

93

square meters, which corresponds to the area of the base

B

of the pyramid.

Calculating the Slant Height

The base area, the base perimeter, and surface area can now be substituted into the formula for the surface area of a pyramid. Then, the obtained equation can be solved for

ℓ

to find the slant height of the trellises.

S A = \frac{1}{2} p \cdot ℓ + B

SubstituteValues

Substitute values

117 + 93 = \frac{1}{2} \cdot 18 \cdot ℓ + 93

SubEqn

$LHS - 93 = RHS - 93$

117 = \frac{1}{2} \cdot 18 ℓ

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

117 = \frac{1 8 ℓ}{2}

CalcQuot

Calculate quotient

117 = 9 ℓ

RearrangeEqn

Rearrange equation

9 ℓ = 117

DivEqn

$LHS / 9 = RHS / 9$

ℓ = 13

The slant height measures

13

meters. This indicates that LaShay and Vincenzo must choose door number

13

to access their final challenge — the Celestial Observatory.

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Catch-Up and Review

The Potion Challenge

Pyramid

Is It a Pyramid?

Volume of a Pyramid

The Secret Chamber's Pyramid

Hint

Solution

The Puzzle of the Shifting Walls

Hint

Solution

Surface Area of a Pyramid

The Secret Code

Hint

Solution

The Enchanted Garden Challenge

Hint

Solution

Perimeter of the Base

Area of the Base

Calculating the Slant Height

Practice Finding the Surface Area and Volume of Pyramids

The Celestial Observatory

Hint

Solution

Side Length of the Base

Perimeter of the Base

Slant Height of the Pyramid

Height of the Pyramid

Volume of the Pyramid

The Apex of Discovery

Hint

Solution

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