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Frustum of a Regular Pyramid Formula

The frustum is a pyramid that is the result of chopping off the top of a regular pyramid. That is the reason why it is called a truncated pyramid.

The distance between the base and the top of the pyramid is the height and is denoted by h. Similarly, it has a slant height that is denoted by “s” and two bases (top and bottom), whose area is defined by B1 and B2

.We need to find the lateral surface area and the volume of Frustum of regular pyramid formula.

Frustum of a Regular Pyramid

Frustum of a regular pyramid is a portion of right regular pyramid included between the base and a section parallel to the base.

Properties of a Frustum of Regular Pyramid

The slant height of a frustum of a regular pyramid is the altitude of the face.
The lateral edges of a frustum of a regular pyramid are equal, and the faces are equal isosceles trapezoids.
The bases of a frustum of a regular pyramid are similar regular polygons. If these polygons become equal, the frustum will become prism.

Elements of a Frustum of Regular Pyramid
a = upper base edge
b = lower base edge
e = lateral edge
h = altitude
L = slant height
A₁ = area of lower base
A₂ = area of upper base
n = number of lower base edges

Formulas for Frustum of a Regular PyramidArea of Bases, A₁ and A₂
See the formulas of regular polygon for the formula of A₁ and A₂

Volume

Frustum of a Pyramid

If a right pyramid is cut by plane parallel to the base then the portion of the pyramid between the plane and the base of the pyramid is called a frustum of the pyramid.

Let the square WXYZ be the base and P, the vertex of a right pyramid.

If a plane parallel to the base WXYZ of the pyramid cuts it in the plane W’X’Y’Z’ then the portion of pyramid between the planes WXYZ and W’X’Y’Z’ will be a frustum of the given pyramid. The perpendicular distance between this two planes is the height of the frustum. Clearly side-face (viz. WXX’W’, XYY’X’ ect.) are trapeziums; the distance between the parallel sides of this trapeziums is the slant height of the frustum of the pyramid.

Let S₁ and S₂ be the areas of the lower and upper planes respectively of the frustum of a pyramid; if h and l be the height and slant height respectively of the frustum, then

(A) Area of the slant faces of the frustum

= ½ × (perimeter of the lower face + perimeter of the upper face) × l.

(B) Area of whole surface of the frustum

= Area of the slant faces + S₁ + S₂;

Worked-out problems on Frustum of a Pyramid:

A monument has the shape of a frustum of a right pyramid whose lower and upper plane faces are squares of sides 16 meter and 9 meter respectively. If the height of the monument is 21 meter, find its volume.

Solution:

Clearly, the area of the lower faces of the monument = S₁ = (16)² square meter = 256 square meter and the area of the upper face of the monument = S₂ = 9² square meter = 81 square meter.

Therefore, the volume of the monument

= the volume of the frustum of a right pyramid

= 1/3 × (S₁ + S₂ + √S₁S₂) × height of the frustum

= 1/3 × [256 + 81 + √{(16)² × 9²} × 21 cubic meter.

= 1/3 × (256 + 81 + 144) × 21 cubic meter.

= 1/3 × 481 × 21 cubic meter.

= 7 × 481 cubic meter.

= 3367 cubic meter.

Finding the Surface Area and Volume of Frustums of a Pyramid and Cone

Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.

What Is a Frustum of a Right Circular Cone?

The frustum of a right circular cone is a portion of the cone enclosed by its base, a section that is parallel to the base, and the conical surface included between the base of the cone and the parallel section. A frustum of a circular cone has different parts. Listed below are the different elements of a frustum of a right circular cone.

1. Lower Base – a base of a frustum of a right circular cone with a larger radius

2. Upper Base – a base of a frustum of a right circular cone with a smaller radius

3. Altitude – the perpendicular distance between the bases of a frustum of a cone

4. Slant Height – the distance measured along the lateral face of the frustum

5. Radii -the distance from the center of a base to its edge

To determine the slant height (l) from the given radii of the bases and the altitude of the frustum, consider the projected trapezoid of the frustum to a plane. Using the Pythagorean Theorem, the equation will be:

l = √[h² + (r₂ – r₁)²]

Surface Area and Volume of a Frustum of a Right Circular Cone

The area of a conical surface of the frustum is the lateral surface area of the frustum. It is equal to one-half of the sum of the circumferences of the bases multiplied by the slant height. The variables C₁ and C₂ are the circumferences of the bases of the frustum, and l is the slant height. Since C₁ = 2πr₂ and C₂ = 2πr₂, replace C by 2πr. For the total surface area, of a frustum of a right circular cone is given by the sum of the lateral surface area and area of the two bases.

LSA = 1/2 (C₁ + C₂) (l)

LSA = π (r₁ + r₂) (l)

TSA = π (r₁ + r₂) (l) + πr₁² + πr₂²

The volume of a frustum of a circular cone is equal to one-third of the sum of the two base areas and the square root of the two base areas, multiplied by the altitude. Since the two bases of a frustum of a cone are circles, you can substitute πr² to the variable B resulting in a more specific equation of the volume.

V = (1/3) (h) (B₁ + B₂ + √B₁B₂)

V = (1/3) (h) [πr₁² + πr₂² + √(πr₁²*πr₂²)]

V = (π/3) (h) (r₁² + r₂² + r₁r₂)

What Is a Frustum of a Regular Pyramid?

The frustum of a pyramid is a portion of a pyramid obtained by passing a plane parallel to the base, intersecting all the lateral edges. A frustum of a pyramid is a polyhedron enclosed by the lateral surface and the base of the pyramid. If the pyramid is a regular right pyramid, the frustum has congruent isosceles trapezoids as lateral faces. A frustum of a regular pyramid has different parts. Listed below are the different elements of a frustum of a regular pyramid.

1. Lower Base – a base of a frustum of a regular pyramid with a larger area

2. Upper Base – a base of a frustum of a regular pyramid with a smaller area

3. Altitude – the perpendicular distance between the bases of a frustum of a regular pyramid

4. Slant Height – the distance measured along the lateral face of the frustum

Surface Area and Volume of a Frustum of a Regular Pyramid

The sum of the areas of the lateral faces of a frustum of a regular pyramid is the lateral surface area. Since each lateral face is an isosceles trapezoid, then the area of each lateral face is one-half of the sum of the two edge bases multiplied by the slant height. In the equation below, b1 and b2 are the edges of the upper base and lower base of a frustum of a regular pyramid. Since all the lateral faces are congruent, the lateral surface area of the frustum of a regular pyramid is equal to the number of sides of the base multiplied by the area of one trapezoid. In the equation below, P1 and P2 are the perimeters of the bases of the frustum. Lastly, the total surface area of the frustum is the sum of the lateral area and the areas of the two bases. B1 and B2 are the areas of the bases of the frustum.

A = 1/2 (b₁ + b₂) (l)

LSA = 1/2 (P₁ + P₂) (l)

TSA = LSA + B₁ + B₂

The volume of a frustum of a regular pyramid is equal to one-third of the altitude multiplied by the sum of its bases and the geometric mean between them.

V = 1/3 (h) (B₁ + B₂ + √(B₁B₂)

Problem 1: Surface Area and Volume of Frustum of a Right Circular Cone

In a frustum of a right circular cone, the diameter of the lower base is 24 feet, while the diameter of the upper base is 14 feet. If the slant height of the frustum is 13 feet, find the total area and the volume of the frustum.

Solution

a. Given the diameter of the upper base d₁ = 14 feet and diameter of the lower base d₂ = 24 feet, compute for the two base areas using the formula for an area of a circle.

r₁ = 14 / 2

r₁ = 7 feet

r₂ = 24 / 2

r₂ = 12 feet

B₁ = π (7)²

B₁ = 49π ft²

B₂ = π (12)²

B₂ = 144π ft²

b. Solve for the height of the frustum of the right circular cone using the Pythagorean theorem.

h = √(c² – a²)

h = √(13² – 5²)

h = 12 feet

c. Compute the total surface area using the given equation.

TSA = B₁ + B₂ + π (r₁ + r₂) (l)

TSA = 49π + 144π + π (7 + 12) (13)

TSA = 1382 ft²

d. Compute the total volume of the frustum of the right circular cone given above.

V = (1/3) (h) (B₁ + B₂ + √B₁B₂)

V = (1/3) (12) (49π + 144π + √(49π)(144π)

V = 1108π ft³

V = 3481 ft³

Final Answer: The total surface area and volume of the frustum of the right circular cone given above are 1382 ft³ and 3481 ft³, respectively.

Problem 2: Surface Area and Volume of Frustum of a Right Circular Cone

In a given frustum of a right circular cone, the radius of the lower base is 30 feet, while the radius of the upper base is 15 feet. If the altitude of the frustum is 20 feet, find the total surface area and volume of the frustum.

Solution

a. Given the radius of the upper base r₁ = 15 feet and radius of the lower base r₂ = 30 feet, compute for the two base areas using the formula for an area of a circle.

r₁ = 15 feet

r₂ = 30 feet

B₁ = π (15)²

B₁ = 225π ft²

B₂ = π (30)²

B₂ = 900π ft²

b. Solve for the slant height of the frustum of the right circular cone using the Pythagorean theorem.

l = √(b² + a²)

l = √(15² + 20²)

l = 25 feet

c. Compute the total surface area using the given equation.

TSA = B₁ + B₂ + π (r₁ + r₂) (l)

TSA = 225π + 900π + π (15 + 30) (25)

TSA = 7068.58 ft²

d. Compute the total volume of the frustum of the right circular cone given above.

V = (1/3) (h) (B₁ + B₂ + √B₁B₂)

V = (1/3) (20) (225π + 900π + √(225π)(900π)

V = 10500π ft³

V = 32986.72 ft³

Final Answer: The total surface area and volume of the frustum of the right circular cone given above are 7068.58 ft² and 32986.72 ft³, respectively.

Problem 3: Surface Area and Volume of Frustum of a Regular Pyramid

Find the volume and the total surface area of a frustum of a regular hexagonal pyramid with base edges of 8 centimeters and 10 centimeters, and an altitude of 10 centimeters.

Solution

a. Let l be the slant height of the given frustum, a₁ the apothem of the upper base, and a₂ the apothem of the lower base. Using the formula for finding the measure of the apothem given below, solve for a₁ and a₂.

a = (s/2) / (tan[180°/n])

a₁ = (8/2) / (tan[180°/6])

a₁ = 4√3 centimeters

a₂ = (10/2) / (tan[180°/6])

a₂ = 5√3 centimeters

b. In the trapezoidal figure shown above, solve for the value of the slant height.

l =√ (12)² + (5√3 – 4√3)²

l = 7√3 centimeters

c. Compute for the total surface area by finding the lateral surface area first.

LSA = (1/2) (P₁ + P₂) (l)

LSA = (1/2) (6(8) + 6(10)) (7√3)

LSA = 654.72 cm²

d. Compute for the area of the two bases using the formula in finding the area of a regular polygon.

B_area = (ns²) / [4tan (180°/n)]

B₁ = (6(8)²) / [4tan (180°/6)]

B₁ = 96√3 cm²

B₂ = (6(10)²) / [4tan (180°/6)

B₂ = 259.81 cm²

e. Solve for the total surface area using the lateral surface area and area of the two bases obtained in previous steps.

TSA = B₁ + B₂ + LSA

TSA = 96√3 + 259.81 + 654.72

TSA = 1080.81 cm²

f. Compute for the volume of the frustum of the regular pyramid given above.

V = (1/3) (h) (B₁ + B₂ + √B₁B₂)

V = (1/3) (12) (96√3 + 259.81 + √(96√3)(259.81)

V = 2535.74 cm³

Final Answer: The total surface area and volume of the frustum of the regular pyramid given above are 1080.81 cm² and 2535.74 cm³, respectively.

Problem 4: Altitude of Frustum of a Regular Pyramid

Find the altitude of a frustum of a pyramid whose volume is 148 ft³.The base areas of the frustum are 9 ft³ and 36 ft³.

Solution

a. In finding the altitude (h) of the given frustum of a pyramid, substitute the given values into the equation. Let V = 148 ft³, B₁ = 9 ft², and B₂ = 36 ft².

V = (1/3) (h) (B₁ + B₂ + √B₁B₂)

148 = (1/3) (h) (9 + 36 + √(9)(36)

h = 7.05 feet

Final Answer: The altitude of the given frustum of a regular pyramid is 7.05 feet.

Problem 5: Lateral Area of Frustum of a Right Circular Cone

The volume of a right circular cone is 52π ft³. Its altitude is 3 feet and the measure of its lower radius is three times the measure of its upper radius. Find the lateral area of the frustum.

Solution

a. Express the area of the two bases in terms of the variable x.
B₁ = πr₁²
B₁ = π(x)²
B₂ = πr₂²

B₂ = π(3x)²

B₂ = π(9x²)

b. Given the volume of the right circular cone, compute for the value of x.

V = (1/3) (h) (B₁ + B₂ + √B₁B₂)

52π = (π/3) (3) (x² + 9x² + √(x²)(9x²)

x = 2 feet

r₁ = 2 feet

r₂ = 6 feet

c. Using Pythagorean theorem, compute for the slant height given the radius of the two bases and the altitude.

l = √[h² + (r₂ – r₁)²]

l = √[3² + (6 – 2)²]

l = 5 feet

d. Solve for the lateral area using the given formula below.

LSA = π (r₁ + r₂) (l)

LSA = π (2 + 6) (5)

LSA = 40π ft²

Final Answer: The lateral surface area of the given frustum of a right circular cone is 40π ft².

Questions & Answers

The frustum of a regular triangular pyramid has equilateral triangle for its bases. The lower and upper base edges are 9 m and 3 m respectively. If the volume is 118.2 cu. m, how far apart (m) are the bases?
(a) 9
(b) 8
(c) 7
(d) 10