Solid Mensuration

Solid Mensuration Solid Geometry – Plane Surface and Curved Surface

Edgar Kenneth C. Luna

Pyramids

•Volume of a Pyramid 1 𝑉 = 𝐵ℎ 3

h

B = base

Example •A pyramid has a pentagonal base having one of its sides equal to 6 cm. If the altitude is 12 cm. (a)Compute the volume of the pyramid. (b)Compute the lateral area of the pyramid. (c)Compute the total surface area of the pyramid.

Frustum of a Pyramid

•Volume of Frustum of a Pyramid. 1 𝑉 = ℎ(𝑏 + 𝐵𝑏 + 𝐵) 3

b h B

Example • The volume of a frustum of a pyramid is 140 cu. m. It has a rectangular upper base of 3 m × 4 m and altitude of 5 m. (a)Determine the dimensions of the lower base. (b)Determine the lateral area of the frustum of a pyramid. (c)Determine the total surface area of the frustum of a pyramid.

Cone • Volume of Cone. 1 3 𝑉 = 𝜋𝑟 ℎ 3 • Lateral area of Cone. 𝐿. 𝐴. = 𝜋𝑟𝐿 Where: 2 2 2 𝐿 =𝑟 +ℎ

L

r

h

Example • A closed conical tank has a diameter of 2 m at the top and height of 6 m. It contains water at a depth of 4 m. (a) What is the volume of the water inside the conical tank. (b) What is the weight of the water in quintals if it has a mass density of 1000 kg/m3. (c) If the conical tank is inverted such that the base will be at the bottom, determine the depth of water at this point.

Frustum of a Cone

r

• Volume of Frustum of a circular Cone. 1 2 2 𝑉 = 𝜋ℎ(𝑟 + 𝑅 + 𝑟𝑅) 3 • Lateral area of a Frustum of a Cone. 𝐿. 𝐴. = 𝜋𝐿(𝑅 + 𝑟)

L h

R

Example • The volume of a frustum of a cone is 1176π m3. If the radius of the lower base is 10 m and the altitude is 18 m. (a)Compute the radius of the upper base. (b)Determine the lateral area of the frustum of a cone. (c)Compute the total surface area of the frustum of the cone.

Prism

•Volume of Prisms and Cylinders. 𝑉 = 𝐵ℎ

h

h

B

B

Prism • Truncated Prism. 𝑉 = 𝐵(𝑚𝑒𝑎𝑛 ℎ𝑒𝑖𝑔ℎ𝑡) • Truncated Cylinder. ℎ + ℎ 1 2 h 2 1 𝑉 = 𝜋𝑟 ( ) 2 • Truncated Rectagular Prism. ℎ1 + ℎ2 +ℎ3 +ℎ4 𝑉 = 𝐵( ) 4

h2

h1

h4 B

B

h3

h2

Prismatoid

A2

• Volume of Prismatoid. 1 𝑉 = ℎ(𝐴1 + 4𝐴𝑚 + 𝐴2 ) 6 Am = Area of the midsection A1 = Cross sectional area at section 1 A2 = Cross sectional area at section 2

h/2

Am

h

A1

h/2

Example • A cylindrical tank 12 m long has a diameter of 4 m. It is placed at an angle of 18° 24’ with the horizontal longitudinally. The tank is filled up with gasoline. (a) If the filler opening is found at the top of the center of the tank, determine the maximum capacity of the tank. (b) If the tank is placed horizontally, determine the depth of the gasoline in the tank. (c) If the tank is placed vertically, determine the depth of the gasoline in the tank.

Cube • Volume of Cube. 𝑉 = 𝑎3 • Surface Area. 𝑆. 𝐴. = 6𝑎2 • Radius of Sphere Circumsribing a Cube. 3 𝑅= 𝑎 2

Example • A cubical box has one of its edge equal to 4 cm. (a) Find the radius of the sphere that encloses the cubical box so that the corners of the box touches the sphere. (b) What is the volume between the cubical box and the sphere? (c) Pass a plane to the cube so that the section form would be a regular hexagon whose vertices are the mid point of the sides of the cube. Find the area of the hexagon.

5 Regular Polyhedrons • Tetrahedron Faces

Triangle

No. of faces

4

No. of edges

6

No. of vertices

4

Polygon angle

60°

Sum of angles

180°

Radius of circumscribing sphere Radius of inscribed sphere Total area Volume

𝑎 6 4 𝑎 6 12 𝑎2 3 1 3 𝑎 2 12

5 Regular Polyhedrons • Hexahedron Faces

Square

No. of faces

6

No. of edges

12

No. of vertices

8

Polygon angle

90°

Sum of angles

270°

Radius of circumscribing sphere Radius of inscribed sphere

𝑎 3 2 𝑎 2

Total area

6𝑎2

Volume

𝑎3

5 Regular Polyhedrons • Octahedron Faces

Triangle

No. of faces

8

No. of edges

12

No. of vertices

6

Polygon angle

60°

Sum of angles

240°

Radius of circumscribing sphere Radius of inscribed sphere Total area Volume

𝑎 2 2 𝑎 6 6 2𝑎2 3 1 3 𝑎 2 3

5 Regular Polyhedrons • Dodecahedron Faces

Pentagon

No. of faces

12

No. of edges

30

No. of vertices

20

Polygon angle

108°

Sum of angles

324°

Radius of circumscribing sphere

Radius of inscribed sphere

𝑎 3(1 + 5) 4 𝑎 50 + 22 5 4 5

Total area 15𝑎2

Volume

3+ 5 5− 5

5𝑎3 3

5 Regular Polyhedrons • Icosahedron Faces

Triangle

No. of faces

20

No. of edges

30

No. of vertices

12

Polygon angle

60°

Sum of angles

300°

Radius of circumscribing sphere Radius of inscribed sphere

Total area Volume

𝑎 2(5 + 5) 4 𝑎 7+3 5 2 6 5𝑎 2 3 5 3 𝑎 (3 + 5) 12

Example • A regular tetrahedron has one of each side equal to 20 cm. (a)Compute the surface area of the tetrahedron. (b)Compute the volume of the tetrahedron. (c)Compute the volume of sphere inscribed in the regular tetrahedron.

Plates (A4)

1.) The inside diameters of the bases of a flowerpot are 10 inches ans 7 inches and slant height of 9 in. How many cubic of soil does it contain when it is completely filled? (Ans: 508.79 in3)

Plates (A4)

2.)A pile of sand is in a form of right circular cone of altitude 7 ft and slant height 25 ft. What is the weight of the 3 sand, if its density is 100 lb/ft . (Ans: 422,230 lbs)

Plates (A4)

3.) A steam pipe is 10 m long and has an internal diameter of 50 cm. If the pipe has 10 cm thick, find the volue of metal in the pipe. (Ans: 1,885,000 cm3)

Plates (A4)

4.) A cylindrical tank has its height 150% of its diameter. If the tank volume is 200 3 m , find the tank height. (Ans: 8.30 m)

Plates (A4)

5.) Find the volume of a cube that can be cut from a log that has a radius of 20 cm. 3 (Ans: 22,627.417 cm )

Plates (A4)

6.) A cube has edges of 12 inches. If the edge will be decreased by 4 inches, find the percent decrease in volume. (Ans: 70.37%)

Plates (A4) 7.) Determine the volume of a right truncated prism with the following dimensions: Lot the corners of the triangular base be defined by A, B, and C are perpendicular to the triangle base and have the height of 8.6 ft, 7.1 ft, and 5.5 ft respectively. (Ans: 311 ft3)

Plates (A4)

8.) If the cone has a base radius of 35 cm and an altitude of 45 cm, solve for the total surface area. 2 (Ans: 10,115.93 cm )

Plates (A4)

9.) A pipe line material of silicon carbide used in the conveyance of pulverized coal to fuel boiler, has thickness of 2 cm and inside diameter of 10 cm. Find the volume of the material for pipe length of 6 m. 3 (Ans: 45,239 cm )

Plates (A4)

10.) A circular piece of cardboard with a diameter of 1 m will be made into a conical hat 40 cm high by cutting a sector off and joining the edges to form a cone. Determine the cone diameter and the angle subtended by the sector removed. (Ans: 60 cm, 144 degrees)

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