solid mensuration (solids with volume equals mean BH)
Solid Mensuration ***** Frustum of a PyramidFrustum of a Right Circular Cone Prismatoid Truncated Prism *****
Meryl Mae R. Nelmida UNIT I Frustum of a Pyramid UNIT IIFrustum of a Right Circular Cone UNIT III Prismatoid UNIT IV
Truncated Prism UNIT I FRUSTUM OF A REGULAR PYRAMID If a pyramid is cut by a plane parallel to its base,two solids are formed. (see fig. 1) The solid above thecutting plane is a pyramid which is similar to theoriginal pyramid and the other solid formed is a frustumof the original pyramid. In general, a frustum of apyramid is that portion of the pyramid between its baseand a section parallel to the base. The frustum of aregular pyramid is also called pyramidal frustum. Fig. 2 Fig. 1
Note: figure 2 represents the unfold of a frustum of apyramidProperties: • The bases of the frustum are the base of the original pyramid and the base of the parallel section. • The altitude/height of the frustum is the perpendicular distance between its bases. • The lateral faces of a frustum of a pyramid are trapezoids. • If the frustum is cut from a regular pyramid, then its lateral edges are equal and its lateral faces are congruent isosceles trapezoids. • The slant height of the frustum of the regular pyramid is the altitude of a lateral face. • The bases of a frustum of a regular pyramid are similar regular polygons. If these polygons become equal, the frustum will become prism. Figure 3 represents the frustum of a regular pyramid.MNPQR and M’N’P’Q’R’ are its bases; AA’ is its altitudeand SS’ is the slant height. The segments MM’, NN’, PP’, …are the lateral edges; MN, NP, PO, … are the lower edges;M’N’, N’P, P’Q’, .. are the upper base edges; and MNN’M’,NPP’M’, PQQ’P’, … are the lateral faces. Note thatrelative to the frustum of a pyramid, five important linesegments are involved, namely: 1. Altitude
2. Slant height 3. Lateral edge 4. Lower base edge 5. Upper base edge Fig. 3 The lateral area S of the frustum of a regularpyramid is equal to one-half of the product of the slantheight l and the sum of the perimeters (p1 and p2) of thebases. In symbol, Eqn. 1 The total area of the frustum of a regular pyramid isthe sum of the lateral area and the areas of the bases. The volume V of the frustum of a regular pyramidwhose bases are b and B ( B > b) and with the altitude his given by
Eqn. 2 In words, the volume of the frustum of a regularpyramid is equal to one-third the product of its altitudeand the sum of the upper bases, the lowers base, and themean proportional between the bases. To prove, considerthe pyramid P-MNQR in Figure 4.
Fig. 4Let H = LP + altitude of pyramid P-MNRQ H = LL’ = altitude of the frustum with bases MNRQ and M’N’R’Q’ b = area of the upper base M’N’R’Q’ B = area of the lower base MNRQ V = volume of the frustum P-MNRQ V1 = volume of the pyramid P-M’N’R’Q’ V2 = volume of the pyramid P-MNRQThen
By the equation Volume = (B × h)/3Substituting (2) and (3) in (1) and rearranging the terms,we get s = l2 S L2Also, by the equation which states that the area (s,S) of similar surfaces have the same ratio as the squares of any two corresponding lines.Or solving for H, we obtain
Substituting (6) in (40 and simplifying, we get Equation 2 which isEXAMPLE: 1. Find the volume of the frustum of a regular square pyramid whose altitude is 10 cm and whose base edges are 4 cm and 8 cm. Solution: We have the following data based on the given:
b = 42 = 16 B = 82 = 64 H = 10 Then by Equation 2,2. Calculate the lateral area, surface area and volume of the truncated square pyramid whose larger base edge is 24, smaller base edge is 14 cm and whose lateral edge is 13 cm. h2 = 132 - 52 = 12 cm p1 = 24 * 4 = 96 cm P2 = 14 * 4 = 56 cm
UNIT II FRUSTUM OF A RIGHT CIRCULAR CONE The frustum of a right circular cone is that portionbetween the base and a section parallel to the base of thecone. The terms slant height and altitude are used in thesame sense as with the frustum of a regular pyramid. Fig. 5Properties:
• The altitude of a frustum of a right circular cone is the perpendicular distance between the two bases. • All the elements of a frustum of a right circular cone are equal. • In figure 5, we have a frustum of a right circularcone with slant height l, altitude h, lower base radius Rand upper base radius r. it is proved in elementary solidgeometry that the lateral area of the frustum of a rightcircular cone is equal to one-half the product of the sumof the circumferences of its bases and the slant height.That is, Eqn. 3 Where: c = circumference of the upper base C = circumference of the lower base l = slant height of the cone S = lateral area But c = 2πr and C = 2πr. Substituting these values,we get Eqn. 3.1 Where:
r = upper base radius R = lower base radius l = slant height of the cone The volume of the frustum of a circular cone is usedin the same sense as with the volume of a regular pyramid.That is, But for a right circular cone, b = πr2 and B = πR2.Substituting these values in the above equation, we get = Eqn. 4 Where: V = volume of frustum h = altitude of the frustum r = upper base radius R = lower base radiusEXAMPLE:
1. Find the volume of the frustum of a rightcircular cone whose slant height is 10 cm and whoseradii are 3 cm and 9 cm.Solution: See we are given that r = 3, R = 9, and l = 10. From the figure below, we see that the altitude is Hence, by Equation 4, we obtain = = V π(8)(9 + 81 = 27) = 312 cm32. The volume of a frustum of a right circular cone is 1176π cu. cm. The altitude of the frustum of a cone is 18 cm. find the radii of the upper and lower base if the product of their radii is 60 sq. cm.
3. Find the volume and surface area of a frustum of a cone having radius of the upper base equal to 4 cm and radius of lower base equal to 6 cm, if it has a height of 8 cm. UNIT III PRISMATOID A prismatoid is a polyhedron havingfor bases two polygons in parallel planes, and for lateral
faces triangles or trapezoids with one side lying in onebase, and the opposite vertex or side lying in other baseof the polyhedron.Properties: • The altitude of a prismatoid is the perpendicular distance between the planes of the bases. • The mid-section of a prismatoid is the section parallel to the bases and midway between them. The volume of a prismatoid equals the product of one-sixth the sum of the upper base, the lower base, and fourtimes the mid-section by the altitude.
EXAMPLE: 1. A trapezoidal canal having a base 6 m wide and 8 m wide at the top at one end and a base width of 6 m wide and 10 cm width at the top at the other end of the canal which is 50 m long. Find the volume of
the earth excavated for the canal. The depth of the canal is 4 m depth at one end and 5 m depth at the other end.2. Find the volume of the prismatoid shown.
ABOUT THE AUTHOR Meryl Mae Rabut Nelmida is the present Vice-President of the Louisian Mathematics Society in Saint Louis College (City of San Fernando, La Union). She shares her unique intelligence in Mathematics through the club’s program such as remedial and tutorials in Lingsat Community School and Poro-San Agustin Elementary School. She finished her BasicEducation in Christ the King College, City of SanFernando, La Union. She is an active member of the PantasCircle during her high school years. The Pantas Circleprovides opportunities for students to hone theirknowledge in Mathematics. She is presently in her secondyear of studying Bachelor of Secondary Education Major inMathematics.
UNIT I Frustum of a Pyramid UNIT IIFrustum of a Right Circular Cone UNIT III Prismatoid UNIT IV Truncated Prism