SELECTED TOPICS IN MODERN GEOMETRY
Master of Arts in Teaching Mathematics
Surface Area and Volume of Three-Dimensional Objects/Solids
Surface Area – the sum of the areas of the surfaces.
Volume – measure of the three-dimensional space that an object occupies.
Solids and their properties
• Prisms – It has two congruent parallel faces called the bases. The lateral faces of a prism are
parallelograms.
In a right prism, the lateral edges are perpendicular to the bases and the lateral edges are rectangles.
Also, the altitude lies on, or is parallel to a lateral edge. In oblique prisms, the lateral edges are not
perpendicular to the bases. The altitude does not lie on, or is not parallel to, a lateral edge.
The distance between the bases is the height. The sum of the lateral faces of a prism is called lateral
area. You can think of the lateral area of a right prism as three or more rectangles joined together.
Prisms are classified by both the shape of their bases and the relation (either right or oblique) of their
lateral edges to the planes containing their bases.
- Cube have six surfaces that are each square and have the same measures. A cube is a square right
prism and its lateral edges are equal in length to the sides of the base.
• Cylinder – has two parallel congruent circular bases and a lateral surface that connects the boundaries of
the bases. The lateral surface of a right cylinder as a rectangle.
• Pyramid – has one polygon as a base and triangles for lateral faces. If the base is a regular polygon, the
pyramid is also regular. A regular pyramid has both a height and a slant height. The slant height is the
height of one of the lateral faces.
• Cone – the lateral surface connects each point on the boundary of the base to the vertex. A right cone has
both a height and a slant height.
• Sphere – the radius of a sphere is the distance from the center to any point on the surface.
Postulates and Theorems for Solids
✓ Change Position Postulate – Any geometric figure can be moved or relocated to a new position without
changing the figure’s size or shape.
• Prisms
- Right Prism Lateral Area theorem, the lateral area LA of a right prism equals the product of the
perimeter p of a base and the height h of the prism or LA = ph.
- Right Prism Surface Area theorem, the surface area SA of a right prism equals the sum of its
lateral area and twice the area of one of its bases B, or SA = LA + 2B = ph + 2B.
- Prism Volume theorem, the volume V of a prism equals the product of the area of a base B and
the height h, or V = Bh.
- Cube Volume theorem, the volume of a cube is the cube of the length e of one edge, or V = e 3.
• Cylinder
- Right Cylinder Lateral Area theorem, the lateral area LA of a right cylinder with radius r equals
the product of the circumference of a base and the height h of the cylinder or LA = 2πrh.
- Right Cylinder Surface Area theorem, the surface area SA of right circular cylinder is equal to the
lateral area LA plus the area of two congruent circle bases, or SA = LA + 2πr 2 = 2πrh + 2πr2.
- Right Cylinder Volume theorem, the volume V of a right cylinder with radius r equals the product
of the area of a base B and the height h, or V = Bh = πr 2h.
• Pyramid
- Regular Pyramid Lateral Area theorem, the lateral area LA of a regular pyramid equals half the
1
product of the slant height l of the pyramid and the perimeter p of its base, or 𝐿𝐴 = 𝑙𝑝.
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SURFACE AREA AND VOLUME OF THREE-DIMENSIONAL OBJECTS/SOLIDS
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SELECTED TOPICS IN MODERN GEOMETRY
Master of Arts in Teaching Mathematics
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•
3
Cone
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•
Regular Pyramid Surface Area theorem, the surface area SA of a regular pyramid of slant height l
and the base perimeter p equals the sum of its lateral area LA and the area of its base B, or SA =
1
LA + B = 𝑙𝑝 + 𝐵.
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Pyramid Volume theorem, the volume V of a pyramid equals one third the product of the area of
1
the base B and the height h, or 𝑉 = 𝐵ℎ.
Right Cone Lateral Area theorem, the lateral area of a right cone of radius r equals half the
1
product of the circumference of its base and the slant height l of the cone, or 𝐿𝐴 = ∙ 2𝜋𝑟𝑙 =
2
𝜋𝑟𝑙.
Right Cone Surface Area theorem, the surface area SA of a right cone of radius r and slant height
l equals the sum of its lateral area LA and the area of its base B, or SA = LA + B = πrl + πr 2.
Cone Volume theorem, the volume V of a cone of radius r equals one third the product of the area
1
1
of the base B and the height h, or 𝑉 = 𝐵ℎ = 𝜋𝑟 2 ℎ.
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3
Sphere
- Sphere Surface Area theorem, the surface area SA of a sphere equals the product of 4π and the
square of the radius r, SA = πr2.
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- Sphere Volume theorem, the volume V of a sphere equals the product of 𝜋 and the cube of the
3
4
𝜋𝑟 3 .
3
radius r, 𝑉 =
✓ Similar solids surface area theorem: The ratio of the surface areas of two similar solids is equal to the
square of the ratio of any pair of corresponding lengths, so that if the ratio of lengths (scale factor) is x : y,
then the ratio of areas is x2 : y2.
✓ Similar solids volume theorem: The ratio of the volumes of two similar solids is equal to the cube of the
ratio of lengths (scale factor) is x : y, then the ratio of volumes is x 3 : y3.
✓ Cavalieri’s postulate: For two geometric solids in a plane, if every plane parallel to the base plane that
intersects the other solid at the same distance from the base, and the resulting cross sections have the
same area, then the two solids have the same volume.
References
Gerver, e. a. (1998). Geometry, An Integrated Approach.
Cincinnati, OH: International Thomson Publishing.
Ross, D. A. (2010). Master Math: Geometry. Boston, MA:
Cengage Learning.
SURFACE AREA AND VOLUME OF THREE-DIMENSIONAL OBJECTS/SOLIDS
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SELECTED TOPICS IN MODERN GEOMETRY
Master of Arts in Teaching Mathematics
NAME: __________________________________________
I.
PRACTICE. Solve and give the exact answer and an approximate answer to the nearest hundredth, when
applicable.
a. SURFACE AREA and VOLUME (5 points each)
1.
2.
3.
4.
5. If the radius of a sphere is 2 feet, determine the surface area and volume.
b. VOLUME of COMBINED GEOMETRIC FIGURES (3 points each)
1.
SURFACE AREA AND VOLUME OF THREE-DIMENSIONAL OBJECTS/SOLIDS
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SELECTED TOPICS IN MODERN GEOMETRY
Master of Arts in Teaching Mathematics
2.
II.
APPLICATIONS. Solve each problem. Give the exact answer and an approximate answer to the
nearest hundredth, when applicable. (3 points each)
a. Determine the lateral area of the pencil shown.
b. A triangular prism separates white light into a visible spectrum composed of primary colors. In
the figure shown, the height of the prism is 10mm, and each edge of its base is 2mm long. Find
the surface area.
c. A sugar cube is ½ inch on each edge. How much volume does it occupy?
d. A cylindrical oil tank has a diameter of 6 feet and a length of 7 feet. Find the volume of the tank.
e. The lifting power of a spherical balloon depends on its volume. How many cubic feet of gas will
a balloon hold if its 40 feet in diameter?
SURFACE AREA AND VOLUME OF THREE-DIMENSIONAL OBJECTS/SOLIDS
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SELECTED TOPICS IN MODERN GEOMETRY
Master of Arts in Teaching Mathematics
ANSWER’S KEY
(45 POINTS)
I.
PRACTICE
a. SURFACE AREA and VOLUME
5 points
complete (both SA and V) and correct answer with solution
4 points
incomplete (either SA or V) and correct answer with solution
3 points
complete incorrect answer with solution
2 points
incomplete incorrect answer with solution
1 point
incorrect answer without solution
1. SA = 89 cm2, V = 51 cm3
2. SA = 52 cm2, V = 14 cm3
3. SA = 115 in2, V = 72 in3
4. SA = 108.8 cm2, V = 75 cm3
5. SA = 50.26 ft2, V = 33.5 ft3
b. VOLUME of COMBINED GEOMETRIC FIGURES
3 points
correct answer with solution
2 points
incorrect answer with solution
1 point
incorrect answer without solution
1. 576 cm3
2. 335.10 in3
II.
APPLICATION
3 points
2 points
1 point
a. 4,080 m3
b. 173.21 mm3
1
c. 8 𝑖𝑛3
d. 192.92 ft3
e. 33,510.32 ft3
correct answer with solution
incorrect answer with solution
incorrect answer without solution
SURFACE AREA AND VOLUME OF THREE-DIMENSIONAL OBJECTS/SOLIDS
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