Geometric Solids
SEME 104 – Plane and Solid Geometry
Learning Objectives
At the end of the lesson, you will be able to:
• compute the lateral area and the total surface area for a right
prism, right circular cylinder, a regular pyramid, a right
circular cone, and a sphere;
• compute the volume of a right prism, a right circular cylinder,
a regular pyramid, a right circular cone, and a sphere.
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Prisms
Prisms are solids (three-dimensional figures) that, unlike planar
figures, occupy space. They come in many shapes and sizes.
Every prism has the following characteristics:
1. Bases: a prism has two bases, which are congruent polygons
lying in parallel planes.
2. Lateral Edges: The lines formed by connecting the
corresponding vertices, which form a sequence of parallel
segments.
3. Lateral Faces: The parallelograms formed by the lateral
exges.
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Naming a Prism
A prism is named by the polygon that forms its base, as follows:
• Altitude: A segment perpendicular to the planes of the bases
with an endpoint in each plane.
• Oblique prism: A prism whose lateral edges are not
perpendicular to the base.
• Right Prism: A prism whose lateral edges are perpendicular to
the bases. In a right prism, a lateral edge is also an altitude.
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Different Types of Prisms
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Right Prisms
In certain prisms, the lateral faces are each perpendicular
to the plane of the base. The lateral area of a right prism is the
sum of the areas of the areas of all the lateral faces.
Theorem: The lateral Area, 𝐿𝐴, of a right prism of altitude ℎ
and the perimeter 𝑝 is given by the following equation.
𝐿𝐴!"#$% '!"() = 𝑝 & ℎ units *
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Example 1
Find the lateral area of the right hexagonal prism, as shown in
the figure.
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Total Area of a Right Pris
The total area of a right prism is the sum of the lateral
area and the areas of the two bases. Because the bases are
congruent, their areas are equal.
Theorem: The total area, 𝑇𝐴, of a right prism with lateral area
𝐿𝐴 and a base area 𝐵 is given by the following equation.
𝑇𝐴!"#$% '!"() = 𝐿𝐴 + 2𝐵 or 𝑇𝐴!"#$% '!"() = 𝑝ℎ + 2𝐵
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Example 2
Find the total area of the triangular prism, as shown in the
figure.
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Interior Space of a Solid
Lateral area and total area are measurements of the
surface of a solid. The interior space of a solid can also be
measures.
A cube is a square right prism whose lateral edges are the
same length as a side of the base.
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Interior Space of a Solid
The volume of a solid is the number of cubes with unit
edge necessary to entirely fill the interior of the solid. In the
accompanying figure, the right rectangular prism measures 3
inches by 4 inches by 5 inches.
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Interior Space of a Solid
This prism can be filled with cubes 1 inch on each side,
which is called a cubic inch. The top layer has 12 such cubes,
because the prism has 5 such layers, it takes 60 of these to fill
the solid, thus the volume of
this prism is 60 cubic inches.
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Interior Space of a Solid
Theorem: The volume, 𝑉, of a right prism with a base area 𝐵
and an altitude of ℎ is given by the following equation.
𝑉!"#$% '!"() = 𝐵 & ℎ 𝑢𝑛𝑖𝑡𝑠 *
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Try this!
The accompanying figure is an isosceles trapezoidal right prism.
Find a. 𝐿𝐴, b. 𝑇𝐴 and c.𝑉.
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Right Circular Cylinders
A prism-shaped solid whose bases are circles is a cylinder.
If the segment joining the centers of the circles of a cylinder is
perpendicular to the planes of the bases, the cylinder is a right
circular cylinder.
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Lateral Area, Total Area, and Volume of a Cylinder
If a cylinder is pictured as a soup can, its lateral area is
the area of the label. If the label is carefully peeled off, the
label becomes a rectangle, as shown in the figure.
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Lateral Area, Total Area, and Volume of a Cylinder
Theorem: The lateral area, 𝐿𝐴, of a right circular cylinder with
base circumference 𝐶 and an altitude ℎ is given by the following
equation.
𝐿𝐴!"#$% +"!+,-.! +/-"012! = 𝐶 & ℎ 𝑢𝑛𝑖𝑡𝑠 *
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Lateral Area, Total Area, and Volume of a Cylinder
Theorem: The total area, 𝑇𝐴, of a right circular cylinder with
lateral area 𝐿𝐴 and a base area 𝐵 is given by the following
equation.
𝑇𝐴!"#$% +"!+,-.! +/-"012! = 𝐿𝐴 + 2𝐵 𝑢𝑛𝑖𝑡𝑠 *
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Lateral Area, Total Area, and Volume of a Cylinder
Theorem: The volume, 𝑉, of a right circular cylinder with base
area 𝐵 and altitude ℎ is given by the following equation.
𝑉!"#$% +"!+,-.! +/-"012! = 𝐵 & ℎ 𝑢𝑛𝑖𝑡𝑠 3
or
𝑉!"#$% +"!+,-.! +/-"012! = 𝜋𝑟 * & ℎ 𝑢𝑛𝑖𝑡𝑠 3
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Try This!
Find the lateral area, total area, and volume of the right
circular cylinder.
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