3. a. Prisms
b. Platonic solid
c. Pyramid
01. Polyhedra 02. Non-Polyhedra
03. Quiz
a. Cylinders
b. Cone
c. Sphere
Table of Contents
a. Exercise 1
b. Exercise 2
c. Exercise 3
4. Definition
• Space geometry is defined by the three-dimensions: a length,
a width, and a height.
• Space geometry has three geometric parameters: 𝑥 𝑦 and
𝑧 that each parameter is perpendicular to the other two, and
cannot lie in the same plane.
• Space geometry is about three dimensional objects such as
cube, prisms, cylinders, and spheres.
6. 1. Polyhedra
A polyhedra is a 3-dimensional solid made by joining together
polygons. The polyhedras are defined by the number of faces it has.
Component of polyhedra are:
1. Face: the flat surfaces that make up a polyhedron are called its
faces. These faces are regular polygons.
2. Edge: the regions where the two flat surfaces meet to form a
line segment are known as the edges.
3. Vertex: It is the point of intersection of the edges of the
polyhedron. A vertex is also known as the corner of a
polyhedron. The plural of vertex is called vertices.
8. A. Prisms
A prism is a polyhedron that has two equal, parallel sides called
“bases” and its lateral sides are “parallelograms”.
The prisms are named after the base’s polygon.
Principal of prisms:
● Bases: All prisms have two bases that are equal and parallel.
● Lateral sides: Lateral sides are the parallelograms that link the
2 bases.
● Height: Height is the distance between the two bases.
There are two types of prisms, such as:
● Irregulars: Prisms that have irregular polygons as bases.
● Regulars: Prisms that have regular polygons as bases
9. A. Prisms
Generally, the formula of volume
and surface area of prisms are:
• Lateral Area : LA = p x h
• Volume : V = B x h
• Surface Area : A = 2B + LA
Where:
V = Volume of prism
A = Surface area of prism
LA = Lateral area of prism
B = area of prism base
h = height of prism
p = Perimeter of prism base
10. Triangular Prism Cuboid
• A cuboid has 6 faces, 12 edges and
8 vertices.
• The faces of the cuboid are all
rectangular in shape.
• Angles formed at the vertices of
the cuboid are all 90 degrees.
Cube
Some kind of prisms
A. Prisms
• Had a total of 9 edges, 5
faces, and 6 vertices (which
are joined by the rectangular
faces).
• Had two triangular bases and
three rectangular sides.
• It has 6 faces, 12 edges, and 8
vertices.
• All faces are in the shape of a
square.
• All sides have the same length.
• Each vertex meets three faces
and three edges.
11. A. Prisms
b
t
• Formula of Triangular Prism
h
Perimeter of base prism
p = a + b + c
Lateral area
LA = p 𝑥 ℎ
Surface area
A = 2B + (p 𝑥 ℎ)
A = 2(
𝑏 𝑥 𝑡
2
) + (p 𝑥 ℎ)
Volume
V = B x h
V = (
𝑏 𝑥 𝑡
2
) 𝑥 ℎ
a
c
12. A. Prisms
Perimeter of base prism
p = 2(l + 𝑤)
Lateral area
LA = p 𝑥 ℎ
Surface area
A = 2B + (p 𝑥 ℎ)
A = 2(𝑙 𝑥 𝑤) + (p 𝑥 ℎ)
A = 2(𝑙 𝑥 𝑤) + 2(h 𝑥 𝑤) +
2(𝑙 𝑥 ℎ)
Volume
V = B x h
V = (𝑙 𝑥 𝑤) x h
l
w
h
• Formula of Cuboid
13. A. Prisms
• Formula of Cuboid
Perimeter of base prism
p = 4 x a
Lateral area
LA = p 𝑥 𝑎
Surface area
A = 2B + (p 𝑥 ℎ)
A = 2(𝑎 𝑥 𝑎) + (p 𝑥 𝑎)
A = 6 (𝑎 𝑥 𝑎)
Volume
V = B x h
V = (𝑎 𝑥 𝑎) 𝑥 𝑎
a a
a
15. B. Platonic solid
kel 7 BIM..pptx
Tetrahedron Cube Octahedron Decahedron icosahedron
Platonic Solid is a three dimensional shape whose faces are
all equal and identical and the same number of faces meet
at each other vertex.
16. Properties of Platonic Solids
• All the faces are regular and congruent.
• Platonic shapes are convex polyhedrons.
• Faces of platonic solid do not intersect except at their
edges.
• The same number of faces meet at each vertex.
• Platonic solids have polygonal faces that are similar in form,
height, angles, and edges.
• Platonic solids are three-dimensional, convex, and regular
solids shapes.
B. Platonic sloid
18. C. Pyramid
Pyramid is a shape which has a polygonal base with the upright sides in
the from of a triangular shape with a vertex at the top.
Characteristics of pyramid:
1. The top of pyramid has an acute
point.
2. The bottom of pyramid is a flat
shape.
3. The perpendicular side of the
pyramid is triangular in shape
19. C. Pyramid
Generally, the formula of volume
and surface area of pyramid are:
• Lateral Area : 𝐿𝐴 =
𝑝 × 𝑠
2
• Volume : 𝑉 =
𝐵×ℎ
3
• Surface Area : 𝐴 = 𝐵 + 𝐿𝐴
Where:
V = Volume of pyramid
A = Surface area of pyramid
LA = Lateral area of triangular
h = Height of pyramid
p = Perimeter of pyramid base
s = Slant height
20. Triangular Pyramid Square Pyramid
Some kind of pyramids
C. Pyramid
• The triangular pyramid has 4 faces.
• The 3 side faces of the triangular pyramid
are triangles.
• The base is also triangular in shape.
• It has 4 vertices (corner points)
• It has 6 edges.
• Had 5 Faces, 5 Vertices (corner
points), 8 Edges.
• The 4 Side Faces are Triangles.
• The Base is a Square.
21. C. Pyramid
s
• Formula of Triangular Pyramid
Perimeter of base prism
p = a + b + c
Lateral area
LA =
𝑝 𝑥 𝑠
2
Surface area
A = B +
𝑝 𝑥 𝑠
2
A = (
𝑏 𝑥 𝑡
2
) +
𝑝 𝑥 𝑠
2
Volume
V =
𝐵×ℎ
3
V =
(
𝑏×𝑡
2
)×ℎ
3
b
t
h
22. C. Pyramid
a
• Formula of Square Pyramid
Perimeter of base prism
p = 4a
Lateral area
LA =
4𝑎 𝑥 𝑠
2
LA = 2as
Surface area
A = B +
𝑝 𝑥 𝑠
2
A = (𝑎2) +2as
Volume
V =
𝐵×ℎ
3
V =
(𝑎2)×ℎ
3
a
s
h
25. A. Cylinder
A cylinder is a three-dimensional shape consisting of two
parallel circular bases, joined by a curved surface. The center of
the circular bases overlaps each other to form a right cylinder.
Characteristics of cylinder:
• It has a flat base and a flat top.
• The base is the same as the top.
• From base to top the shape stays the same.
• It has one curved side.
Oblique Cylinder
When the two ends are directly aligned on each other it is a
Right Cylinder otherwise it is an Oblique Cylinder.
26. Volume and Surface Area Of Cylinder
For:
𝑉 = Volume of cylinder
𝐴 = Surface area of cylinder
𝐿𝐴 = Lateral area of cylinder
𝑟 = radius
ℎ = height
Volume
𝑉 = π × r2 × h
Surface Area
A = 2 × π × r × (r+h)
Lateral Area
LA = 2 × π × r ×h
A. Cylinder
28. B. Cone
A cone is a three-dimensional shape in geometry that narrows
smoothly from a flat base (usually circular base) to a
point(which forms an axis to the centre of base) called the apex
or vertex.
Characteristics of cone:
• It has one circular face at one end and a point at
the other end
• It has zero edges.
• It has one vertex (corner)
29. Volume and Surface Area Of Cone
For:
𝑉 = Volume of cone
𝑆𝐴 = Surface area of cone
𝐿𝐴 = Lateral area of cone
𝑟 = radius
ℎ = height
𝑠 = slant length
Volume
𝑽 =
1
3
π × r2 × h
Surface Area
𝐒𝑨 = π × r × (r + s)
Lateral Area
𝑳𝑨 = π × r × s
𝒔 = 𝒓𝟐 + 𝒉𝟐
B. Cone
31. C. Sphere
A sphere is symmetrical, round in shape. It is a three
dimensional solid, that has all its surface points at equal
distances from the center. It has surface area and volume based
on its radius.
Characteristics of sphere:
• It is perfectly symmetrical.
• All points on the surface are the same
distance "r" from the center.
• It has no edges or vertices (corners).
• It has one surface (not a "face" as it isn't flat).
32. Volume and Surface Area Of sphere
For:
𝑉 = Volume of sphere
𝐴 = Surface area of sphere
𝑟 = radius
C. Sphere
34. The volume of a pyramid is 520 cm³ with a square base. If
the base is a parallelogram with a base length of 12 cm
and a height of 10 cm, then the height of the pyramid is...
01
36. Doni is given a ball by his father who has a
radius of 30 cm, then find the volume of the
ball and the surface area of the ball?
02
37. Answer:
•Volume:
•V = 4/3 x π x r3
•V = 4/3 x 3,14 x 30cm x 30cm x 30cm
•V = 113.040cm3
•Surface Area:
L = 4π r2
•L = 4 x 3,14 x 30cm x 30cm
•L = 11.304cm2
38. Anya eats an ice cream while wondering the
volume of the ice cream that she eats. Then,
Anya measure the ice cream and found out that
the total height of the ice cream is 17 cm and the
diameter of the cream is 14 cm. Knowing all of
this, Anya don’t know how to calculate it. So, can
you help Anya finds the volume of the ice cream
that she eats?
03
17 cm
14 cm
39. Answer:
•Total height : 17 cm
•r : ½(14) = 7 cm
•Height of cone :
•17 cm – 7 cm = 10 cm
•Volume of ice cream :
•Volume of Cone + Volume of ½ Sphere
•Volume of cone :
1
3
𝜋 × 𝑟2 × ℎ
•=
1
3
×
22
7
× (7)2 × 10 =
1
3
× 22 × 7 𝑥 10
•=
1
3
× 22 × 7 × 10= 513.33 𝑐𝑚2
•Volume of ½ sphere :
1
2
×
4
3
𝜋 × 𝑟3
•=
1
2
×
4
3
×
22
7
× (7)3 =
2
3
× 22 × (7)2
•=
2
3
× 22 × 49 = 718.67 𝑐𝑚2
•Volume of ice cream :
•Volume of Cone + Volume of ½ Sphere
•513.33 𝑐𝑚2 + 718.67 𝑐𝑚2
•1232 𝒄𝒎𝟐
