Surface Area of Three-dimensional Shapes

Surface Area of Prisms
SURFACE AREA = The total area of the surface of a three-
dimensional object
(Or think of it as the amount of paper you’ll need to wrap the
shape.)
PRISM = A solid object that has two identical ends and all flat
sides.
We will start with 2 prisms – a rectangular prism and a triangular
prism.

Rectangular
Prism
Triangular
Prism

Surface Area (SA) of a
Rectangular Prism
Like dice, there
are six sides (or 3
pairs of sides)

Add the area of all 6 sides to find the Surface
Area.
10 - length
5 - width
6 - height

Formula: SA = 2lw + 2lh + 2wh
10 (length)
5 (width)
6 (height)
SA = 2lw + 2lh + 2wh
SA = 2 (10 x 5) + 2 (10 x 6) + 2 (5 x 6)
= 2 (50) + 2(60) + 2(30)
= 100 + 120 + 60
= 280 units squared

Practice
10 ft
12 ft
22 ft
SA = 2lw + 2lh + 2wh
= 2(22 x 10) + 2(22 x 12) + 2(10 x 12)
= 2(220) + 2(264) + 2(120)
= 440 + 528 + 240
= 1208 ft2

Surface Area of a Triangular Prism
formula:
Surface Area = (Perimeter of the base × Length
of the prism) + (2 × Base Area)
SA = (S1 +S2 + S3)L + bh
where,
b is the bottom edge of the base triangle,
h is the height of the base triangle,
L is the length of the prism and
S1, S2, and S3 are the three edges (sides) of the
base triangle
bh is the combined area of the two triangular
faces
[2 × (1/2 × bh)] = bh

Step 1: Find the area of the top
and the base triangles using the
formula, Area of the two base
triangles =
2 × (1/2 × base of the triangle ×
height of the triangle)
which simplifies to 'base × height’
(bh).
Step 2: Find the product of the
length of the prism and the
perimeter of the base triangle

Step 3: Add all the areas together
to get the total surface area of a
right triangular prism in square
units. This means, total surface
area of a right triangular prism =
(S1 + S2 + h) × l + bh

15ft
15ft
12 ft
6 ft
Area of two
triangles
= 2 x [½ (b x h)]
= 2 x [½ (12ft x
15ft)]
= 180ft2

Side 1 = 20 ft
Side 2 = 20 ft
Side 3 = 12 ft
Length = 25 ft
Area of rectangles
= (S1 + S2 + S3) x L
= (20 + 20 + 12) x 25
= 1300 ft2

Total Surface Area
=
2(area of triangle)
+ Area of
Rectangles
= 180ft2
+ 1300ft2
= 1480 ft2

Practice
Solve for the
Surface Area
of this
Triangular
Prism

Area (Triangle)
= 2 x [1/2 x (b x h)]
= b x h
= 10cm x 12cm
= 120cm2

Area (Rectangles)
= (s1+s2+s3) x L
= (13+13+10) x 20
= 36cm x 20cm
= 720cm2

Total Surface Area
= 120cm2 +
720cm2
= 840cm2

Surface Area of a Cylinder
The surface area of a cylinder is the
area occupied by its surface in a three-
dimensional space. A cylinder is a
three-dimensional structure having
circular bases which are parallel to
each other. It does not have any
vertices. Generally, the area of the
three-dimensional shapes refers to the
surface area.

The Soup Can
Think of the Cylinder as a soup can.
You have the top and bottom lid (circles) and
you have the label (a rectangle – wrapped
around the can).
The lids and the label are related.
The circumference of the lid is the same as the
length of the label.

Solve for the Surface
Area of a Cylinder
Formula for Area of Circles (Top
and bottom: A= 2 r2
Given: r = 4cm; h = 24cm
A =  x r2
A =  x (4cm)2
A = 50.2655 cm2
But there are 2 of them so
50.2655 cm2
x 2 = 100.531 cm2

Area of a Cylinder
Formula for Area of Cylinder:
A= 2 rh
Given: r = 4cm; h = 24cm
A = 2 x 4cm x 24cm
A = 2 x 96cm2
A = 603.1858 cm2

Area of a Cylinder
Total Surface Area of Cylinder
A = 2 r2
+ 2 rh
= 100.531 cm2
+ 603.1858 cm2
= 703.72 cm2

Practice
Solve for the
surface area of
the Cylindrical
Cheese

Surface Area of a Pyramid
The surface area of a pyramid is obtained by adding the
area of all its faces. A pyramid is a three-dimensional
shape whose base is a polygon and whose side faces (that
are triangles) meet at a point which is called the apex (or)
vertex. The perpendicular distance from the apex to the
center of the base is called the altitude or height of the
pyramid. The length of the perpendicular drawn from the
apex to the base of a triangle (side face) is called the 'slant
height'. Let us learn more about the surface area of a
pyramid along with its formula, a few solved examples,

Pyramid Nets
A pyramid has 2
shapes:
One (1) square
&
Four (4) triangles

you can use a formula…
SA = ½ l x p + B
Where l is the Slant Height and
p is the perimeter and
B is the area of the Base

SA = ½ l x p + B
Perimeter = (2 x 7) + (2 x 6) = 26
Slant height l = 8 ;
SA = ½ l x p + B
= ½ (8 x 26) + (7 x 6) *area of the base*
= ½ (208) + (42)
= 104 + 42
= 146 units 2
6
7
8
5

Practice
6
6
18
10
SA = ½ l x p + B
= ½ (18 x 24) + (6 x 6)
= ½ (432) + (36)
= 216 + 36
= 252 units2
Slant height = 18
Perimeter = 6x4 = 24
What is the extra information in the diagram?

Surface area of a Cone
Surface area of a cone is the complete area
covered by its two surfaces, i.e., circular base
area and lateral (curved) surface area. The
circular base area can be calculated using area
of circle formula. The lateral surface area is the
side-area of the cone. Let us see the formula to
calculate the surface area of cone.

Formula: A =  r2
+  r L
Given: d = 6cm; L = 16.2cm
r = d/2 = 6cm/2 = 3cm
A = ( x r2
) + ( x r x L)
A = ( x (3cm)2
) + ( x 3cm x
16.2 cm)
A = ( x 9cm2
)+ ( x 48.6cm2
)
A = 180.96 cm2

Practice
Solve for the
Surface Area
of the Sweet
Cone

Surface area of a Sphere
A sphere is a solid figure bounded by
a curved surface such that every point
on the surface is the same distance
from the center. In other words, a
sphere is a perfectly round
geometrical object in three-
dimensional space, just like a surface
of a round ball.

Surface area of a Sphere
The distance from the center to the outer
surface of sphere is called its radius. The
surface area of a sphere is defined as the
amount of region covered by the surface
of a sphere and is equal to 4πr². The
volume of sphere is the space occupied
by it in 3d space.

Surface Area of Three-dimensional Shapes

Surface Area of Three-dimensional Shapes

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