Geometric Formulas for Area of Various Shapes
Area of Any Triangle
Area of Parallelogram
Area of Kite & Rhombus
Volume of Solids
Volume of Solids
Area of Trapezium
Composite Area
Volume & Surface Area
Surface Area of a Cylinder
Volume of a Cylinder
Composite Volume
Exam Type
Questions
2
Simple Areas
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A few types of special Areas
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Learning Intention
Learning Intention
Success Criteria
Success Criteria
1.
To know the formula for the
To know the formula for the
area of
area of
ANY
ANY
triangle.
triangle.
1. To develop a formula for the
area of
ANY
triangle.
2.
Use the formula to solve
problems.
2.
Apply formula correctly.
Apply formula correctly.
(showing working)
(showing working)
3.
Answer containing
Answer containing
appropriate units
appropriate units
Any Triangle Area
4
Any Triangle Area
h
Sometimes
called the
altitude
5
Any Triangle Area
10cm
Example 2
: Find the area of the triangle.
Altitude
h
outside triangle this time.
6
Any Triangle Area
5cm
Example 3
: Find the area of the isosceles triangle.
www.mathsrevision.com
Hint : Use
Pythagoras Theorem first !
4
c
m
7
Parallelogram Area
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h
=
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Learning Intention
Learning Intention
Success Criteria
Success Criteria
1.
To know the formula for the
To know the formula for the
area of
area of
ANY
ANY
rhombus and kite.
rhombus and kite.
1. To develop a single formula for
the area of
ANY
rhombus and
Kite.
2.
Use the formula to solve
problems.
2.
Apply formulae correctly.
Apply formulae correctly.
(showing working)
(showing working)
3.
Answer containing
Answer containing
appropriate units
appropriate units
Rhombus and Kite Area
9
Area of a Rhombus
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10
Area of a Kite
Exactly the same process as the rhombus
11
Rhombus and Kite Area
Example 2
: Find the area of the V – shape kite.
7cm
4cm
12
Learning Intention
Learning Intention
Success Criteria
Success Criteria
1.
To know the formula for the area
To know the formula for the area
of a trapezium.
of a trapezium.
1. To develop a formula for the
area of a trapezium.
2.
Use the formula to solve
problems.
2.
Apply formula correctly.
Apply formula correctly.
(showing working)
(showing working)
3.
Answer containing
Answer containing
appropriate units
appropriate units
Trapezium Area
13
Trapezium Area
W
X
Y
Z
1
2
a cm
b cm
h
cm
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W
X
Y
a
n
d
W
Y
Z
14
Trapezium Area
Example 1
: Find the area of the trapezium.
6cm
4cm
5cm
15
Learning Intention
Learning Intention
Success Criteria
Success Criteria
1.
To know the term composite.
To know the term composite.
1. To show how we can apply basic
1. To show how we can apply basic
area formulae to solve more
area formulae to solve more
complicated shapes.
complicated shapes.
2.
2.
To apply basic formulae to
To apply basic formulae to
solve composite shapes.
solve composite shapes.
3.
Answer containing
Answer containing
appropriate units
appropriate units
Composite Areas
16
Composite Areas
We can use our knowledge of the basic areas
to work out more complicated shapes.
4cm
3cm
5cm
6cm
Example 1
: Find the area of the arrow.
17
Composite Areas
Example 2
: Find the area of the shaded area.
11cm
10cm
8cm
4cm
Summary Areas
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Learning Intention
Learning Intention
Success Criteria
Success Criteria
1.
To know the volume formula
To know the volume formula
for any prism.
for any prism.
1.
To understand the
To understand the
prism formula for calculating
prism formula for calculating
volume.
volume.
2.
Work out volumes for
Work out volumes for
various prisms.
various prisms.
3.
Answer to contain
Answer to contain
appropriate units and working.
appropriate units and working.
Volume of Solids
Volume of Solids
Prisms
Prisms
Definition : A prism is a solid shape with
uniform cross-section
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Volume = Area of Cross section x length
Volume of Solids
Volume of Solids
Definition : A prism is a solid shape with
uniform cross-section
Triangular Prism
Volume = Area x length
Q. Find the volume the triangular prism.
20cm
2
10cm
=
2
0
x
1
0
=
2
0
0
c
m
3
www.mathsrevision.com
Definition : A prism is a solid shape with
uniform cross-section
Volume = Area x length
Q. Find the volume the hexagonal prism.
43.2cm
2
20cm
=
4
3
.
2
x
2
0
=
8
6
4
c
m
3
H
e
x
a
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a
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P
r
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m
Volume of Solids
Volume of Solids
This is a NET for the triangular prism.
5 faces
3 congruent rectangles
2 congruent triangles
Net and Surface Area
Net and Surface Area
Triangular Prism
Triangular Prism
= 2 x3 =6cm
2
Example
Find the surface area of the
right angle prism
Working
Rectangle 1 Area = l x b
= 3 x10 =30cm
2
Rectangle 2 Area = l x b
= 4 x 10 =40cm
2
Total Area
= 6+6+30+40+50 = 132cm
2
2 triangles the same
1 rectangle
3cm by 10cm
1 rectangle
4cm by 10cm
3cm
10cm
1 rectangle
5cm by 10cm
Rectangle 3 Area = l x b
= 5 x 10 =50cm
2
5cm
This is a NET for the cuboid
Net and Surface Area
Net and Surface Area
The Cuboid
The Cuboid
6 faces
Top and bottom congruent
Front and back congruent
Left and right congruent
5cm
4cm
3cm
Front Area = l x b
= 5 x 4 =20cm
2
Example
Find the surface area
of the cuboid
Working
5cm
4cm
3cm
Top Area = l x b
= 5 x 3 =15cm
2
Side Area = l x b
= 3 x 4 =12cm
2
Total Area
= 20+20+15+15+12+12
= 94cm
2
Front
and
back
are the same
Top
and
bottom
are the same
Right and
left
are the same
Learning Intention
Learning Intention
Success Criteria
Success Criteria
1.
To know split up a cylinder.
To know split up a cylinder.
1.
To explain how to calculate the
surface area of a cylinder by
using basic area.
2.
2.
Calculate the surface area of
Calculate the surface area of
a cylinder.
a cylinder.
Surface Area
Surface Area
of a Cylinder
of a Cylinder
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l
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u
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f
a
c
e
A
r
e
a
=
2
π
r
2
+
2
π
r
h
The surface area of a cylinder is made up of 2 basic
shapes can you name them.
C
u
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v
e
d
A
r
e
a
=
2
π
r
h
Cylinder
(circular Prism)
h
Surface Area
Surface Area
of a Cylinder
of a Cylinder
R
o
l
l
o
u
t
c
u
r
v
e
s
i
d
e
2
π
r
T
o
p
A
r
e
a
=
π
r
2
B
o
t
t
o
m
A
r
e
a
=
π
r
2
Example
: Find the surface area of the cylinder below:
=
2
π
(
3
)
2
+
2
π
x
3
x
1
0
3cm
Cylinder
(circular Prism)
10cm
=
1
8
π
+
6
0
π
Surface Area
Surface Area
of a Cylinder
of a Cylinder
S
u
r
f
a
c
e
A
r
e
a
=
2
π
r
2
+
2
π
r
h
=
7
8
π
c
m
Example
:
A net of a cylinder is given below.
Find the diameter of the tin and the total
surface area.
2
r
=
Surface Area
Surface Area
of a Cylinder
of a Cylinder
2
π
r
=
2
5
25cm
9cm
D
i
a
m
e
t
e
r
=
2
r
S
u
r
f
a
c
e
A
r
e
a
=
2
π
r
2
+
2
π
r
h
=
2
π
(
2
5
/
2
π
)
2
+
2
π
(
2
5
/
2
π
)
x
9
=
6
2
5
/
2
π
+
2
5
x
9
=
3
2
4
.
5
c
m
V
o
l
u
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e
=
A
r
e
a
x
h
e
i
g
h
t
The volume of a cylinder can be thought as being a pile
of circles laid on top of each other.
=
π
r
2
Volume of a Cylinder
Volume of a Cylinder
Cylinder
(circular Prism)
x
h
h
=
π
r
2
h
V
=
π
r
2
h
Example
: Find the volume of the cylinder below.
=
π
(
5
)
2
x
1
0
=
2
5
0
π
c
m
Volume of a Cylinder
Volume of a Cylinder
Other Simple Volumes
Composite volume
is simply volumes that are made up from basic volumes.
C
y
l
i
n
d
e
r
=
π
r
2
h
Learning Intention
Learning Intention
Success Criteria
Success Criteria
1.
To know what a composite
To know what a composite
volume is.
volume is.
1.
To calculate volumes for
composite shapes using
knowledge from previous
sections.
2.
Work out composite volumes
Work out composite volumes
using previous knowledge of
using previous knowledge of
basic prisms.
basic prisms.
3.
Answer to contain
Answer to contain
appropriate units and working.
appropriate units and working.
Volume of Solids
Volume of Solids
Prisms
Other Simple Volumes
Composite volume
is simply volumes that are made up from basic volumes.
C
y
l
i
n
d
e
r
=
π
r
2
h
Volume of a Solid
Q. Find the volume the composite shape.
Composite volume
is simply volumes that are made up from basic volumes.
Volume = Cylinder + half a sphere
h = 6m
Volume of a Solid
Q. This child’s toy is made from 2 identical cones.
Calculate the total volume.
Composite Volumes
are simply volumes that are made up from basic volumes.
Volume = 2
x
cone
r = 10cm
h = 60cm
Explore formulas for finding the area of different geometric shapes such as triangles, parallelograms, rhombuses, kites, and more. Learn how to calculate areas efficiently and apply them to solve problems with units. Utilize diagrams and examples to grasp the concepts effectively.
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Presentation Transcript
Volume of Solids Area of Any Triangle Area of Parallelogram Area of Kite & Rhombus Area of Trapezium Composite Area Volume & Surface Area Surface Area of a Cylinder Volume of a Cylinder Composite Volume Exam Type Questions
Simple Areas Definition : Area is how much space a shape takes up A few types of special Areas Any Type of Triangle Parallelogram Rhombus and kite Trapezium 2
Any Triangle Area Learning Intention Success Criteria 1. To know the formula for the 1. To develop a formula for the area of ANY triangle. area of ANY triangle. 2. Apply formula correctly. (showing working) 2. Use the formula to solve problems. 3. Answer containing appropriate units 3
Any Triangle Area h = vertical height Sometimes called the altitude h b 1 2 Area b h = 4
Any Triangle Area Example 2 : Find the area of the triangle. Altitude h outside triangle this time. 1 2 Area b h = 1 2 20 10cm Area 4 10 = Area cm 2 = 4cm 5
Any Triangle Area Pythagoras Theorem first ! Hint : Use Example 3 : Find the area of the isosceles triangle. www.mathsrevision.com a b b c 2 2 2 1 2 + + = = Area b h = 4 5 2 2 2 5cm 1 2 12 b b 5 9 4 2 2 2 Area 8 3 = = = 4cm 2 Area cm 2 = 8cm b b 9 = = 3 6
Parallelogram Area Important NOTE h h = vertical height b Parallelogram Area =b h 7
Rhombus and Kite Area Learning Intention Success Criteria 1. To know the formula for the 1. To develop a single formula for the area of ANY rhombus and Kite. area of ANY rhombus and kite. 2. Apply formulae correctly. (showing working) 2. Use the formula to solve problems. 3. Answer containing appropriate units
This part of the rhombus Area of a Rhombus is half of the small rectangle. d D Rectangle Area = (D d) 1 2 Rhombus Area= (D d) 9
Area of a Kite Exactly the same process as the rhombus d D Rectangle Area = (D d) 1 2 Kite Area= (D d) 10
Rhombus and Kite Area Example 2 : Find the area of the V shape kite. 1 2 = D d Kite Area ( ) 4cm 1 2 Area = (7 4) 7cm 2 Area = 14cm 11
Trapezium Area Learning Intention Success Criteria 1. To know the formula for the area 1. To develop a formula for the area of a trapezium. of a trapezium. 2. Apply formula correctly. (showing working) 2. Use the formula to solve problems. 3. Answer containing appropriate units 12
Trapezium Area Two triangles WXY and WYZ a cm X Y 1 1 Area 2 = 2b h Area 1 = 2a h 1 h cm 2 1 1 + Total Area = 2 2 a h b h Z W 1 2 b cm Trapezium Area = (a+b) h 13
Trapezium Area Example 1 : Find the area of the trapezium. 5cm 1 2 + Trapezium Area = ( ) a b h 4cm 1 2 + Trapezium Area = (5 6) 4 2 Trapezium Area = 22cm 6cm 14
Composite Areas Learning Intention Success Criteria 1. To show how we can apply basic area formulae to solve more complicated shapes. 1. To know the term composite. 2. To apply basic formulae to solve composite shapes. 3. Answer containing appropriate units 15
Composite Areas We can use our knowledge of the basic areas to work out more complicated shapes. Example 1 : Find the area of the arrow. l b = = 2 Rectangle Area = 3 4 12 cm 5cm 6cm 3cm 1 2 1 2 b h = 6 5 15 = 2 Triangle Area = cm 4cm 2 Total Area = 15 + 12 = 27cm 16
Composite Areas Example 2 : Find the area of the shaded area. Trapezium Area - Triangle Area 8cm 1 2 1 2 a b + Trapezium Area = ( ) h 11cm = (10 8) 11 99 + = 2 cm 4cm 1 2 1 4 11 = 2 = 22 cm Triangle Area = 2bh 10cm = 2 Shaded Area = 99 - 22 77cm 17
Summary Areas Rhombus and kite Any Type of Triangle 1 2 1 2 = D d Area ( ) Area b h = Trapezium Parallelogram = 1 2 Area = (a+b)h Area b h
Volume of Solids Prisms Learning Intention Success Criteria 1. To know the volume formula 1. To understand the prism formula for calculating volume. for any prism. 2. Work out volumes for various prisms. 3. Answer to contain appropriate units and working.
Volume of Solids Definition : A prism is a solid shape with uniform cross-section Hexagonal Prism Cylinder (circular Prism) Triangular Prism Pentagonal Prism Volume = Area of Cross section x length
Definition : A prism is a solid shape with uniform cross-section Q. Find the volume the triangular prism. Triangular Prism Volume = Area x length 3 = 20 x 10 = 200 cm 10cm 2 20cm
Volume of Solids Definition : A prism is a solid shape with uniform cross-section www.mathsrevision.com Q. Find the volume the hexagonal prism. 2 43.2cm Volume = Area x length 20cm Hexagonal Prism 3 = 43.2 x 20 = 864 cm
Net and Surface Area Triangular Prism Bottom 4cm 4cm FT BT Back 10cm 4cm Front 4cm 5 faces 3 congruent rectangles 2 congruent triangles 10cm This is a NET for the triangular prism.
Example Find the surface area of the right angle prism Working 1bh 2 Triangle Area = 2 = 2 x3 =6cm Rectangle 1 Area = l x b = 3 x10 =30cm Rectangle 2 Area = l x b = 4 x 10 =40cm Rectangle 3 Area = l x b = 5 x 10 =50cm 2 5cm 3cm 2 10cm 4cm 2 2 triangles the same 1 rectangle 3cm by 10cm 1 rectangle 4cm by 10cm 1 rectangle 5cm by 10cm Total Area = 6+6+30+40+50 = 132cm 2
Net and Surface Area The Cuboid 3cm Bottom Back LS 4cm RS 4cm Top 3cm 3cm 5cm Front 4cm 6 faces Top and bottom congruent Front and back congruent Left and right congruent 5cm This is a NET for the cuboid
Example Find the surface area of the cuboid Working Front Area = l x b 2 = 5 x 4 =20cm Top Area = l x b 2 = 5 x 3 =15cm 4cm Side Area = l x b 2 = 3 x 4 =12cm 3cm Total Area 5cm Front and back are the same Top and bottom are the same Right and left are the same = 20+20+15+15+12+12 = 94cm 2
Surface Area of a Cylinder Learning Intention Success Criteria 1. To explain how to calculate the surface area of a cylinder by using basic area. 1. To know split up a cylinder. 2. Calculate the surface area of a cylinder.
Surface Area of a Cylinder The surface area of a cylinder is made up of 2 basic shapes can you name them. Cylinder (circular Prism) 2 r Curved Area =2 rh Top Area = r2 Bottom Area = r2 h Roll out curve side Total Surface Area = 2 r2 + 2 rh
Surface Area of a Cylinder Example : Find the surface area of the cylinder below: 3cm Surface Area = 2 r2 + 2 rh 10cm = 2 (3)2 +2 x 3 x 10 = 18 + 60 = 78 cm Cylinder (circular Prism)
Surface Area of a Cylinder Diameter = 2r Example : A net of a cylinder is given below. Find the diameter of the tin and the total surface area. 2 r = 25 25 9cm 2r = 25cm Surface Area = 2 r2 + 2 rh = 2 (25/2 )2 + 2 (25/2 )x9 = 625/2 + 25x9 = 324.5 cm
Volume of a Cylinder The volume of a cylinder can be thought as being a pile of circles laid on top of each other. Volume = Area x height = r2 = r2h h x h Cylinder (circular Prism)
Volume of a Cylinder Example : Find the volume of the cylinder below. 5cm V = r2h 10cm = (5)2x10 = 250 cm Cylinder (circular Prism)
Other Simple Volumes Composite volume is simply volumes that are made up from basic volumes. r r h D h Cylinder (circular Prism) r Cylinder = r2h 4 3 Volume = r 3 1 3 Volume Cone = r h 2
Volume of Solids Prisms Learning Intention Success Criteria 1. To know what a composite 1. To calculate volumes for composite shapes using knowledge from previous sections. volume is. 2. Work out composite volumes using previous knowledge of basic prisms. 3. Answer to contain appropriate units and working.
Other Simple Volumes Composite volume is simply volumes that are made up from basic volumes. r r h D h Cylinder (circular Prism) r Cylinder = r2h 4 3 Volume = r 3 1 3 Volume Cone = r h 2
Volume of a Solid Composite volume is simply volumes that are made up from basic volumes. Q. Find the volume the composite shape. Volume = Cylinder + half a sphere 1 2 2 3 4 3 1 2 4 3 V= r h + ( r ) 2 3 = (2) 6 + ( (2) ) 2 3 16 3 88 3 r = 24 + (8) = 24 + = h = 6m 2m
Volume of a Solid Composite Volumes are simply volumes that are made up from basic volumes. Q. This child s toy is made from 2 identical cones. Calculate the total volume. Volume = 2 x cone 1 V = 2 r h 3 2 = (10) 3 2 r = 10cm h = 60cm 30 2 = 6283cm 3
