Area Formulas
Figure
Area formula
Why does this formula make sense?
Rectangle
Square
Parallelogram
Triangle
Rhombus
(Alternative to the
parallelogram formula)
Trapezoid
Regular Polygon
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Surface Area & Volume Formulas
SURFACE AREA:
h=
VOLUME:
PRISM
B
B=
P=
h
LA =
TSA =
V=
B
CYLINDER
h=
r=
h
LA =
V=
TSA =
r
PYRAMID
l=
B=
h=
l
h
LA =
TSA =
B
V=
2
SURFACE AREA:
VOLUME:
CONE
l=
h=
r=
l
h
LA =
V=
r
TSA =
SPHERE
r=
r
TSA =
V=
3
9.2 & 9.3 Areas of Plane Figures (Day 1)
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End
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Day 2 Warm Up
The Heesch number of a shape in the plane is the maximum number of times that shape can be completely
surrounded by copies of itself.
Hexagonal Tiles
In the center of Memorial Park is a fountain in the shape of a hexagon. Workers are ringing the fountain
in hexagonal tiles. The first ring of tiles is made of black tiles. The next ring is made of white tiles. The
next black, and so on.
•
How many hexagonal tiles will the workers use in the fourth ring?
•
In all, the workers completed 15 rings. What color were the tiles in the fifteenth ring?
•
How many did they need for this ring?
•
How many rings can be made if the workers keep going?
•
Are the hexagons regular?
•
What information would you need to know to find the area of the hexagonal tiles?
Note: The Heesch number of a triangle, quadrilateral, regular hexagon, or any other shape that can tile the plane, is infinity.
Conversely, a shape with infinite Heesch number must tile the plane. The Heesch number of a circle is zero, because it can't
even be surrounded once by copies of itself without leaving some uncovered space.
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9.4 Area of Regular Polygons (Day 2)
Vocabulary:
radius of a polygon
apothem
central angle of a polygon
The polygon below is a regular hexagon. Complete the table below with EXACT values.
1)
r
a
s
A
radius
apothem
side
Area of polygon
4
2)
5 3
3)
6
4)
2 3
5) Find the exact area of a regular hexagon with perimeter 18 3 .
6) The figure to the right is a regular pentagon with radius 5.
(a) Find the measure of the central angle shown.
(b) Draw in the apothem, a.
(c) Use trigonometry to find the values of a and s.
(d) Find the area and perimeter of the pentagon.
5
s
7) Find the area of a regular pentagon with perimeter 60.
7
8) The octagon to the right is regular and has radius 1.
Follow the steps of #6 to find its area and perimeter.
9) Consider a regular 1000-sided polygon with radius 1.
Keep all answer accurate to 5 decimal places!
(a) One of the 1000 triangles is shown to the right.
Find the measure of the central angle.
(b) Draw in the apothem. Find the length of the apothem and side.
Remember to round to 5 decimals.
1
(c) Find the area of the triangle.
(d) Find the area of the entire polygon.
(e) If you got 3.1415, you’re correct! If you did not get 3.1415, try again.
Why is the answer so close to 𝜋? Explain in a complete sentence.
10) Find the area of a 20-sided polygon with apothem 6.
11) Write the formula for the area of a polygon using …
(a) the number of sides, n, the apothem, a, and the side length, s.
A=
(b) the apothem, a, and the perimeter of the polygon, p.
A=
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9.7 Areas of Similar Figures (Day 3)
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10.1 Prisms (Day 4)
Prisms are 3D figures that start with a pair of congruent and parallel polygons, called bases. When you connect corresponding
corners you get a prism.
Bases:
Lateral Faces:
Lateral Edge:
Altitude:
• An altitude is a segment between ________________________________.
• The length of the altitude is the _____________ of the prism.
Right Prism
• In a right prism each lateral edge is
to the bases.
Oblique Prism
• A prism that is not _________.
Prisms are named by the shape of their bases. Some names are…
• Triangular prism
• Isosceles triangular prism
• Right regular hexagonal prism
• Oblique pentagonal prism
Prism Area:
Surface area/Total area = Combined area of all the bases and faces.
Lateral Area = Combined area of the lateral faces only.
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Ex1 Consider the right equilateral triangular prism.
(The word “right” refers to the prism, not the triangle.)
(a) Find the lateral area.
(b) Find the area of the two bases.
(c) Find the total surface area.
Ex2 Find the Lateral Area of the right irregular octagonal prism below without
a calculator.
Fact: The lateral area of a right prism is
LA=
Prism Volume
Right rectangular prism
Volume =
Base Area =
Volume =
12
General prism:
Volume =
For Ex3, 4, and 5, assume lengths given are cm. Include units
with your answers.
Ex3 Consider the prism to the right.
(a) Find the combined area of the bases.
(b) Find the lateral area of the prism.
(c) Find the total surface area of the prism.
(d) Find the volume of the prism.
Ex4 Consider the prism to the right.
(a) Find the combined area of the bases.
(b) Find the lateral area of the prism.
(c) Find the total surface area.
(d) Find the volume of the prism.
Ex5 Consider the right regular hexagonal prism.
(a) Find the combined area of the bases.
(b) Find the lateral area of the prism.
(c) Find the total surface area.
(d) Find the volume of the prism.
End
13
Quiz Review (Day 5)
1. The area of a rhombus is 100 cm2. Find the length of each diagonal if one is twice as long as the
other.
2. A rectangle has perimeter 50 meters and area 144 m2. Find its length and width.
3. Find the area of the trapezoid below. Express your answer as an exact value.
12 in.
30°
30°
30 in.
4. A regular hexagon has area 96 sq. units. Find its apothem. Round your answer to 2 decimal places.
5. The diagonals of a parallelogram divide the parallelogram into four triangles. Explain why the
areas of these four triangles are all equal even though they are not all congruent.
6. ABCD is a parallelogram with area 48. Also, AE : BE = 2 : 1.
D
C
a. Explain why the area of I is 24.
I
III
A
E
V
b. Find the areas of II and III.
II
IV
B
c. Find the areas of IV and V.
F
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Quiz Review (continued)
The Smiths have a cozy little summer home on the Cape. After a rough winter they decided it was
time to paint. They plan to paint the entire outside of the house, except for the doors and windows
(shaded). All measurements are in feet.
(a) If a 1-gallon can of paint covers 100 ft2, and costs $10.50, how much will it cost to paint the house?
•
•
•
•
•
Keep each calculation accurate to at least two decimals.
The left side of the house looks the same as the right side (3 windows).
Do not forget the triangular parts of the roof.
You can’t buy 3.737054 cans. Round up to the nearest whole can.
BE ORGANIZED!!!
(b) After painting their house they decide to paint the outside of their above-ground lobster tank. If the
pool has a 12 ft radius and is 5 ft high, how much paint will they need?
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Geometric Probability (Day 5 cont.)
Find the probability that a point selected randomly from each figure below lies in the shaded portion.
Write each answer as a fraction and as a percent.
1.
2. A circle is centered at P
4. Each circle is inscribed in a square
(a)
(b)
7.
8.
5.
3. The circles are concentric.
6.
9. The triangle is equilateral.
End
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10.2 Pyramids (Day 7)
1. Do p. 365 Classwork #2-7
2.)
3.)
4.)
5.)
6.)
7.)
8.)
9.)
10.)
11.)
12.)
13.)
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10.3 Cylinders & Cones I (Day 8)
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20
10.3 Cylinders & Cones II (Day 9)
21
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22
10.4 Spheres (Day 10)
23
End
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10.5 Areas and Volumes of Similar Solids (Day 11)
25
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Ch. 9 & 10 Area & Volume REVIEW (Day 12)
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