The document discusses different types of three-dimensional shapes studied in solid geometry. It provides definitions and examples of cubes, rectangular prisms, cylinders, spheres, cones, and pyramids. It also gives the formulas for calculating the volume and surface area of these shapes. For each shape, it provides examples of applying the formulas to solve volume and surface area problems.

1. Solid Geometry: http://www.onlinemathlearning.com/solid-geometry.html
What is solid geometry?
Solid geometry is concerned with three-dimensional shapes. Some examples of three-dimensional shapes are
cubes, rectangular solids, prisms, cylinders, spheres, cones and pyramids. We will look at the volume formulas
and surface area formulas of the solids. We will also discuss some nets of solids.
The following figures show some examples of shapes in solid geometry. Scroll down the page for more
examples, explanations and worksheets for each shape.
The following table gives the volume formulas and surface area formulas for the following solid shapes: Cube,
Rectangular Prism, Prism, Cylinder, Sphere, Cone, and Pyramid.

3. Cubes
A cube is a three-dimensional figure with six matching square sides.
The figure above shows a cube. The dotted lines indicate edges hidden from your view.
If s is the length of one of its sides, then the volume of the cube is s × s × s
Volume of the cube = s3
The area of each side of a cube is s2. Since a cube has six square-shape sides, its total surface area is 6 times
s2.
Surface area of a cube = 6s2
What is Volume?
The volume of a three-dimensional shape is a measurement of the space occupied by the shape.
Volume is measured in cubic units.
The volume of a unit cube
= 1 unit × 1 unit × 1 unit
= 1 unit3( Read as one cubic unit )
The volume of a cube with sides 1 cm × 1 cm × 1 cm

4. Volume = 1 cm × 1 cm × 1 cm = 1 cm3( Read as one cubic cm )
Some important units of conversion for volume are:
1 cm3 = 1,000 mm3
1 m3 = 1,000, 000 cm3
Volume of a Cube
A cube is a three-dimensional figure with six matching square sides.
The figure above shows a cube. The dotted lines indicate edges hidden from your view.
If s is the length of one of its sides, then the volume of the cube is s × s × s
Volume of the cube = s3
Since the cube has six square-shape sides, the
Surface area of a cube = 6s2
Surface Area of a Cube
A cube is a three-dimensional figure with six equal square faces.
The surface area of a cube is the sum of the area of the six squares that cover it.
The following figure shows the surface area of a cube. Scroll down the page for more examples and solutions

5. of finding the surface area of a cube.
If s is the length of one of its sides, then the area of one face of the cube is s2.
Since a cube has six faces the surface area of a cube is six times the area of one face.
Surface area of a cube = 6s2
Example
Find the surface area of a cube with a side of length 3 cm
Solution:
Given that s = 3
Surface area of a cube = 6s2 = 6(3)2 = 54 cm2
How to use the net of a cube to find its surface area?
Another way to look at the surface area of a cube is to consider a net of the cube. The net is a 2-dimensional
figure that can be folded to form a 3-dimensional object.
Imagine making cuts along some edges of a cube and opening it up to form a plane figure. The plane figure is
called the net of the cube.
The following net can be folded along the dotted lines to form a cube.

6. We can then calculate the area of each square in the net and then multiply the area by 6 to get the surface area
of the cube.
There are altogether 11 possible nets for a cube as shown in the following figures. Notice that the surface area
of each of the net is the same.
Surface Area of a Cube in terms of its Volume
How to find the surface area of a cube in terms of volume
S = 6V2/3
Rectangular Prisms or Cuboids
A rectangular prism is also called a rectangular solid or a cuboid.
In a rectangular prism, the length, width and height may be of different lengths.
The volume of the above rectangular prism would be the product of the length, width and height that is

7. Volume of rectangular prism = lwh
Total area of top and bottom surfaces is lw + lw = 2lw
Total area of front and back surfaces is lh + lh = 2lh
Total area of the two side surfaces is wh + wh = 2wh
Surface area of rectangular prism = 2lw + 2lh + 2wh = 2(lw + lh + wh)
Volume of a Rectangular Prism
A rectangular prism is a 3-dimensional object with six rectangular faces. All its angles are right angles and
opposite faces are equal.
In a rectangular prism, the length, width and height may be of different lengths. A rectangular prism is also
called a cuboid or rectangular solid. A cube is a special case of a cuboid in which all six faces are squares.
The volume of the above rectangular prism would be the product of the length, width and height that is
Volume of rectangular prism = lwh
Surface area of rectangular prism = 2(lw + wh + lh)
Example :
Find the volume of the following rectangular prism or cuboid.

8. Solution:
Volume = l × w × h
= 6 cm × 3 cm × 4 cm
= 72 cm3
Surface Area of a Cuboid or Rectangular Prism
A cuboid is a 3-dimensional object with six rectangular faces. All its angles are right angles and opposite faces
are equal. A cuboid is also called a rectangular prism or a rectangular solid.
In a cuboid, the length, width and height may be of different lengths. A cube is a special case of a cuboid in
which all six faces are squares.
To calculate the surface area of the cuboid we need to first calculate the area of each face and the add up all
the areas to get the total surface area.
Total area of top and bottom surfaces is lw + lw = 2lw
Total area of front and back surfaces is lh + lh = 2lh
Total area of the two side surfaces is wh + wh = 2wh
Surface area of cuboid = 2lw + 2lh + 2wh = 2(lw + lh + wh)
Volume of rectangular prism = lwh
Example
Find the surface area of the following cuboid.

9. Solution:
Total area of top and bottom surfaces is 2 × 5 × 6 = 60 in2
Total area of front and back surfaces is 2 × 5 × 3 = 30 in2
Total area of the two side surfaces is 2 × 6 × 3 = 36 in2
Surface area of cuboid = 60 + 30 + 36 = 126 in2
or
l = 6 in, w = 5 in and h = 3 in
Surface area of cuboid = 2(lw + lh + wh) = 2 (6 × 5 + 6 × 3 + 5 × 3) = 126 in2
Nets of a Cuboid
Another way to look at the surface area of a cuboid is to consider a net of the cuboid. The net is a 2-
dimensional figure that can be folded to form a 3-dimensional object.
Imagine making cuts along some edges of a cuboid and opening it up to form a plane figure. The plane figure
is called the net of the cuboid.
The following net can be folded along the dotted lines to form the cuboid.

10. We can then calculate the area of each rectangle in the net and add them up to get the surface area of the
cuboid.
Prisms
A prism is a solid that has two congruent parallel bases that are polygons. The polygons form the bases of the
prism and the length of the edge joining the two bases is called the height.
Triangle-shaped base Pentagon-shaped base
The above diagrams show two prisms: one with a triangle-shaped base called a triangular prism and another
with a pentagon-shaped base called a pentagonal prism.
A rectangular solid is a prism with a rectangle-shaped base and can be called a rectangular prism.
The volume of a prism is given by the product of the area of its base and its height.

11. Volume of prism = area of base × height
The surface area of a prism is equal to 2 times area of base plus perimeter of base times height.
Surface area of prism = 2 × area of base + perimeter of base × height
Prisms
A prism is a solid that has two parallel faces which are congruent polygons at both ends. These faces form the
bases of the prism. A prism is named after the shape of its base.
The other faces are in the shape of parallelograms. They are called lateral faces.
The following diagrams show a triangular prism and a rectangular prism.
A right prism is a prism that has its bases perpendicular to its lateral surfaces. If the bases are not
perpendicular to its lateral bases then it is called an oblique prism.
When we cut a prism parallel to the base, we get a cross section of a prism. The cross section has the same
size and shape as the base.

12. Example:
Volume of a Prism
The volume of a right prism is given by the formula:
Volume = Area of base × height = Ah
whereA is the area of the base and h is the height or length of the prism.
Worksheet to calculate volume of prisms and pyramids.
Example:
Find the volume of the following right prism.

13. Solution:
Volume = Ah
= 25 cm2 × 9 cm
= 225 cm3
Example:
Find the volume of the following right prism
Solution:
First, we need to calculate the area of the triangular base.

14. We would need to use Pythagorean theorem to calculate the height of the triangle.
h2 + 32 = 52
Area of triangle =
= × 6 × 4
= 12 cm2
Volume of prism = Ah
= 12 cm2× 8 cm
= 96 cm3
Surface area of prisms:
What is a Prism?
A prism is a solid that has two parallel faces which are congruent polygons at both ends. These faces form the
bases of the prism. A prism is named after the shape of its base. The other faces are in the shape of
parallelograms. They are called lateral faces.
The following diagrams show a triangular prism and a rectangular prism.

15. A right prism is a prism that has its bases perpendicular to its lateral surfaces.
When we cut a prism parallel to the base, we get a cross section of a prism. The cross section is congruent
(same size and shape) as the base, as can be seen in the following diagram.
How to calculate the surface area of a prism?
The surface area of a prism is the total area of all its external faces.
Step 1 : Determine the shape of each face.
Step 2 :Calculate the area of each face.
Step 3 : Add up all the areas to get the total surface area.
We can also use the formula
Surface area of prism = 2 × area of base + perimeter of base × height
Example:

16. Calculate the surface area of the following prism.
Solution:
There are 2 triangles with the base = 4 cm and height = 3 cm.
Area of the 2 bases
= 12 cm2
1 rectangle with length = 7 cm and width = 5 cm
Area = lw = 7 × 5 = 35 cm2
1 rectangle with length = 7 cm and width 3 m
Area = lw = 7 × 3 = 21 cm2
1 rectangle with length = 7 cm and width 4 m
Area = lw = 7 × 4 = 28 cm2

17. The total surface area is 12 + 35 + 21 + 28 = 96 cm2
We can also use the formula
Surface area of prism = 2 × area of base + perimeter of base × height
= 2 × 6 + (3 + 4 + 5) × 7 = 96 cm2
Example:
The diagram shows a prism whose base is a trapezoid. The surface area of the trapezoidal prism is 72 cm2.
Find the value of x.
Solution:
There are 2 rectangles with length = 5 cm and width = 3 cm
Area = 2 × 5 × 3 = 30 cm2
There is one rectangle with length = 5 cm and width = 4 cm
Area = 5 × 4 = 20 cm2
There is one rectangle with length = 5 cm and width = 2 cm
Area = 5 × 2 = 10 cm2
There are two trapezoids.
Area = cm2 = 6x cm2
Sum of area
30 + 20 + 10 + 6x = 72
60 + 6x = 72
x = 2
The value of x is 2.

18. Cylinders
A cylinder is a solid with two congruent circles joined by a curved surface.
In the above figure, the radius of the circular base is r and the height is h. The volume of the cylinder is the
area of the base × height.
The net of a solid cylinder consists of 2 circles and one rectangle. The curved surface opens up to form a
rectangle.

19. Surface area = 2 × area of circle + area of rectangle
Surface area of cylinder = 2πr2 + 2πrh = 2πr (r + h)
How to find the Volume of Solid Cylinders?
A cylinder is a solid with two congruent circles joined by a curved surface.
In the above figure, the radius of the circular base is r and the height is h.
The volume of the cylinder is the area of the base × height. Since the base is a circle and the area of a circle is
πr2 then the volume of the cylinder is πr2 × h.
Surface Area of cylinder = 2πr2+ 2πrh
Example:
Calculate the volume of a cylinder where:
a) the area of the base is 30 cm 2 and the height is 6 cm.
b) the radius of the base is 14 cm and the height is 10 cm.

20. Solution:
a)
b)
How to find the volume of Hollow Cylinders?
Sometimes you may be required to calculate the volume of a hollow cylinder or tube or pipe.
Volume of hollow cylinder:
= πR2 h – πr2 h
= πh (R2– r2)
Example:
The figure shows a section of a metal pipe. Given the internal radius of the pipe is 2 cm, the external radius is
2.4 cm and the length of the pipe is 10 cm. Find the volume of the metal used.

21. Solution:
The cross section of the pipe is a ring:
Area of ring = [ π (2.4)2– π (2)2]= 1.76 π cm2
Volume of pipe = 1.76 π × 10 = 55.3 cm3
Volume of metal used = 55.3 cm3

22. Spheres
A sphere is a solid with all its points the same distance from the center.
What is a sphere?
A sphere is a solid with all its points the same distance from the center. The distance is known as the radius of
the sphere. The maximum straight distance through the center of a sphere is known as the diameter of the
sphere. The diameter is twice the radius.
The following figure gives the formula for the volume of sphere. Scroll down the page for examples and
solutions.

23. How to find the volume of a sphere?
The volume of a sphere is equal to four-thirds of the product of pi and the cube of the radius.
The volume and surface area of a sphere are given by the formulas:
wherer is the radius of the sphere.
Example:
Calculate the volume of sphere with radius 4 cm.
Solution:
Volume of sphere
We can also change the subject of the formula to obtain the radius given the volume.
Example:
The volume of a spherical ball is 5,000 cm3. What is the radius of the ball?
Solution:

24. Volume of a hemisphere
What is a hemisphere?
A hemisphere is half a sphere, with one flat circular face and one bowl-shaped face.
How to find the volume of a hemisphere?
The volume of a hemisphere is equal to two-thirds of the product of pi and the cube of the radius.
The volume of a hemisphere is given by the formula:
wherer is the radius
Surface Area of a Sphere
A sphere is a solid in which all the points on the round surface are equidistant from a fixed point, known as
the center of the sphere. The distance from the center to the surface is the radius.

25. Surface area of a sphere is given by the formula
Surface Area of sphere = 4πr2
wherer is the radius of the sphere.
Example:
Calculate the surface area of a sphere with radius 3.2 cm
Solution:
Surface area of sphere
= 4π r2
= 4π (3.2)2
= 4 × 3.14 × 3.2 × 3.2
= 128.6 cm2
Surface Area of a hemisphere
A hemisphere is half a sphere, with one flat circular face and one bowl-shaped face.

26. The surface area of a hemisphere is equal to the area of the curve surface plus the area of the circular base.
Surface area of hemisphere
= × surface area of sphere + area of base
= × 4π r2 + π r2 = 3πr2
Example:
Calculate the surface area of a hemisphere with radius 4 cm
Solution:
Surface area of hemisphere
= × surface area of sphere + area of base
= × 4π r2 + π r2
= 3πr2
= 3π (4)2
= 150.86 cm2

27. Cones
A circular cone has a circular base, which is connected by a curved surface to its vertex. A cone is called a
right circular cone, if the line from the vertex of the cone to the center of its base is perpendicular to the base.
The net of a solid cone consists of a small circle and a sector of a larger circle. The arc of the sector has the
same length as the circumference of the smaller circle.

28. Surface area of cone = Area of sector + area of circle
= πrs + πr2 = πr(r + s)
Volume of a Cone
The following diagram shows the formula for the volume of a cone. Scroll down the page for more examples
and solutions on how to use the formula.
Cones
A cone is a solid with a circular base. It has a curved surface which tapers (i.e. decreases in size) to a vertex
at the top. The height of the cone is the perpendicular distance from the base to the vertex.

29. A right cone is a cone in which the vertex is vertically above the center of the base. When the vertex of a cone
is not vertically above the center of the base, it is called an oblique cone.
The following diagrams show a right cone and an oblique cone.
In common usage, cones are assumed to be right and circular. Its vertex is vertically above the center of the
base and the base is a circle. However, in general, it could be oblique and its base can be any shape. This
means that, technically, a pyramid is also a cone.
Volume of a Cone
The volume of a right cone is equal to one-third the product of the area of the base and the height. It is given
by the formula:
where r is the radius of the base and h is the perpendicular height of the cone.
Example:
Calculate the volume of a cone if the height is 12 cm and the radius is 7 cm.

30. Solution:
Volume
Surface Area of a Cone
A cone is a solid with a circular base. It has a curved surface which tapers (i.e. decreases in size) to a vertex
at the top. The height of the cone is the perpendicular distance from the base to the vertex.
The net of a solid cone consists of a small circle and a sector of a larger circle. The arc of the sector has the
same length as the circumference of the smaller circle.
The following figures show the formula for surface area of a cone. Scroll down the page if you need more
examples and explanations.
Surface area of cone = Area of sector + area of circle

31. Surface area of a cone when given the slant height
Example:
A cone has a circular base of radius 10 cm and a slant height of 30 cm. Calculate the surface area.
Solution:
Area = πr(r + s)
=
= 1,257.14 cm2
Pyramids
A pyramid is a solid with a polygon base and connected by triangular faces to its vertex. A pyramid is a
regular pyramid if its base is a regular polygon and the triangular faces are all congruent isosceles triangles.
Pyramids
A pyramid is a solid with a polygon base and connected by triangular faces to its vertex. The lateral faces meet
at a common vertex. The height of the pyramid is the perpendicular distance from the base to the vertex.

32. A pyramid is a regular pyramid if its base is a regular polygon and the triangular faces are all congruent
isosceles triangles. The pyramid is named after the shape of its base. A rectangular pyramid has a rectangle
base. A triangular pyramid has a triangle base.
A right pyramid is a pyramid in which the vertex is vertically above the center of the base. If the vertex is not
vertically above the center of the base then it is an oblique pyramid.
Volume Of Pyramids
The volume of a pyramid is equal to one-third the product of the area of the base and the height.
The volume of a pyramid is given by the formula:

33. Example:
Find the volume of a pyramid with a rectangular base measuring 6 cm by 4 cm and height 10 cm.
Solution:
Volume
= 80 cm3
Example:
The following figure is a right pyramid with an isosceles triangle base. Find the volume of the pyramid if the
height is 20 cm.
Solution:
First, we have to calculate the area of the base.

34. To do that, we would need to get the height of the isosceles triangle that forms the base.
Using Pythagorean theorem,
Area of triangle
=
= 108 cm2
Volume of pyramid
= 720 cm3

35. Nets Of A Solid
An area of study closely related to solid geometry is nets of a solid. Imagine making cuts along some edges of
a solid and opening it up to form a plane figure. The plane figure is called the net of the solid.
The following figures show the two possible nets for the cube.
How to calculate the volume of prisms, cylinders, pyramids and cones?
Volumes of Prisms and Cylinders = Area of Base × Height
Volumes of Pyramids and Cones = 1/3 × Area of Base × Height

38. http://www.onlinemathlearning.com/geometry-help.html
Geometry Topics
Angles Triangles Polygons
Circles Circle Theorems Solid Geometry
Geometric Formulas Coordinate Geometry & Graphs Geometric Constructions
Transformations Geometric Proofs Practice Questions
Videos have been included in almost all the following topics to help reinforce your understanding.

41. Free GRE Solutions and Explanations to Sample 1
Solutions and detailed explanations to questions and problems similar to the questions in the GRE test.
sample 1
1. A car covered 130 miles using 4 gallons of diesel. What distance would the same car cover, under
similar conditions, on 6.7 gallons?
Solution
The number of miles covered using 1 gallon is given by
(130 / 4) = 32.5 miles/gallon
Using 6.5 gallons, the car will cover
6.7 * 32.5 = 217.75 miles
2. If plotted in the same rectangular system of axis, the graphs of f(x) = | |x| - 4 | and g(x) = 2 will have
A) no points of intersection
B) 1 point of intersection
C) 2 points of intersection
D) 3 points of intersection
E) 4 points of intersection
Solution
Shown below is the graph of ||x|-4| by steps. First top left is the V-shaped graph of |x|, then below is
the graph of |x|-4 which is the graph of |x| shifted 4 units down. Top right is the graph of ||x| - 4| which
is the graph of |x|-4 where the part of the graph below the x axis is reflected on the x axis to make it
positive or zero. Finally the graph of ||x| - 4| and that of y = 2 shows that there are 4 points of
intersection.

42. .
3. Which of the following is the largest?
A) 125%
B) 1.25
C) 1 + 1/3
D) 4/3
E) 0.0015/0.001
Solution
Convert all given quantities into decimal numbers
A) 125% = 125/100 = 1.25
B) 1.25 = 1.25
C) 1 + 1/3 = 1.333...
D) 4/3 = 1.333...
E) 0.0015/0.001 = 1.5/1 = 1.5

43. The largest of all given quantities is 0.0015/0.001
4. | |- 10 - 19| - 20 | = ?
Solution
First calculate |- 10 - 19|
|- 10 - 19| = |-29| = 29
Substitute |- 10 - 19| by 29 in the given expression and calculate
| |- 10 - 19| - 20 | = |29 - 20| = |9| = 9
5. The algebraic expression x / (x + 2) is undefined if x =
Solution
Since division by 0 is not allowed in mathematics, the given expression is undefined for any value
that makes its denominator equal to 0. The denominator of the given expression is equal to 0 if
x + 2 =0 or x = - 2
The given expression is undefined if x = -2
6. (√5 - √7)(√5 + √7) = ?
Solution
We first expand the given expression using the identity (x - y)(x + y) = x2 - x2 and then simplify
(√5 - √7)(√5 + √7) = (√5)2 - (√7)2 = 5 - 7 = - 2
7. In the figure below the sides AB and AC of triangle ABC have equal lengths. Find the size of angle

44. ABC.
.
Solution
The size of angle BAC is equal to 46° since it is a vertical angle to the given angle of size 46°. We
now use the fact that the sum of all angles in a triangle is equal to 180°.
size of angle ABC + size of angle ACB + 46° = 180°
Since AB and AC have equal lengths, triangle ABC is isosceles and therefore angles ABC and ACB
are equal in size. Hence
2 (size of angle ABC) + 46° = 180°
size of angle ABC = (180 - 46) / 2 = 67°
8. If y = 10° in the figure below, what is the value of x?
.
Solution
The sum of all three angles in a triangle is equal to 180°. Hence

45. (4x + 4y) + (x + 3y) + (x + 2y) = 180
Group like terms
6x + 9y = 180
Substitute y by 10 in the equation and solve for x.
6x + 9(10) = 180
6x = 180 - 90 = 90
x = 15
9. By what percent will the volume of a rectangular solid increase if its length, width and height increase
by 25% each?
Solution
Let L, W and H be the length, width and height of the rectangular solid. The volume V is given by
V = L W H
When L, W and H are increased by 25%, they become
L = L + 25%L = L(1 + 25/100) = L(1 + 0.25) = 1.25 L
W = 1.25 W
H = 1.25 H
The volume V2 after the increase is given by
V2 = (1.25 L) (1.25 W) (1.25 H) = 1.253 L W H
Find the percentage P of increase of the volume as follows
P = (V2 - V) / V = (1.253 L W H - L W H) / L W H =
L W H (1.253 - 1) / L W H
= 1.253 - 1 = 0.95 (rounded to the nearest hundredth)
= 95%

46. 10. What is the average of all prime numbers between 20 and 40?
Solution
The prime numbers between 20 and 40 are
23, 29, 31 and 37
Their average is equal to
(23 + 29 + 31 + 37) / 4 = 30
Free GRE Practice Questions with Solutions
Sample 2
Solutions and detailed explanations to questions and problems similar to the questions in the GRE test.
sample 2
1. If w is the average (arithmetic mean) of the numbers a, b, c and d, then the average of m(a + k), m(b +
k), m(c + k) and m(d + k) is given by
Solution
w is the mean of a, b, c and d is written as
w = (a + b + c + d) / 4
w is the average W of m(a + k), m(b + k), m(c + k) and m(d + k) is given by
W = [ m(a + k) + m(b + k) + m(c + k) + m(d + k) ] / 4
W = m [ a + b + c + d + 4 k] / 4 = m [a + b + c + d ] / 4 + m k
= m (w + k)
If w is the average of a, b, c and d, then the average W of m(a + k), m(b + k), m(c + k) and m(d + k) is
given by

47. W = m (w + k)
2. What is the ratio of the area of the larger circle to the area of the smaller circle such that the radius of
the larger circle is three times the radius of the smaller circle?
Solution
Let r and R be the radii of the smaller and larger circles respectively. The radius of the larger circle is
three times the radius of the smaller circle leads to
R = 3r
Areas A1 of smaller and A2 of larger circles are given by
A1 = Pi r2
A2 = Pi R2 = Pi (3r)2 = 9 Pi r2
ratio R of areas larger / smaller is equal to
R = 9 Pi r2 / Pi r2 = 9
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3. A group of 20 employees in a company have an average (arithmetic mean) salary of $35,000 while a
second group of 30 employees in the same company have an average salary of $40,000. What is the
average salary of the 50 employees making the two groups?
Solution
Let S1 be the total salary of the group of 20 employess. Hence
35,000 = S1 / 20
S1 = 20 * 35,000 = $700,000
Let S2 be the total salary of the group of 30 employess. Hence
40,000 = S2 / 30
S2 = $1,200,000

50. The average of all 50 employers is given by
(700,000 + 1,200,000) / 50 = $38,000
4. Which of the following is equal to √48
A) 16
B) 3√4
C) 4√3
D) 18√3
E) 24
Solution
Rewrite the given expression using the fact that 48 = 3 × 16
√48 = √(3 × 16)
Use the formula √(a × b) = √a × √a to rewite √(3 × 16) as
√48 = √(3 × 16) = √3 × √16
= 4 √3
5. The sizes of the interior angles A, B and C of a triangle are in the ratio 2:4:3. What is the measure, in
degrees, of the smallest angle?
Solution
Since the three angles are in the ration 2:4:3, their sizes they may be written in the form
Size of A = 2 k , Size of B = 4 k and size of C = 3 k , where k is a constant.
The sum of the angles of a ny triangle is equal to 180°; hence
2 k + 4 k + 3 k = 180
Solve for k
9 k = 180 , k = 20
The smallest angle is A and its size is equal to 2 k

51. 2 k = 2 × 20 = 40°
6. If n is even and m is odd, then which of the following is true?
A) n + m is even
B) n - m is even
C) n * m is odd
D) n2 + m2 + 1 is even
E) 2n + 3m + 1 is odd
Solution
If n is even, it can be written as follows
n = 2 k , where k is an integer
If m is odd, it can be written as follows
m = 2 K + 1 , where K is an integer
We now express n + m in terms of k and K
n + m = 2 k + 2 K + 1 = 2(k + K) + 1
n + m is odd
We now express n - m in terms of k and K
n - m = 2 k - (2 K + 1) = 2 k - 2 K - 1
n - m = 2 (k - K) - 1
n - m is odd
We now express n * m in terms of k and K
n * m = (2 k)(2 K + 1) = 2( k(2K + 1) )
n * m is even
We now express n2 + m2 + 1 in terms of k and K
n2 + m2 + 1 = (2 k)2 + (2 K + 1)2 + 1 = 4 k2 + 4 K2 + 4 K + 1 + 1
= 2 ( 2 k2 + 2 K2 + 2 K + 1)

52. n2 + m2 + 1 is even
Statement D is true.
7. 5100 + 2550 + 3(12534 / 25) = Solution
Use the facts that 25 = 52 and 125 = 53 to rewrite the given expression as follows
5100 + 2550 + 3(12534 / 25) = 5100 + (52)50 + 3( (53)34 / (52))
Use formula for exponents to simplify
= 5100 + 5100 + 3( 5102 / 52)
= 5100 + 5100 + 3( 5100)
= 5 * 5100
= 5101
8. [ 6x10 - 2x9 ] / (9x2 - 1) =
Solution
Factor numerator as follows
6x10 - 2x9 = 2x9 (3x - 1)
Factor denominator as follows
9x2 - 1 = (3x - 1)(3x + 1)
Substitute numerator and denominator by their factored forms and simplify the given expression
[ 6x10 - 2x9 ] / (9x2 - 1) = [2x9 (3x - 1) ] / [(3x - 1)(3x + 1)]
= 2x9 / (3x + 1)

53. 9. (- 2x + 6)2 =
Solution
Expand by multiplication or using the identity (x + y)2 = x2 + 2 x y + y2.
(- 2x + 6)2 = (-2x)2 + 2 (-2x)(6) + 62
= 4 x2 - 24 x + 36
10. The sum of all interior angle of a regular polygon is 1800°. How many sides does this polygon have?
Solution
The sum of all interior angles of a polygon of n sides is given by.
(n - 2) * 180
and is equal to 1800°. Hence
(n - 2) * 180 = 1800
Solve for n
(n - 2) = 10
n = 12