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References: Circle, Cone, Cross Section, Cube, Cuboid, Cylinder, Edge, Ellipse, Eulerâ€™s Formula, Face, Hemisphere, Plane, Polyhedron, Pyramid, Tetrahedron, Sphere, Vertex.
SOLVING AN EQUATION
References: Balancing an Equation, Equations, Linear Equation, Quadratic Equations.
This is a solid shape in which every point on its surface is the same distance from its center. The â€śsame distanceâ€ť is the radius of the sphere. A well-known example of a sphere is a ball. The earth is approximately a sphere. Any cross section of a sphere is a circle. In the figure, the large dot is at the center, and the length of the arrow is the radius (R) of the sphere.
The formula for the volume of a sphere of radius R is
Volume = fjri?3 The formula for the surface area of a sphere of radius R is
Area = 4ttR2 It is not possible to draw a net for a sphere.
Example 1. Find the volume of a soccer ball if its radius is 11.3 cm (to 1 dp).
V = ^7tR3 Formula for the volume of a sphere
V = | x it x 11.33 Substituting R = 11.3 cm
V = 6044 to nearest whole number Using the calculator value for tt
The volume of the soccer ball is 6044 cm3.
Example 2. What is the surface area of the earth if it has a diameter of 12,700 km?
Solution. The radius of the earth is half its diameter, so R = 6350 km. Write
A = 4ttR2 Formula for surface area of a sphere
A = 4 x it x 63502 Substituting R = 6350 km
A = 507,000,000 (to 3 sf) Using the calculator value for tt
The surface area of the earth is approximately 507 million km2, which is 5.07 x 108 in standard form.
A spiral is a curve traced out by a point that moves around a fixed point and at an ever-increasing distance from the fixed point (see the figure). It is the effect produced when
a thick piece of rubber or carpet is rolled up. Spirals occur in nature; for example, some seashells are in the shape of spirals.
References: Box and Whisker Graph, Quartiles, Range, Standard Deviation.
A square is a quadrilateral with each of its four angles equal to 90Â° and all four sides equal in length. The square has four axes of symmetry and has a rotational symmetry of order four (see figure a). The diagonals of the square bisect each other at right angles, forming four congruent right-angled isosceles triangles. The diagonals make angles of 45Â° with the sides of the square.
The length of a side of a square is one-fourth the length of its perimeter. The length of the side of the square is the square root of its area. For example, if the area of a square is 36 cm2, then the length of the side of the square is V36 = 6 cm.
Example. If the length of the diagonal of a square is 8 cm, find the length of a side of the square.
Solution. Let the length of each side of the square be x cm. The diagonal of the square divides it up into two right-angled isosceles triangles (see figure b). We can use the theorem of Pythagoras in one of these triangles:
x2 + x2 = 82 Pythagorasâ€™ theorem
2x2 =64 x2 + x2 = 2x2 and 82 = 64
x2 = 32
Dividing both sides of the equation by 2
SQUARE ROOT 421
x = V32
Taking the square root of both sides of the equation
x =5.66 (to 2 dp)
The length of one side of the square is 5.66 cm.
The construction of a square, which is similar to the construction of a rectangle, is explained under the entry Constructions.
References: Axis of Symmetry, Bisect, Constructions, Equiangular Triangle, Isosceles Triangle, Right Angle, Rotational Symmetry, Squaring, Pythagorasâ€™ Theorem.
This is the set of numbers that are the squares of the natural numbers. To square a number, we multiply the number by itself. For example, 52 = 5 x 5 = 25. The squares of the first eight natural numbers are as follows:
Natural numbers 1 2 3 4 5 6 7 8
Square numbers 12 = 1 22 =4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64
None of the square numbers is a prime number.
It is interesting to note that a square number n2 is equal to the sum of the first n odd numbers. For example, 62= 1+3 + 5-1-7-1-9 + 11.
References: Natural Numbers, Prime Number.
This is an operation that occurs frequently in mathematics, especially in the use of the theorem of Pythagoras. When we take the square root of a number (or a term) we undo the â€śsquaringâ€ť operation. We say that taking the square root is the inverse operation of squaring. For example, 52 = 25, and the square root of 25 is 5. The square root of 25 is written as V25- The square root of a number can also be written in index form: for example, ^/25 can be written as 251/2. Square roots of numbers can be found using a scientific calculator.
When we take the square root of terms in algebra that are expressed as indices we halve the index; for example: VP* = x4. The reason why this rule works is explained as follows:
Multiplying the powers using laws of indices