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Geometry: Praxis I Exam (page 3)

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Updated on Jul 5, 2011

Area of a Triangle

Area is the amount of space inside a two-dimensional object. Area is measured in square units, often written as unit2. So, if the length of a triangle is measured in feet, the area of the triangle is measured in feet2.

A triangle has three sides, each of which can be considered a base of the triangle. A perpendicular line segment drawn from a vertex to the opposite base of the triangle is called the altitude, or the height. It measures how tall the triangle stands.

Area of a Triangle

It is important to note that the height of a triangle is not necessarily one of the sides of the triangle. The correct height for the following triangle is 8, not 10. The height will always be associated with a line segment (called an altitude) that comes from one vertex (angle) to the opposite side and forms a right angle (signified by the box). In other words, the height must always be perpendicular to (form a right angle with) the base. Note that in an obtuse triangle, the height can be outside the triangle, and in a right triangle the height is usually one of the sides.

Area of a Triangle

The formula for the area of a triangle is given by , where b is the base of the triangle, and h is the height.

Example

Determine the area of the following triangle.

Area of a Triangle

Volume Formulas

A prism is a three-dimensional object that has matching polygons as its top and bottom. The matching top and bottom are called the bases of the prism. The prism is named for the shape of the prism's base, so a triangular prism has congruent triangles as its bases.

Volume Formulas

Note: This can be confusing. The base of the prism is the shape of the polygon that forms it; the base of a triangle is one of its sides.

Volume is the amount of space inside a three-dimensional object. Volume is measured in cubic units, often written as unit3. So, if the edge of a triangular prism is measured in feet, the volume of it is measured in feet3.

The volume of ANY prism is given by the formula V = Abh, where Ab is the area of the prism's base, and h is the height of the prism.

Example

Determine the volume of the following triangular prism:

Volume Formulas

The area of the triangular base can be found by using the formula bh, so the area of the base is (15)(20) = 150. The volume of the prism can be found by using the formula V = Abh, so the volume is V= (150)(40) = 6,000 cubic feet.

A pyramid is a three-dimensional object that has a polygon as one base, and instead of a matching polygon as the other, there is a point. Each of the sides of a pyramid is a triangle. Pyramids are also named for the shape of their (non-point) base.

The volume of a pyramid is determined by the formula Abh.

Example

Determine the volume of a pyramid whose base has an area of 20 square feet and stands 50 feet tall.

Because the area of the base is given to us, we only need to replace the appropriate values into the formula.

The pyramid has a volume of 33 cubic feet.

Polygons

A polygon is a closed figure with three or more sides—for example, triangles, rectangles, and pentagons.

Polygons

Terms Related to Polygons

  • Vertices are corner points, also called endpoints, of a polygon. The vertices in the above polygon are A, B, C, D, E, and F, and they are always labeled with capital letters.
  • A regular polygon has congruent sides and congruent angles
  • An equiangular polygon has congruent angles.

Interior Angles

To find the sum of the interior angles of any polygon, use this formula:

    S = (x – 2)180°, where x = the number of sides of the polygon.

Example

Find the sum of the interior angles in this polygon:

Interior Angles

The polygon is a pentagon that has five sides, so substitute 5 for x in the formula:

    S = (5 – 2) × 180°
    S = 3 × 180°
    S = 540°

Exterior Angles

Similar to the exterior angles of a triangle, the sum of the exterior angles of any polygon equals 360 degrees.

Similar Polygons

If two polygons are similar, their corresponding angles are congruent and the ratios of the corresponding sides are in proportion.

Example

Similar Polygons

These two polygons are similar because their angles are congruent and the ratios of the corresponding sides are in proportion

Quadrilaterals

A quadrilateral is a four-sided polygon. Because a quadrilateral can be divided by a diagonal into two triangles, the sum of its interior angles will equal 180 + 180 = 360°.

Quadrilaterals

Parallelograms

A parallelogram is a quadrilateral with two pairs of parallel sides.

Parallelograms

In the figure, . Parallel lines are symbolized with matching numbers of triangles or arrows

A parallelogram has:

  • opposite sides that are congruent
  • opposite angles that are congruent (A = C and B = D)
  • consecutive angles that are supplementary (A + B = 180°, B + C = 180°, C + D = 180°, D + A = 180° )
  • diagonals (line segments joining opposite vertices) that bisect each other (divide each other in half)

Special Types of Parallelograms

  • A rectangle is a parallelogram that has four right angles
  • Rectangle

  • A rhombus is a parallelogram that has four equal sides.
  • Rhombus

  • A square is a parallelogram in which all angles are equal to 90 degrees and all sides are congruent. A square is a special case of a rectangle where all the sides are congruent. A square is also a special type of rhombus where all the angles are congruent
  • square

Diagonals of Parallelograms

Diagonals of Parallelograms

In this diagram, parallelogram ABCD has diagonals and that intersect at point E. The diagonals of a parallelogram bisect each other, which means that .

In addition, the following properties hold true:

  • The diagonals of a rhombus are perpendicular.
  • The diagonals of a rectangle are congruent.
  • The diagonals of a square are both perpendicular and congruent.

Example

In parallelogram ABCD, the diagonal = 5x + 10 and the diagonal = 9x. Determine the value of x.

Because the diagonals of a parallelogram are congruent, the lengths are equal. We can then set up and solve the equation 5x + 10 = 9x to determine the value of x.

5x + 10 = 9x Subtract x from both sides of the equation.
10 = 4x Divide 4 from both sides of the equation.
2.5 = x

Area and Volume Formulas

The area of any parallelogram can be found with the formula A = bh, where b is the base of the parallelogram, and h is the height. The base and height of a parallelogram is defined the same as in a triangle.

Note: Sometimes b is called the length (l) and h is called the width (w) instead. If this is the case, the area formula is A = lw.

Area and Volume Formulas

A rectangular prism (or rectangular solid) is a prism that has rectangles as bases. Recall that the formula for any prism is V = Abh. Because the area of the rectangular base is A = lw, we can replace lw for Ab in the formula giving us the more common, easier to remember formula, V = lwh. If a prism has a different quadrilateralshaped base, use the general prisms formula for volume.

Note: A cube is a special rectangular prism with six congruent squares as sides. This means that you can use the V = lwh formula for it, too.

Area and Volume Formulas

Circles

Terminology

A circle is formally defined as the set of points a fixed distance from a point. The more sides a polygon has, the more it looks like a circle. If you consider a polygon with 5,000 small sides, it will look like a circle, but a circle is not a polygon. A circle contains 360 degrees around a center point.

Terminology

  • The midpoint of a circle is called the center.
  • The distance around a circle (called perimeter in polygons) is called the circumference.
  • A line segment that goes through a circle, with its endpoints on the circle, is called a chord.
  • A chord that goes directly through the center of a circle (the longest line segment that can be drawn) in a circle is called the diameter.
  • The line from the center of a circle to a point on the circle (half of the diameter) is called the radius.
  • A sector of a circle is a fraction of the circle's area.
  • An arc of a circle is a fraction of the circle's circumference.

Circumference, Area, and Volume Formulas

The area of a circle is A = πr2, where r is the radius of the circle. The circumference (perimeter of a circle) is 2πr, or πd, where r is the radius of the circle and d is the diameter.

      Example
      Determine the area and circumference of this circle:

    Circumference, Area, and Volume Formulas

      We are given the diameter of the circle, so we can use the formula C = πd to find the circumference.
      C = πd
      C = π(6)
      C = 6π = 18.85 feet
      The area formula uses the radius, so we need to divide the length of the diameter by 2 to get the length of the radius: 6 ÷ 2 = 3. Then we can just use the formula.
      A = π(3)2
      A = 9π = 28.27 square feet.

Note: Circumference is a measure of length, so the answer is measured in units, whereas the area is measured in square units.

Area of Sectors and Lengths of Arcs

The area of a sector can be determined by figuring out what the percentage of the total area the sector is, and then multiplying by the area of the circle.

The length of an arc can be determined by figuring out what the percentage of the total circumference of the arc is, and then multiplying by the circumference of the circle.

      Example
      Determine the area of the shaded sector and the length of the arc AB.

    Area of Sectors and Lengths of Arcs

      Because the angle in the sector is 30°, and we know that a circle contains a total of 360°, we can determine what fraction of the circle's area it is: of the circle.
      The area of the entire circle is A = πr2, so A = π(4)2 = 16π.
      So, the area of the sector is square inches.
      We can also determine the length of the arc AB, because it is of the circle's circumference.
      The circumference of the entire circle is C = 2πr, so C = 2π(4) = 8π.
      This means that the length of the arc is inches.

A prism that has circles as bases is called a cylinder. Recall that the formula for the volume of any prism is V = Abh. Because the area of the circular base is A = πr2, we can replace πr2 for Ab in the formula, giving us V = πr2h, where r is the radius of the circular base, and h is the height of the cylinder.

Area of Sectors and Lengths of Arcs

A sphere is a three-dimensional object that has no sides. A basketball is a good example of a sphere. The volume of a sphere is given by the formula V = πr3.

    Example
    Determine the volume of a sphere whose radius is 1.5'.
    Replace 1.5' in for r in the formula V = πr3.
    V = πr3.
    V = π(1.5)3
    V = (3.375)π
    V = 4.5π ≈ 14.14
    The answer is approximately 14.14 cubic feet.
    Example
    An aluminum can is 6" tall and has a base with a radius of 2". Determine the volume the can holds.
    Aluminum cans are cylindrical in shape, so replace 2" for r and 6" for h in the formula V = πr2h.
    V = πr2h
    V = π(2)2(6)
    V = 24π ≈ 75.40 cubic feet
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