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# Geometry Study Guide for McGraw-Hill's ASVAB (page 3)

By Dr. Janet E. Wall
McGraw-Hill Professional

### Quadilaterals

A quadrilateral is a polygon with four sides and four angles. The sum of the four angles is always 360°.

Types of Quadrilaterals There are several different kinds of quadrilaterals. Each type is classified according to the relationships among its sides and angles. The square, rectangle, parallelogram, and rhombus are all types of quadrilaterals.

Parallelogram A parallelogram is a quadrilateral with both pairs of opposite sides parallel and congruent. The opposite angles are also congruent. Around the edge of the parallelogram, each pair of consecutive angles is supplementary; that is, their sum is 180°. Diagonal lines drawn from opposite vertices bisect each other (divide each other exactly in half), but the diagonals themselves are not equal in length.

Like triangles, quadrilaterals have bases and height. Any side of this parallelogram can be a base. Dashed line is a height of this parallelogram. The height is a line originating at a vertex and drawn perpendicular to the opposite base. The height forms two right angles where it meets the base.

Rhombus A rhombus is a parallelogram with four congruent sides. Opposite angles are also congruent.

Rectangle A rectangle is a parallelogram with four right angles. Diagonal lines drawn from opposite vertices of a rectangle bisect each other (divide each other exactly in half) and are equal in length.

Square A square is a rectangle with four congruent sides. Diagonal lines drawn from opposite vertices of a rectangle bisect each other (divide each other exactly in half) and are equal in length.

Trapezoid A trapezoid is a quadrilateral with only one pair of parallel sides. Like other quadrilaterals, it has bases and height. In the example below, dashed line CE is the height. Sides AB and CD are parallel, but sides AC and BD are not parallel.

### Circles

A circle is a closed figure with all points the same distance from a center. A circle with its center at point A is called circle A.

Parts of a Circle A chord is a line segment that has endpoints on a circle. A diameter is a chord that passes through the center of a circle. A radius is a line segment that connects the center of a circle and a point on the circle. Its length equals half the length of the diameter. In the figure below, A is the center of the circle. EF is a chord. BC is a diameter of the circle. AC and AB are each a radius of the circle.

An arc is two points on a circle and the part of the circle between the two points. In the figure below, CD is an arc of the circle. A central angle is an angle whose vertex is the center of a circle. In the figure below, CAD is a central angle. Its measure is 50°. The sum of the measures of the central angles in a circle is 360°.

### Perimeter and Area

The perimeter is the distance around a closed twodimensional figure. The area is the amount of surface a two-dimensional figure covers. Area is measured in square units such as square inches (in2) or square centimeters (cm2). A square inch is the area of a square with sides 1 inch long.

Finding the Perimeter of a Polygon To find the perimeter of a polygon, just add the length of each side to find the total.

Examples

ΔABC is an isosceles triangle. If side AB has a length of 25 cm and side BC has a length of 15 cm, what is the perimeter?

Since ΔABC is an isosceles triangle, side AB = side AC. So if side AB has a length of 25 cm, side AC also has a length of 25 cm. Thus the perimeter is 25 + 25 + 15 = 65 cm.

Finding the Area of a Polygon There are special formulas you can use to calculate the areas of various types of polygons. You will want to memorize these, as you will almost certainly be asked a question about area on the ASVAB.

Area of a Triangle The area (A) of a triangle is onehalf the base (b) multiplied by the height (h), or

Examples

If a triangle has a height that measures 30 cm and a base that measures 50 cm, what is its area?

b = 50 cm

h = 30 cm

Area of a Square or Rectangle The area (A) of a square or rectangle is its length (l) multiplied by its width (w), or A = lw

Examples

If this rectangle is 10 miles long and 6 miles wide, what is its area?

A = lw

A = (10)(6)

A = 60 square miles (60 mi2)

Area of a Parallelogram (Including a Rhombus) The area (A) of a parallelogram is its base (b) multiplied by its height (h), or A = bh.

Examples

If a parallelogram has a base of 6 meters and a height of 4 meters, what is the area?

A = bh

A = (6)(4)

A = 24 square meters (24 m2)

Area of a Trapezoid The area (A) of a trapezoid is one-half the sum of the two bases (b1 and b2) multiplied by the height (h), or

Examples

If a trapezoid has one base of 30 meters and another base of 60 meters, and its height is 20 meters, what is the area?

Finding the Circumference and Area of a Circle The circumference of a circle is the distance around the circle. The circumference (C) divided by the diameter (d) always equals the number π (pi). Pi is an infinite decimal, meaning that its decimal digits go on forever. When you use it to solve problems, you can approximate as π 3.14 or 22/7.

To find the circumference of a circle, use the formula C = πd.

Examples

If a circle has a radius of 3 inches, what is the circumference?

Since the diameter is twice the radius (2r), the diameter is 6 inches.

C = πd

C = 3.14(d)

C = 3.14(6)

C = 18.84 in.

To find the area of a circle, multiply π times the square of the radius: A = πr2

Examples

If a circle has a radius of 4 centimeters, what is its area?

A = πr2

A = 3.14(4)2

A = 3.14(16)

A = 50.24 cm2

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