SOLID MENSURATION: UNDERSTANDING THE 3D SPACE

RICHARD T. EARNHARTSOLID MENSURATION:UNDERSTANDING THE 3D SPACE1Chapter IPlane FiguresRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceIntroductionPoint, line, and plane are undefined terms in geometry. Using these undefined terms, other geometric figures are defined. Plane geometry is the study of geometric figures that can be drawn on a two-dimensional surface called plane. Figures that lie on a plane are called two-dimensional figures or simply plane figures. This chapter deals with different plane figures, and their properties, relations, and measurement. The most common plane figures are the polygons.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceSolid Mensuration: Understanding the 3-D SpacePolygonsA polygon is a closed plane figure formed by line segments.Parts of a PolygonThe side or edge of a polygon is one of the line segments that make up the polygon. Adjacent sides are pairs of sides that share a common endpoint.The vertices of a polygon are the end points of each side of the polygon. Adjacent vertices are endpoints of a side.A diagonal of a polygon is a line segment joining two non-adjacent vertices of the polygon.An interior angle is the angle formed by two adjacent sides inside the polygon.An exterior angle is an angle that is adjacent to and supplementary to an interior angle of the polygon.Side or EdgeVertexDiagonalInterior AngleExterior AngleA polygon may also be defined as a union of line segments such that:Each vertex is a common end point of two adjacent line segments;no two adjacent line segments intersect except at an endpoint; andno two segments with the same endpoint are collinear.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceTypes of Polygons1. Equiangular PolygonA polygon is equiangular if all of its angles are congruent.2. Equilateral PolygonA polygon is equilateral if all of its sides are equal.3. Regular polygonRegular polygons are both equiangular and equilateral.4. Irregular PolygonA polygon that is neither equiangular nor equilateral is said to be an irregular polygon.5. Convex Polygon Every interior angle is less than 180. If a line is drawn through the convex polygon, the line will intersect at most two sides.6. Concave PolygonA concave polygon has at least one interior angle that measures more than 180. If a line is drawn through a concave polygon the line mat intersect more than two sides.An example of a convex polygonAn example of a concave polygonRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceNaming PolygonsPolygons are named according to their number of sides. Generally, a polygon with n sides is called an n-gon. To form the name of polygons with 13 to 99 sides, begin with the prefix of the tens digit, followed by kai (the Greek word for and) and the prefix for the units digit.Number of Sides Name of Polygonnn-gon3triangle or trigon4quadlerateral or tetragon5pentagon6hexagon7heptagon8octagon9nonagon or enneagon10decagon11undecagon or hendecagon12dodecagon13tridecagon or triskaidecagon14tetradecagon or tetrakaidecagon15pentadecagon or pentakaidecagonRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceNumber of SidesName of Polygon16hexadecagon or hexakaidecagon17heptadecagon or heptakaidecagon18octadecagon or octakaidecagon19enneadecagon or enneakaidecagon20isosagon30triacontagon40tetracontagon50pentacontagon60hexacontagon70heptacontagon80octacontagon90enneacontagon100hectogon or hecatontagon1,000chiliagon10,000myriagon108megagon10100googolgonRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceSidesPrefixandSides(Ones Digit)Suffix20icosi or icosakai+1henagon30triaconta2digon40tetraconta3trigon50pentaconta4tetragon60hexaconta5pentagon70heptaconta6hexagon80octaconta7heptagon90enneaconta8octagon9enneagonRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceFor numbers from 100 to 999, form the name of the polygon by starting with the prefix for the hundreds digit taken from the ones digit, affix the word hecta, then follow the rule on naming polygons with 3 to 99 sides. However, one may use the form n-gon, as in 24-gon for a polygon with 24 sides, instead of using the above method.Example 1A 54-sided polygon is called a pentacontakaitetragon.50and4pentacontakaitetragonExample 2A 532-sided polygon is called a pentahectatriacontakaidigon.50030and2pentahectatriacontakaidigonRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceSimilar PolygonsThe ratio of two quantities is the quotient of one quantity divided by another quantity. Note, however, that the two quantities must be of the same kind. For example, the ratio of the measure of a side and an interior angle is meaningless because they are not quantities of the same kind. A proportion is an expression of equality between two ratios. That is, if two ratios a:b and c:d are equal, then the equation a/b=c/d is a proportion. Thus, you can say that a and b are proportional to c and d.

Two polygons are similar if their corresponding interior angles are congruent and their corresponding sides are proportional. Similar polygons have the same shape but differ in size.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceConsider the similar polygons below.y1y2x1x2A1A2The following relations between the two polygons are obtained using the concept of ratio and proportion:1. The ratio of any two corresponding sides of similar polygons are equal.

Richard T. EanhartSolid Mensuration: Understanding the 3D Space2. The ratio of the areas of similar polygons is the square of the ratio of any two corresponding sides.

3. The ratio of the perimeters of similar polygons is equal to the ratio of any of any two corresponding sides.

Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceProperties of a Regular PolygonA regular polygon of n sides can be subdivided into n congruent isosceles triangles, whose base is a side of the polygon. The common vertex of these triangles is the center of the polygon. sa/2Perimeter

To find a perimeter of a polygon, add the lengths of the sides of the polygon. Since regular polygons are equilateral, the formula in finding the perimiter of a regular polygon isP = ns,

Where n is the number of sides and s is the length of each side.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceCentral Angle

The angle that is opposite a side of a regular polygon is a central angle of the polygon. It is the angle formed by two lines drawn from the center of the polygon to two adjacent vertices. Regular polygons are equiangular. Thus, the measure of each angle is given by

Apothem

The altitude of the isosceles triangles that can be formed from a regular polygon is the apothem of the regular polygon. The apothem bisects the central angle and its opposite side. Thus, we can compute for the apothem as follows:

a

Solving for a,

Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceInterior Angle

In each isosceles triangle, the measure of the base angles can be denoted by , and each interior angle of the regular polygon by 2. Thus, the measure of each interior angle is solved as follows:

Thus,

Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceSum of Interior Angles

Since the number of sides equals the number of interior angles, then the sum of interior angles is n times the measure of the interior angle. Hence,

Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceThe formula for area of a regular polygon can be expressed in terms of its number of sides and the measure of one side as follows:Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceExample 3

Find the area of a regular nonagon whose sides measure 3 units. Determine the number of distinct diagonals that can be drawn from each vertex and the sum of its interior angles.

Solution:

A nonagon is a 9-sided polygon. Thus, n = 9. Given s = 3, solve as follows:

Area of the polygon:Number of diagonals:Sum of interior angles: Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceExercises1. Use the diagram below to answer questions (a) to (d)AIs the polygon convex or concave?How many diagonals can be drawn from vertex A?How many sides does the polygon have?What is the name of the polygonRichard T. EanhartSolid Mensuration: Understanding the 3D Space2. Use the diagram below to answer questions (a) to (d)AIs the polygon convex or concave?How many diagonals can be drawn from vertex A?How many sides does the polygon have?What is the name of the polygonRichard T. EanhartSolid Mensuration: Understanding the 3D Space3. Find the measure of an interior angle of a regular tridecagon.

4. That is the measure of an interior angle of a regular pentacontakaitrigon?

5. Find the sum of the interior angles of a regular trcontakaitetragon.

6. What is the sum of the interior angle of a regular icosagon?

7. Name each polygon with the given number of sides. Also, find the number of diagonal of each polygon.2418147653

8. Name each polygon with the given number of sides.3912782186

9. How many sides does each polygon have?IcosikaihenagonEnneacontakaidigonOctahectatetracontakaiheptagonRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceHow many sides does each polygon have? How many distinct diagonals can be drawn from a vertex of each polygon?TrihectatriacontakaitrigonPentacontakaioctagonHeptacontakaiheptagonThe number of diagonals of a regular polygon is 35. Find the area of the polygon if its apothem measures 10 centimitersThe number of diagonals a regular polygon is 65. Find the perimiter of the polygon if its apothem measures 8 inches.The Sum of the interior angles of a regular polygon is 1,260. Find the area of the polygon if its perimeter is 45 centimeters.The measure of an interior angle of a regular polygon is 144. Find the apothem if one side of the polygon measures 5 units.Find the number of sides of each of the two polygons if the total number of sides of the polygons is 13, and the sum of the number of diagonals of the polygons is 25.Find the number of sides of each of the two polygons if the total number of sides of the polygons is 15, and the sum of the number of diagonals of the polygon is 36.What is the name of a regular polygon that has 90 diagonals?What is the name of a regular polygon that has 135 diagonals?Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceFind the number of diagonals of a regular polygon whose interior angle measures 144Find the sum of the interior angles and the number of diagonals of a regular polygon whose central angle measures 6.The ratio of areas between two similar triangles is 1:4. If one side of the smaller triangle is 2 units, find the measure of the corresponding side of the other triangle.One side of a polygon measures 10 units. If the measure of the corresponding side of a similar polygon is 6 units, find the ratio of their areas. What is the area of the larger polygon if the area of the smaller polygon is 12 square units?A regular hexagon A has the midpoints of its edges joined to form a smaller hexagon B. This process is repeated by joining the midpoints of the edges of hexagon B to get a third hexagon C. What is the ration of the area of hexagon C to the area of hexagon A?What is the ratio of the area of hexagon B to the area of hexagon A in number 23?If ABCDE is a regular pentagon and diagonals EB and AC intersect at O, then what is the degree measure of angle EOC?Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceTrianglesThe most fundamental subset of polygons is the set of triangles. Although triangles are polygons with the least number of sides, these polygons are widely used in the field of mathematics and engineering. In this section, some important formulas which are used extensively in solving geometric problems will be introduced.Classification of Triangles According to SidesEquilateral a triangle with three congruent sides and three congruent angles. Each angle measures 60.Isosceles a triangle with two congruent sides and two congruent angles.Scalene a triangle with no congruent sides and no congruent angles.EquilateralIsoscelesScaleneRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceClassifications of Triangles According to AnglesRight a triangle with a right angle (90 angle).Oblique a triangle with no right angle.Acute a triangle with three acute angles (less than 90)Equiangular a triangle with three congruent angles. Each angle measures 60.Obtuse a triangle with one obtuse angle (more than 90 but less than 180Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceSimilar Triangles

Two triangles are similar if their corresponding sides are proportional. Similar triangles have the same shape but differ in size. Look at the similar triangles below.a1a2b1b2c1c2Since the two triangles are similar, then the relations that exist between two similar polygons also hold. Thus, it follows that:Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceParts of a TriangleA triangle has three possible bases and three possible vertices. Any of the three sides of a triangle may be considered as the base of the triangle. The angle opposite the base is called vertex angle. The two angles adjacent to the base are called base angles.

A line segment drawn from a vertex perpendicular to the opposite side is called altitude. The point of intersection of the altitudes of a triangle is called orthocenter. A median of a triangle is the line segment connecting the midpoint of a side and the opposite vertex. The centroid is the point of intersection of the medians of a triangle. An angle bisector divides an angle of the triangle into two congruent angles and has endpoints on a vertex and the opposite side. The point of intersection of the angle bisectors of a triangle is called incenter.OrthocenterIncenterCentroidAltitudesMediansACBB/2B/2C/2A/2A/2C/2Angle BisectorsRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceA perpendicular bisector of a side of a triangle divides the side into two congruent segments and is perpendicular to the side. The circumcenter is the point of intersection of the perpendicular bisectors of the sides of a triangle. The Euler line is the line which contains the orthocenter, centroid, and circumcenter of a triangle. The centroid is located between the orthocenter and the circumcenter. However, in an equilateral triangle, the centroid, circumcenter, incircle, and orthocenter are coincident. CircumcenterPerpendicular BisectorsOrthocenterCircumcenterCentroidEuler LineRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceProperties of Triangle Centers

Orthocenter The orthocenter is not always in the interior of the triangle. In an obtuse triangle, the two sides of the obtuse angle and the corresponding altitudes are extended to meet at a point outside the triangle. In a right triangle, the orthocenter is on a vertex of the triangle.Centroid The centroid is known as the center of mass of the triangle. Unlike the orthocenter, the centroid is always inside the triangle and for right, isosceles and equilateral triangles, the centroid is located one-third of the altitude from the base.Incenter The incenter is the center of the largest circle that can be inscribed in the triangle.Circumcenter The circumcenter is the center of the circle circumscribing a triangle. It is not always inside the triangle. The vertices of the triangle lie on the circle and are equidistant from the circumcenter.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceAltitude, Median, and Angle Bisector Formulas

Consider an arbitrary triangle with sides a, b, and c, and angles A, B, and C/ Let hc, mc and Ic be the lengths of the altitude, median, and angle bisector from vertex C, respectively. Then,CABbachcAltitude:CABbacmcMedian:Angle Bisector:CABbacmcRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceFacts About Triangles

The sum of the lengths of any two sides of a triangle is always greater than the third side. The difference between the lengths of any two sides is always less the third side of a triangle.The sum of the measures of the interior angles of a triangle is 180.Two equiangular triangles are similar.Two triangles are similar if their corresponding sides are parallel. Two triangles are similar if their corresponding sides are perpendicular.In any right triangle, the longest side opposite the right angle is called hypotenuse.If any two sides of a right triangle are given, the third side can be obtained by the Pythagorean Theorem c2=a2+b2.Two triangles are equal if the measures of the two sides and the included angle of one triangle are equal to the measures of the two sides and the included angle of the other triangle.The line segment which joins the midpoints of two sides of a triangle is parallel to the third side and equal to one-half the length of the third side.In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceThe altitude h to the hypotenuse c of a right triangle divides the triangle into two similar triangles. Each of the triangles formed by this altitude is similar to the original triangle.

Each leg of a right triangle is the geometric mean between the hypotenuse and the projection of the leg on the hypotenuse.hxc - xapcRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceFormulas for the Area of the Triangle

In general, the area of any triangle is one-half the product of its base and its altitude.To solve for the area of a triangle given the measures of two sides and an included angle, use the SAS formula.SAS (Side-Angle-Side) FormulaabThe area of a triangle is one-half the product of any two sides and the sine of their included angle.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceWhen the measure of the three sides of a triangle are given, the area of the triangle is determined by Herons Formula.Herons Formula or SSS (Three Sides) Formula:CABbacRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceExample 4

The measures of the three sides of a triangle are AB = 30 in., AC = 50 in., and BC = 60in.. From a point D on side AB, a line DE is drawn through a point E on side AC such that angle AED is equal to angle ABC. If the perimeter of the triangle ADE is equal to 56 in., find the sum of the lengths of line segments BD and CE.Solution:

Draw the figure and label the parts with the given measures.BAEDC603050Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceHence, BD + EC = 10 + 38 = 48 in.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceExample 5Derive formulas for the height and area of an equilateral triangle with side s.

Solution:In an equilateral triangle, the altitude divides the triangle into two congruent right triangles. Thus, by the Pythagorean Theorem,60shRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceExample 6If one side of a triangle is 20 units and the perimeter is 72 units, what is the maximum area that the triangle can have?Solution:Imagine the side of the length 20 units as the base of the triangle. Thus, the sum of the lengths of the other two sides is P 20 = 52 units. Since the area of the triangle is maximum when the height is also maximum, the triangle is isosceles and the two sides measure 26 units each. By Pythagorean Theorem,Hence, the area is26h102610Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceExample 7 Derive the formula for the median of triangle ABC drawn from vertex C to side AB using the Cosine Law.Solution:Draw and label the triangle.by the Cosine Law, you get:CABbaRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceEXERCISESIs it possible to form a triangle with sides 20, 30, and 50 units? Explain.Is it possible to form a triangle with sides 2, 4, and 8 units? Justify your answer.Find the altitude and the area of an equilateral triangle whose side is 8 cm long.One side of an isosceles triangle whose perimeter is 42 units measures 10 units. Find the area of the trianglesFind the area of an equilateral if its altitude is 5 cm.The ratio of the base of an isosceles triangle to its altitude is 3:4. Find the measures of the angles of the triangle.The base of an isosceles triangle and the altitude drawn from one of the congruent sides are equal to 18 cm and 15 cm, respectively. Find the length of the sides of the triangle.Two altitudes of an isosceles triangle are equal to 20 cm and 30 cm. Determine the possible measures of the base angles of the triangle.In a right triangle, the bisector of the right angle divides the hypotenuse in the ratio of 2 is to 5. Determine the measures of the acute angles of the triangle.The area of a triangle is equal to 48 cm2 and two of its sides measure 12 cm and 9 cm, respectively. Find the possible measures of the included angles of the given sides.The lengths of the sides of a triangle are in the ratio 17:10:9. Find the lengths of the three sides if the area of the triangle is 576 cm2Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceSuppose that AD, BC, AC and BD are line segments with line AD parallel to line BC as shown in the figure on the right. If AD = 3 units, BC = 1 unit, and the distance from AD to BC is 5 units, find the altitude of the smaller triangle. A3DB15CRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceWhat is the sum of the areas of the two triangles formed in number 16?2

ACEBDFIn triangle ABC, E is the midpoint of AC and D is the midpoint of CB. If DF is parallel to BE, find the length of side AB.ABCDEF345Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceThe measure of the base of an isosceles triangle is 24 cm, and one of its sides is 20 cm long. Find the distance between the centroid and the vertex opposite the base.

The two sides of a triangle are 17 cm and 28 cm long, and the length of the median drawn to the third side is equal to 19.5 cm. Find the distance from an endpoint of this median to the longest side.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceQUADRILATERALS

A quadrilateral, also known as tetragon or quadrangle, is a general term for a four-sided polygon. There are six types of quadrilaterals. They are square, parallelogram, rectangle, rhombus, trapezoid, and trapezium. Each type of quadrilateral has unique properties that make it distinct from other types. A square is the most unique quadrilateral because it possess all those unique properties.The common parts of a quadrilateral are described as follows:Side A side is a line segment which joins any two adjacent vertices.Interior angle An interior angle is the angle formed between two adjacent sides.Height or Altitude It is the distance between two parallel sides of a quadrilateral.Base This is the side that is perpendicular to the altitude.Diagonal This is the line segment joining any two non-adjacent vertices.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceClassification of QuadrilateralsThe classification of quadrilaterals is based on the number of pairs of its parallel sides as shown in the figure below.Classifications of Quadrilaterals

Parallelogram has two pairs of parallel sides.Trapezoid has only one pair of parallel sides.Trapezium does not have any pair of parallel sides.Rectangle, rhombus, and square are special types of parallelograms.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceGeneral Formulas for the Area of QuadrilateralsConsider the quadrilateral below.ABCDabcde1e2There are several useful formulas for the area of a planar convex quadrilateral in terms of sides a, b, c, and d, and diagonal lengths e1 and e2. Among them are the following:Formula 1: Formula 2:where the four sides are labeled such that a2 + c2 > b2 + d2.Formula 3:where s is the semi-perimeter and angles A and C are any two opposite angles of the quadrilateral.Richard T. EanhartSolid Mensuration: Understanding the 3D SpacePARALLELOGRAM

A parallelogram is a quadrilateral in which the opposite sides are parallel. The figure below illustrates an example of a parallelogram.ABCDb (base)h (height)Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceABCDhabdahRichard T. EanhartSolid Mensuration: Understanding the 3D SpacePerimeter of a Parallelogram

Opposite sides of a parallelogram are equal. Thus, its perimeter is given byArea of a ParallelogramThe area of a parallelogram can be obtained by any of the following formulas:Formula 1:

where b is the length of the base, and h is the height.Formula 2:where a and b are the lengths of the sides of the parallelogram and is any interior angle.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceKinds of ParallelogramThe next three quadrilaterals that will be discussed-rectangles, rhombuses, and squares-are all special types of parallelograms. You can classify each shape depending on the congruent sides and angles. Given a shape, you can work backwards to find out its sides or angles. Coordinate geometry is an effective way to measure the angles and the sidesRECTANGLEA rectangle is essentially a parallelogram in which the interior angles are all right angles. Since a rectangle is a parallelogram, all of the properties of a parallelogram also hold for a rectangle. In addition to these properties, the diagonals of a rectangle are equal. However, the sides are not necessarily all equal.ABCDhbRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceDiagonals of a RectangleA diagonal of a rectangle cuts the rectangle into two congruent right triangles. In the figure on page 26, the diagonal AC divides the rectangle ABCD into congruent right triangles ADC and ABC. Since the diagonal of the rectangle forms right triangles that include the diagonal and two sides of the rectangle, one can always compute for the third side with the use of the Pythagorean Theorem, if any two of these parts are given. Thus, the diagonal d=AC may be determined using the equationPerimeter of a RectangleThe perimeter is the sum of the four sides. Thus,Area of a RectangleIf b is the length of the base and h is the height, then the formula for the area of a rectangle isA=bhRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceSQUAREA square is a special type of a rectangle in which all the sides are equal. Since all sides and interior angles are equal, a square is classified as a regular polygon of four sides.adawhere a is the length of one side of the square. Note that, if the length of the diagonal is given, one can always compute for the length of the sides of the square using the same formula.Richard T. EanhartSolid Mensuration: Understanding the 3D SpacePerimeter of a Square

Since all the sides of a square are equal, it is also possible to provide a simple formula for the perimeter of the square. Thus, the simplified form of the perimeter is Area of a square

The formula for the area of a square is given byRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceRHOMBUS

A rhombus is a parallelogram in which all sides are equal.bd1d2hA rhombus may also be defined as an equilateral parallelogram. The terms rhomb and diamond are sometimes used instead of rhombus. A rhombus with an interior angle of 45 is sometimes called a lozenge.The Diagonal of a RhombusbhRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceAlso, the diagonals of the rhombus are angle bisectors of the vertices. By the Cosine Law, the diagonals may be obtained in a similar manner like that of a parallelogram. Thus,andOne can also verify that the angle opposite the shorter diagonal d1, may be obtained by the formulawhere d2 is the longer diagonal and is the angle opposite the shorter diagonal. The Perimeter of a Rhombus

If b is the measure of one side of a rhombus, then the perimeter is given by Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceArea of a Rhombus

The area of a rhombus may be determined by any of the following ways: The area is one-half the product of its two diagonals.Note that this expression follows from Formula 1 for the area of quadrilateral, where =90

Since a rhombus is a parallelogram, the area is also the product of the base times the height.The area is twice the area of one of the two congruent triangles formed by one of its diagonals. This is the same method used in finding the area of a parallelogram.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceTRAPEZOID

A trapezoid is a quadrilateral with one pair of parallel sides.abhIn the trapezoid shown above, the parallel sides a and b are called bases and h is the height or the perpendicular distance between the two bases. If the non-parallel sides are congruent, the trapezoid is called an isosceles trapezoid. The base angles of an isosceles trapezoid are also congruent. One can observe that the relationship among the sides, height, and base angles of an isosceles trapezoid may be obtained from the right triangle formed by constructing a line from one vertex perpendicular to the opposite side (lower base).Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceA trapezoid which contains two right angles is called a right trapezoid. The trapezoid on the right is an example of a right trapezoid.ab-ahbArea of a TrapezoidThe area of a trapezoid is equal to the product of the mean of the bases and the height. In symbols, the area is given by the formulaRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceTRAPEZIUM

A trapezium is a quadrilateral with no parallel sides. In finding the area of a trapezium, you may use any of the three formulas for the area of a quadrilateral.Example 8

Find the area and perimeter of a square whose diagonal is 15 units long.aa15Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceExample 9

The side of a square is x meters. The midpoints of its sides are joined to form another square whose area is 16 m2. Find the value of x and the area of the portion of the bigger square that is outside the smaller square. ABCRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceExample 10

If ABCD is a rhombus, AC=4, and ADC is an equilateral triangle, what is the area of the rhombus?

Solution:If ADC is an equilateral triangle, the then length of a side of the rhombus is 4, and angle ADC is 60.

Thus, the area of the rhombus isABCD4Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceExample 11Find the diagonal of the rectangle inscribed in the isosceles right triangle shown in the figure if the upper two vertices of the rectangle lie at the midpoints of the two legs of the triangle.ABCDEFG12mSolution:Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceExample 12Find the area and the perimeter of the right trapezoid shown in the figure.8116083608hhzRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceExample 13A vacant lot has the shape of a trapezium with sides 8m, 12m, 18m, and 20m. If the sum of the opposite angles is 230, find the area of the lot.8122018Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceABCDEd1208hRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceQaSccxh=4RRichard T. EanhartSolid Mensuration: Understanding the 3D SpacexxRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceEXERCISES

The diagonal of a rectangle is 25 meters long and makes an angle of 36 with one side of the rectangle. Find the area and the perimeter of the parallelogram.Determine the area of a rectangle whose diagonal is 24cm and the angle between the diagonals is 60.A side of a square is 16 inches. The midpoints of its sides are joined to form an inscribed square. Another square is drawn in such a way that its vertices would lie also at the midpoints of the sides of the second square. This process is continued infinitely. Find the sum of the areas of these infinite squares.A rectangle and square have the same area. If the length of the side of the square is 6 units and the longest side of the rectangle is 5 more than the measure of the shorter side, find the dimensions of the rectangle.Determine the sides of the rectangle if they are in the ratio of 2 is to 5, and its area is equal to 90cm2.Find the height of a parallelogram with sides 10 and 20 inches long, and an included angle of 35. Also, calculate the area of the figure.A certain city block is in the form of a parallelogram. Two of its sides measure 32 ft. and 41ft. If the area of the land in the block is 656ft.2,what is the length of its longer diagonal?The area of an isosceles trapezoid is 246m2. If the height and the length of one of its congruent sides measure 6m and 10m, respectively, find the lengths of the two bases.

Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceAn isosceles trapezoid has an area of 40m2 and an altitude of 2m. Its two bases have a ratio of 2 is to 3. What are the lengths of the bases and one diagonal of the trapezoid?A piece of wire of length 52m is cut into two parts. Each part is when bent to form a square. It is found that combined area of the two square is 109m2. Find the measures of the sides of the two squares.A rhombus has diagonals of 32 and 20 inches. Find the area and the angle opposite the longer diagonal.If you double the length of the side of a square, by how much do you increase the area of that square?If the diagonal length of a square is tripled, how much is the increase in the perimeter of that square?If the length and width of a rectangle are doubled, by what factor is the length of its diagonal multiplied?The area of the rhombus is 156m2. If its shorter diagonal is 13m, find the length of the longer diagonal.A garden plot is to contain 240 sq. ft. If its length is to be three time its width, what should its dimension be?The altitude BE of parallelogram ABCD divides the side AD into segments in the ratio 1:3. Find the area of the parallelogram if the length of its shorter side AB is 14cm, and one of its interior angle measures 60.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceRichard T. EanhartSolid Mensuration: Understanding the 3D SpaceChapter TestI. Completion of StatementsIf three sides of one triangle are equal respectively to three sides of another, the triangles are said to be _________.Corresponding parts of congruent triangles are ________.If the median of a triangle is also the altitude, the triangle is _______.In a right triangle, the side opposite the right angle is called _______.The _______ of a triangle is the line connecting a vertex and the midpoint of the opposite side of the triangle.The sum of the three angles in any triangle is _______.A triangle is _______ if it has two congruent altitudes.A regular polygon of three sides is called a/an _______.A regular polygon of four sides is called a/an _______.The sum of the measures of the angles in a quadrilateral in _______.A trapezoid is said to be a/an _______. If two of its angles measure 90.The intersection of the angle bisectors of a triangle is called _______.In an isosceles triangle, the _______ is located one-third of its altitude from the base.In naming of polygons, the word kai means _______.A quadrilateral with no parallel sides is called _______.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceII. True-False Statements______ 1. A line perpendicular to another line also bisects the line.______ 2. An equilateral triangle is also equiangular______ 3. The altitude of a triangle always passes through the midpoint of a side.______ 4. In an isosceles triangle, median to the base is perpendicular to the base.______ 5. The bisector of an angle of a triangle bisects the side opposite of a side.______ 6. The altitude of a triangle intersects the midpoint of a side.______ 7. The bisectors of two angles of a triangle are perpendicular to each other.______ 8. In an equilateral triangle, the altitude is a perpendicular bisector of the base.______ 9. In an equilateral triangle, the base angles are congruent.______ 10. In an isosceles triangle, all three angles are acute.______ 11. If the two diagonals of a quadrilateral are perpendicular, the quadrilateral is a parallelogram.______ 12.A parallelogram is a rectangle.______ 13. A square is a rectangle.______ 14. An isosceles trapezoid has two congruent sides.______ 15. The two diagonals of a rhombus bisects each other at right angles.Richard T. EanhartSolid Mensuration: Understanding the 3D SpaceIII. Place a check mark under the name of each figure that satisfies the given property.PropertyParallelogramRectangleSquareRhombusTrapezoidAll Sides are congruentBoth pairs of opposite sides are parallelBoth pairs of opposite sides are congruentDiagonals are congruentDiagonals bisect each otherDiagonals are perpendicularRichard T. EanhartSolid Mensuration: Understanding the 3D Space