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MK · lesson 11 of 14 · 42 min

Quadrilaterals and Polygons

This core lesson builds the polygon skills that Mathematics Knowledge questions lean on. You start with what makes a figure a polygon and how regular, irregular, convex and concave shapes differ, then move to interior angle sums, one angle of a regular polygon, exterior angles and solving angle equations. The parallelogram family follows: rectangles, squares, rhombi, trapezoids (with a caution about definitions) and kites.

Reading · 42 minVideo · 9 minPractice questions (44)

Video lesson: Quadrilaterals and Polygons

Video transcript

0:00 Hey everyone and welcome to this explainer. Today we're going to absolutely empower you to tackle planar geometry without any math anxiety. And the best part, we're doing it entirely without a calculator. So whether you're gearing up for the ASVAB or you just really want to brush up on those fundamental math skills, this session is custom-built to make these concepts super approachable. We're going to rely on simple rules and mental math to find exactly what you need. All right, let's jump right in. So here is our blueprint for today. We'll quickly cover polygon and quadrilateral rules, move into perimeter and area, tackle some step-by-step examples, highlight the common traps, and then finish up with a quick test. Let's do this. Section one. Let's start with the absolute basics of polygons and how to handle their angles. Okay, so a polygon is just a flat, closed shape made of straight sides. Now knowing whether it's convex or concave is going to help you understand its interior angles. Convex polygons basically push outward. You know, think of a standard stop sign. Concave ones, on the other hand, have at least one angle that dips inward, kind of creating a dent.

1:03 It looks a bit like an arrowhead. Now here is the absolute crucial formula to remember. The sum of the interior angles equals the number of sides minus two times 180 degrees. Why? Well, if you draw lines from one single corner of any polygon to the others, you're basically just splitting it into triangles. And every single triangle holds exactly 180 degrees. Moving on to section two. Let's talk about our four-sided friends, the quadrilaterals. Within quadrilaterals, we have some specific families. A parallelogram has opposite sides that are parallel and equal, right? Rectangles and squares are just special versions of these. But I really want you to pay close attention to the rhombus, which has four equal sides, and the kite, which has two pairs of equal adjacent sides. They actually have their own unique diagonal rules that are going to be a huge shortcut for area calculations later. All right, section three. Now that we know the shapes, let's look at measuring their boundaries. That's the perimeter and the space they take up inside, which is the area. For parallelograms, area is just base times height.

2:08 But, and this is a big but, you absolutely must use the true perpendicular height. That's a straight drop from top to bottom. Never, ever use the slanted side for this calculation. It's a trap. Now, trapezoids might look a bit intimidating with those two different parallel bases, but the math is honestly beautifully simple. You just average your two parallel bases and multiply by the height. It's a super logical step that completely keeps you away from needing a calculator. Section four. Let's head into the workshop and see how this all builds into simple, mental arithmetic using exact examples from our source material. Example one. We need to find a missing quadrilateral angle, and we're given angles of 115, 64, and 93 degrees. Remember, a four-sided shape always totals 360 degrees. So, simply add the three you know to get 272 and subtract that from 360. Boom, your missing angle is exactly 88 degrees. See? Easy mental math. Okay, what if we need to find the number of sides for a regular polygon, and all they give us is a 160-degree interior angle?

3:12 Actually, scratch trying to use that messy interior formula backwards. Here's a massive shortcut. Find the exterior angle first. 180 minus 160 is 20 degrees. Since exterior angles always, always total 360, just divide 360 by 20. That easily tells you the shape has 18 sides. Now, for example three. This is a classic trap you're definitely going to see on the test. We've got an area of 72, a base of nine, and a slanted side of 10. Listen to me. Completely ignore that slanted side of 10. It's literally just there to trick you. Since the area 72 equals the base of nine times the true height, well, nine times what is 72? The true height is 8 meters. This next one is basically made for mental math. We have a trapezoid with bases of 7 and 13 and a height of 6. Just average those bases first. 7 plus 13 is 20, and half of that is 10. Now multiply that 10 by the height of 6, and you've got 60 square centimeters. Easy, right? Didn't even break a sweat.

4:15 Moving on to example five, the rhombus. We have diagonals of 14 and 10. Because the diagonals of a rhombus cross at perfect right angles, finding the area is super fast. It's just half the product of those diagonals. So multiply 14 by 10 to get 140, then cut it right in half. You land perfectly on 70 square inches. Sometimes they'll throw composite L shapes at you. Just remember, subtraction is a totally valid strategy here. Find the outer area of the whole 9 by 7 rectangle first, which is 63. Then simply subtract the 5 by 3 missing block, which is 15. 63 minus 15 leaves you with a final area of 48 square meters. All right, example 7. Questions about frames require a little extra care. Say a 20 by 30 mirror gets a 3-inch outer frame. That uniform border adds 3 inches to both the left and the right, as well as the top and the bottom. That means you're adding 6 inches to each dimension. So your new outer size is 26 by 36. Multiply those to get an outer area of 936 and subtract the 600 inner area.

5:19 The frame itself is 336 square inches. When it comes to scaling shapes, there is one vital rule you cannot forget. If similar shapes have a side ratio of 4 to 7, the area scales by the square of that side ratio. So square them both, and the area ratio is 16 to 49. To find the large area, you multiply the small area 48 by 49 over 16. Now look closely, 48 divided by 16 is exactly 3. So just multiply 3 by 49, and that takes you straight to 147 square feet. Finally, let's run that trapezoid formula backwards. We have an area of 90, a height of 6, and one base is 11. Here's the trick. Group the one half and the height of 6 right away. Half of 6 is 3, so 90 equals 3 times the sum of the bases. Divide 90 by 3, and you get 30. That means 11 plus our missing base has to equal 30. Making the missing base? Exactly, 19. Alright, section 5.

6:21 You've done the math. Now let's do a quick safety inspection of the most common pitfalls the exam writers are going to try to use against you. Always, always look out for these. First, never use the slanted side as your height. It's going to inflate your answer. Second, don't mix up your interior and exterior angles. Third, remember the one half in your trapezoid and diagonal formulas. Fourth, don't forget that frames double the border width on the total dimension. Fifth, scale areas by the ratio squared, not just the base ratio. And lastly, always make sure you convert your lengths to the same unit before you start multiplying. Okay, we've made it to section 6. It's time for the final test to prove you literally don't need a calculator to crush this material. I'm going to read the questions and then I'll pause for a moment so you can think it through. Here are two questions exactly as they appear in our source. Question one, what is the sum of the interior angles of a hexagon? A, 360 degrees, B, 1080 degrees, C, 540 degrees, or D, 720 degrees. And question two, three angles of a quadrilateral measure 88 degrees, 102 degrees, and 79 degrees.

7:31 What is the measure of the fourth angle? A, 91, B, 269, C, 271, or D, 81 degrees. Take a few seconds to lock in your answers. All right, let's check your work. For question one, the answer is D, 720 degrees. A hexagon has six sides, right? So six minus two is four, and four times 180 gives you 720. As for question two, the answer is A, 91 degrees. You know a quadrilateral's interior angles total 360. Add the three angles they gave you to get 269. Subtract that from 360, and you're left with 91. See how mastering these simple logical rules completely removes the intimidation of geometry? Next time you face a tough geometry question on the ASVAB, what simple shortcut will you spot first? Thanks for joining me on this explainer. You've got this.

Contents · 40 sections
  1. Learning Goals and ASVAB Application
  2. A Polygon Is a Closed Figure
  3. Regular, Irregular, Convex and Concave
  4. Quadrilateral Interior Angle Sum
  5. Polygon Interior Angle Sum
  6. Regular Polygon: One Interior Angle
  7. Exterior Angles of Convex Polygons
  8. Solving Angle Equations in Polygons
  9. The Parallelogram Family
  10. Rectangles
  11. Squares
  12. Rhombi
  13. Trapezoids and Definition Caution
  14. Kites
  15. Perimeter as Boundary Length
  16. Missing Perimeter Sides
  17. Rectangle and Square Area
  18. Parallelogram Area and True Height
  19. Trapezoid Area
  20. Area From Perpendicular Diagonals
  21. Regular Polygon: Apothem Formula
  22. Polygon Diagonals
  23. Composite Shapes: Add Areas
  24. Composite Shapes: Subtract a Cutout
  25. Frames and Paths
  26. Similar Polygons: Perimeter and Area Scaling
  27. Unit Conversion Before Area
  28. Reading Diagrams Without Unjustified Assumptions
  29. No-Calculator Formula and Estimation Routine
  30. Worked Example 1 — Missing Quadrilateral Angle
  31. Worked Example 2 — Regular Polygon Angle
  32. Worked Example 3 — Parallelogram Height
  33. Worked Example 4 — Trapezoid
  34. Worked Example 5 — Rhombus Diagonals
  35. Worked Example 6 — Composite L Shape
  36. Worked Example 7 — Frame
  37. Worked Example 8 — Similarity and Area
  38. Worked Example 9 — Inverse Trapezoid Formula
  39. Common Polygon Traps
  40. Quick Review and Decision Tree
Visual lesson

Lesson slides

15 slides
Quadrilaterals and Polygons – slide 1 of 15

Learning Goals and ASVAB Application

In this lesson you will find the angle sum of any polygon, the size of one angle in a regular polygon, and the number of diagonals a polygon has. You will sort quadrilaterals into families by their sides, angles and diagonals.

You will also find perimeter and area for rectangles, parallelograms, trapezoids, rhombi and composite shapes, and see how similar figures scale. On the Mathematics Knowledge test these skills come up as short, no-calculator problems.

A Polygon Is a Closed Figure

A polygon is a flat shape made only of straight segments joined end to end, so the path closes up with no gaps. Each segment is a side, and each meeting point is a vertex. Sides may not cross each other.

A triangle, a stop-sign octagon and a home plate outline are polygons. A circle is not, because it has no straight sides, and an open zigzag is not, because it does not close.

Three shapes: a pentagon with a vertex and one side (highlighted) labelled is a polygon; a circle is not, because it is curved; an open zigzag is not, because it does not close.

Regular, Irregular, Convex and Concave

A regular polygon has all sides equal and all angles equal, like a square or the hexagon cells of a honeycomb. If either condition fails, the polygon is irregular. A rhombus has equal sides but usually unequal angles, so it is irregular.

A convex polygon has every interior angle below 180 degrees, so no corner points inward. A concave polygon has at least one angle above 180 degrees, giving it a dent, like an arrowhead.

A regular hexagon with equal sides and equal angles marked; a rhombus with four equal sides but unequal angles, so it is irregular; and a concave arrowhead with one angle over 180° marked.

Quadrilateral Interior Angle Sum

Draw one diagonal in any simple quadrilateral and it splits into two triangles. Each triangle holds 180 degrees, so the four interior angles always total 360 degrees.

Example: three angles are 70, 95 and 110 degrees. They add to 275, so the fourth angle is 360 minus 275, or 85 degrees.

Quadrilateral with angles 70°, 95°, 110° and 85°; a dashed diagonal splits it into two triangles of 180° each, so the angles total 360° and the fourth angle is 360 − 275 = 85°.

Polygon Interior Angle Sum

From one vertex of an n-sided polygon you can draw diagonals that cut it into n minus 2 triangles. That gives the interior angle sum:

For a heptagon (7 sides), the sum is 5 times 180, or 900 degrees.

Heptagon with dashed diagonals drawn from one vertex, cutting it into 5 numbered triangles: 5 × 180° = 900°.

Regular Polygon: One Interior Angle

In a regular polygon all angles are equal, so divide the total angle sum by the number of angles:

Here n is the number of sides. For a regular decagon, the sum is 8 times 180, or 1440 degrees, and one angle is 1440 divided by 10, or 144 degrees.

Exterior Angles of Convex Polygons

An exterior angle is formed by extending one side past a vertex. At each vertex the interior and exterior angles form a straight line, so they add to 180 degrees. Going once around any convex polygon, the exterior angles always total 360 degrees.

For a regular polygon with n sides, each exterior angle is

A regular 12-sided figure has exterior angles of 30 degrees and interior angles of 150 degrees.

Regular 12-sided polygon with one side extended: the interior angle is 150° and the exterior angle is 30°; 150° + 30° = 180° and 360° ÷ 12 = 30°.

Solving Angle Equations in Polygons

When angles are given as expressions, add them, set the total equal to the angle sum, and solve.

Example: a pentagon has angles x, x, 2x, 2x and 120 degrees. The sum must be 540, so 6x plus 120 equals 540, which gives 6x equal to 420 and x equal to 70. The angles are 70, 70, 140, 140 and 120.

Pentagon with angles x, x, 2x, 2x and 120°; the angles sum to 540°, so 6x + 120 = 540 and x = 70.

The Parallelogram Family

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. That one fact brings several others: opposite sides are equal, opposite angles are equal, neighboring angles add to 180 degrees, and the diagonals cut each other in half.

Rectangles, rhombi and squares are special parallelograms, so they inherit all of these properties. If one angle of a parallelogram is 65 degrees, the others are 115, 65 and 115.

Quadrilateral family tree: quadrilateral branches to parallelogram, trapezoid and kite; parallelogram to rectangle and rhombus; both lead to square, which keeps the properties of the shapes above it. Parallelogram with arrow marks on both pairs of parallel sides and angles 65°, 115°, 65° and 115°; neighboring angles add to 180°.

Rectangles

A rectangle is a parallelogram with four right angles. Besides every parallelogram property, its two diagonals are equal in length.

Opposite sides of a rectangle are equal, but neighboring sides need not be. A door panel 7 feet by 3 feet is a rectangle; its perimeter is 2 times (7 plus 3), or 20 feet.

A 7 ft by 3 ft rectangle with right angles at all four corners and two equal diagonals marked with matching ticks; P = 2 × (7 + 3) = 20 ft.

Squares

A square has four equal sides and four right angles, so it is both a rectangle and a rhombus. Its diagonals are equal, perpendicular and bisect each other. For side s:

A tile with 6-inch sides has a perimeter of 24 inches and an area of 36 square inches.

Square tile with 6 in sides, right angles at all four corners and equal tick marks on the sides; its dashed diagonals are equal, cut each other in half and cross at a right angle; P = 4 × 6 = 24 in, A = 6² = 36 in².

Rhombi

A rhombus is a parallelogram with all four sides equal, like a diamond on a playing card. Its angles do not have to be right angles.

The diagonals of a rhombus cross at right angles, bisect each other, and split each corner angle into two equal parts. If a rhombus has side 5 cm, its perimeter is 20 cm no matter how tilted it is.

Rhombus with four equal 5 cm sides; its dashed diagonals cross at a right angle and cut each other in half (tick marks), and each corner angle is split into two equal halves marked by matching arcs with ticks; perimeter 4 × 5 = 20 cm.

Trapezoids and Definition Caution

A trapezoid is a quadrilateral with a pair of parallel sides, called the bases. The other two sides are the legs. If the legs are equal, the figure is an isosceles trapezoid, and its base angles are equal in pairs.

Isosceles trapezoid with parallel top and bottom bases, equal legs marked with ticks, and equal base angles marked in pairs.

Be careful: some books say a trapezoid has exactly one pair of parallel sides, while others say at least one pair, which would count parallelograms as trapezoids. Test questions normally avoid this conflict, so work from what the problem states.

Kites

A kite has two pairs of equal sides, with each pair next to each other instead of opposite. Think of a classic toy kite: two short upper edges and two long lower edges.

The diagonals of a kite are perpendicular, and the main diagonal cuts the other one in half. The two angles between the unequal sides are equal. A kite is generally not a parallelogram.

Kite with two equal short upper sides and two equal long lower sides; the main diagonal crosses the other at a right angle and cuts it in half, and the two side angles are equal.

Perimeter as Boundary Length

Perimeter is the total distance around a figure, so you add the lengths of all its sides. It is measured in plain length units such as feet or meters, not square units.

Example: a lot with sides 40, 75, 52 and 90 feet has a perimeter of 257 feet. Fencing, trim and edging problems all ask for perimeter.

Missing Perimeter Sides

Figures made only of horizontal and vertical edges often hide some side lengths. The right-facing edges together must equal the left-facing edges, and the top edges must equal the bottom edges.

So a 10 by 6 rectangle with a rectangular notch cut from one corner still has a perimeter of 2 times (10 plus 6), or 32 units.

A 10 by 6 rectangle with a notch cut from its top right corner; arrows show the notch edges sliding out to fill the corner, so P = 2 × (10 + 6) = 32.

Rectangle and Square Area

Area counts how many unit squares cover a region. For a rectangle with length l and width w, and a square with side s:

A 9 m by 4 m deck covers 36 square meters, and a 5 m square covers 25 square meters.

A 9 m by 4 m rectangle gridded into 36 unit squares of 1 m² each; 9 × 4 = 36 m².

Parallelogram Area and True Height

Cut a triangle off one end of a parallelogram and slide it to the other end, and you get a rectangle. So the area is base times height, A = bh.

The height must be the perpendicular distance between the bases, not the slanted side. A parallelogram with base 10, slanted side 7 and height 5 has area 50, not 70.

Parallelogram with base 10, slanted side 7 and dashed perpendicular height 5; a triangle cut from one end slides to the other end to make a rectangle, so A = 10 × 5 = 50, not 10 × 7 = 70.

Trapezoid Area

A trapezoid's area is the average of its two parallel bases times the perpendicular height:

With bases 6 and 10 and height 4, the average base is 8, so the area is 32 square units.

Trapezoid with parallel bases 6 and 10, perpendicular height 4, and a dashed midline showing the average base 8; A = ½(6 + 10) × 4 = 32.

Area From Perpendicular Diagonals

When the diagonals of a quadrilateral cross at right angles, as in a rhombus, kite or square, the area is half the product of the full diagonals:

Diagonals of 10 and 6 give an area of 30 square units.

Rhombus with perpendicular diagonals 10 and 6 drawn inside a dashed 10 by 6 rectangle, filling half of it; A = ½ × 10 × 6 = 30.

Regular Polygon: Apothem Formula

The apothem of a regular polygon is the perpendicular distance from its center to a side. The area is half the apothem times the perimeter:

A regular hexagon with 8-inch sides has perimeter 48 inches and apothem about 6.93 inches, so its area is about 166 square inches.

Regular hexagon with 8-inch sides and a dashed apothem of about 6.93 inches from the center to a side; P = 48 and A ≈ ½ × 6.93 × 48 ≈ 166 square inches.

Polygon Diagonals

A diagonal joins two vertices that are not neighbors. Each of the n vertices connects to n minus 3 others, and each diagonal is counted twice, so

A hexagon has 6 times 3 over 2, or 9 diagonals; an octagon has 20.

Regular hexagon with all 9 diagonals; one vertex and its 3 diagonals (n − 3 = 3) are highlighted; D = 6 × 3 ÷ 2 = 9.

Composite Shapes: Add Areas

A composite shape is built from simpler pieces. Split it with straight cuts into rectangles, triangles or trapezoids, find each area, and add.

Example: a T-shaped sign is an 8 by 3 rectangle with a 4 by 4 square centered below it. The area is 24 plus 16, or 40 square units. Make sure the pieces do not overlap.

T-shaped sign: an 8 by 3 rectangle (24) on top of a 4 by 4 square (16) centred below it, split by a dashed cut; 24 + 16 = 40.

Composite Shapes: Subtract a Cutout

Sometimes it is easier to take the whole outer shape and remove the missing part. Area wanted equals outer area minus cutout area.

Example: a 10 by 7 yard lawn has a 2 by 3 yard shed on it. The grass covers 70 minus 6, or 64 square yards.

A 10 by 7 yard lawn with a hatched 2 by 3 yard shed on it; grass area 70 − 6 = 64 yd².

Frames and Paths

For a border of uniform width around a rectangle, find the outer rectangle, find the inner rectangle, and subtract. A border of width w outside the shape adds 2w to each dimension; a border inside removes 2w.

An 8 by 10 inch photo with a 2-inch frame outside becomes 12 by 14, so the frame area is 168 minus 80, or 88 square inches.

8 by 10 inch photo inside a 2-inch frame; the outer rectangle is 12 by 14 inches (10 + 2 + 2 = 14); 12 × 14 = 168, 8 × 10 = 80, frame = 88 square inches.

Similar Polygons: Perimeter and Area Scaling

Similar polygons have equal angles and proportional sides. If the side ratio is k, perimeters scale by k and areas scale by k squared:

With a side ratio of 2 to 3, a perimeter of 20 becomes 30, while an area of 20 becomes 20 times 9 over 4, or 45.

Similar squares with sides 2 and 3 (k = 3/2), gridded into 4 and 9 unit squares: perimeter scales by 3/2, area by 9/4.

Unit Conversion Before Area

Convert lengths to one unit before multiplying. Square units convert by the square of the length factor: since 1 yard is 3 feet,

A 1 yd by 1 yd square divided into a 3 by 3 grid of 1 ft squares, 3 ft on each side: 1 yd² = 9 ft².

A 15 ft by 12 ft room is 180 square feet, which equals 180 divided by 9, or 20 square yards.

Reading Diagrams Without Unjustified Assumptions

Test diagrams are often not drawn to scale. Do not assume an angle is 90 degrees, sides are parallel, or lengths are equal just because they look that way.

Trust only what is stated or marked: right-angle boxes, tick marks for equal sides, arrow marks for parallel lines, and given numbers. If a needed fact is not given, look for a property of the named shape, such as a parallelogram's equal opposite sides.

Diagram markings: a right-angle box means 90°, matching tick marks mean equal sides, and matching arrow marks mean parallel lines.

No-Calculator Formula and Estimation Routine

First name the shape and decide whether the question wants an angle, a length or an area. Write the matching formula, then plug in.

Keep numbers simple: halve an even factor before multiplying. Before choosing, check units and size; an area answer should be in square units, and an angle in a convex polygon must be under 180 degrees.

Worked Example 1 — Missing Quadrilateral Angle

Problem: A quadrilateral has angles of 115°, 64° and 93°. Find the fourth angle.

Step 1: The four angles of any quadrilateral add to 360°.

Step 2: Add the three known angles, two at a time.

Step 3: Subtract the known total from 360.

Answer: The fourth angle is 88°.

Check:

272 + 88 = 360

Worked Example 2 — Regular Polygon Angle

Problem: Each interior angle of a regular polygon is 160°. How many sides does it have?

Step 1: At each corner, the interior angle and the exterior angle sit on a straight line, so they add to 180°.

So each exterior angle is 20°.

Step 2: The exterior angles of any polygon add to 360°. In a regular polygon they are all equal, so divide 360 by one exterior angle.

Answer: The polygon has 18 sides.

Check: The interior angle sum of an 18-gon is

(18 − 2) × 180 = 16 × 180 = 2880
then
2880 ÷ 18 = 160

°, which matches.

Worked Example 3 — Parallelogram Height

Problem: A parallelogram has area 72 square meters, base 9 meters and slanted side 10 meters. Find its height.

Parallelogram with base 9 m, slanted side 10 m and a dashed perpendicular height h = ? with a right-angle box; Area = 72 m².

Step 1: The area of a parallelogram is base times the perpendicular height. The slanted side is not the height.

Step 2: Substitute the area and base.

Step 3: Divide both sides by 9 to undo the multiplication.

Answer: The height is 8 meters. The 10-meter slanted side is extra information and is not used.

Check:

9 × 8 = 72

the given area. Also the height (8) is shorter than the slanted side (10), as it must be.

Worked Example 4 — Trapezoid

Problem: A trapezoid has parallel bases of 7 cm and 13 cm and a height of 6 cm. Find its area.

Trapezoid with parallel bases 7 cm and 13 cm and a dashed perpendicular height 6 cm; Area = ?.

Step 1: The trapezoid area formula averages the two bases and multiplies by the height.

Step 2: Average the bases: add them, then take half.

Step 3: Multiply the average base by the height.

Answer: The area is 60 square centimeters.

Check: The area must be between a 7 × 6 rectangle (42) and a 13 × 6 rectangle (78). 60 is right between them.

Worked Example 5 — Rhombus Diagonals

Problem: A rhombus-shaped tile has diagonals of 14 inches and 10 inches. Find its area.

Rhombus tile with diagonals 14 in and 10 in crossing at a right angle inside a dashed 14 by 10 rectangle; Area = ?.

Step 1: The diagonals of a rhombus cross at right angles, so the area is half the product of the diagonals.

Step 2: Multiply the diagonals.

Step 3: Take half.

Answer: The area is 70 square inches.

Check: The rhombus fits inside a 14 by 10 rectangle (area 140) and fills exactly half of it, so 70 makes sense.

Worked Example 6 — Composite L Shape

Problem: A 9 by 7 meter rectangle has a 5 by 3 meter rectangle cut out of its upper right corner. Find the area of the L-shaped region that is left.

A 9 m by 7 m rectangle with a dashed 5 m by 3 m piece removed from its upper right corner; the L-shaped area left is marked 'Area = ?'.

Step 1: Find the area of the full rectangle.

Step 2: Find the area of the piece that was removed.

Step 3: Subtract the removed piece from the full rectangle.

Answer: The L-shaped region has an area of 48 square meters.

Check: Split the L into two rectangles instead. The bottom strip is 9 wide and

7 − 3 = 4

tall:

9 × 4 = 36

The upper left block is

9 − 5 = 4

wide and 3 tall:

4 × 3 = 12
then
36 + 12 = 48

again.

Worked Example 7 — Frame

Problem: A 20 by 30 inch mirror gets a frame 3 inches wide around the outside. Find the area of the frame.

A 20 by 30 inch mirror inside a frame 3 inches wide on every side; frame area = ?.

Step 1: Find the outer size. The frame adds 3 inches on both sides, so each dimension grows by

3 + 3 = 6 inches

Step 2: Find the outer area (mirror plus frame).

By hand:

26 × 36 = 26 × 30 + 26 × 6 = 780 + 156 = 936

Step 3: Find the mirror's area.

Step 4: The frame is what is left after removing the mirror.

Answer: The frame covers 336 square inches.

Check: Cut the frame into strips: two long strips

36 × 3 = 108

each (216 total) and two short strips

20 × 3 = 60

each (120 total). Then

216 + 120 = 336

Worked Example 8 — Similarity and Area

Problem: Two similar pentagons have sides in the ratio 4 to 7. The smaller one has area 48 square feet. Find the larger area.

Step 1: For similar figures, areas scale by the square of the side ratio.

Step 2: Multiply the smaller area by this factor.

Step 3: Divide first to keep the numbers small:

48 ÷ 16 = 3

Answer: The larger pentagon has area 147 square feet.

Check: The area ratio should be 16 to 49. Divide both areas by 3:

48 ÷ 3 = 16
and
147 ÷ 3 = 49

It matches.

Worked Example 9 — Inverse Trapezoid Formula

Problem: A trapezoid has area 90, height 6 and one base 11. Find the other base b.

Trapezoid with top base 11, bottom base b = ?, perpendicular height 6 and Area = 90.

Step 1: Write the trapezoid formula and substitute the known values.

Step 2: Simplify: half of 6 is 3.

Step 3: Divide both sides by 3.

Step 4: Subtract 11 from both sides.

Answer: The other base is 19.

Check:

1/2 × (11 + 19) × 6 = 1/2 × 30 × 6 = 15 × 6 = 90

Common Polygon Traps

Using the slanted side as the height. Area formulas for parallelograms and trapezoids need the perpendicular height. A slanted side is longer than the height, so using it inflates the answer. Look for the right-angle mark.

Mixing up interior and exterior angles. Exterior angles of a convex polygon sum to 360 degrees; interior sums depend on n. Remember each interior and exterior pair adds to 180.

Forgetting the one-half. The trapezoid and diagonal formulas both include a factor of one half.

Doubling the border width only once. A uniform border changes each dimension by twice its width.

Scaling area by k instead of k squared. Similar figures with side ratio k have area ratio k squared.

Mixing units. Convert lengths before multiplying, and report area in square units.

Quick Review and Decision Tree

Angles: interior sum is (n minus 2) times 180, one regular angle is that sum over n, and exterior angles total 360.

Lengths: perimeter adds all sides. Diagonal count:

Areas: rectangle lw, parallelogram bh, trapezoid half the base sum times h, rhombus or kite half of d1 times d2. Composite shapes are added or subtracted. For similar figures, area scales by k squared.

Quick review

Interior angle sum of an n-sided polygon?
, because diagonals from one vertex split the polygon into n − 2 triangles. A hexagon has 720°, and every quadrilateral has 360°.
Each interior and exterior angle of a regular n-gon?
Exterior ; interior , which equals 180° minus the exterior angle. Regular octagon: exterior 45°, interior 135°.
Given one interior angle of a regular polygon, how do you find n?
Subtract it from 180° to get the exterior angle, then divide 360° by that, since the exterior angles of a convex polygon total 360°. Interior 150° gives exterior 30°, so n = 12.
Number of diagonals in an n-sided polygon?
: each vertex connects to n − 3 non-neighbors, and each diagonal is counted twice. A pentagon has 5 diagonals; a decagon has 35.
What properties does every parallelogram have?
Opposite sides parallel and equal, opposite angles equal, neighboring angles summing to 180°, and diagonals that bisect each other. Rectangles add equal diagonals; rhombi add perpendicular diagonals; squares have both.
Area of a trapezoid?
: the average of the two parallel bases times the perpendicular height. Bases 5 and 9 with height 4 give 7 4 = 28 square units.
Area of a rhombus or kite from its diagonals?
, using the full lengths of the two perpendicular diagonals. Diagonals of 12 and 8 give 48 square units.
Parallelogram: base 12, slanted side 9, height 7. What is the area?
84 square units, from A = bh with the perpendicular height. Multiplying by the slanted side (getting 108) is the classic trap; a slanted side is always longer than the height.
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Check yourselfPractice questions44 questions, each with a full explanationRemember itFlashcards8 cards from this lessonNext lesson · 12 of 14Circles42 min
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About this lesson

The second half turns to measurement. You work through missing perimeter sides, parallelogram area with its true height, trapezoid area, area from perpendicular diagonals, the apothem formula, composite shapes built by adding or subtracting regions, frames and paths, and how perimeter and area scale for similar polygons. Nine worked examples, a list of common polygon traps and a decision tree tie it together, along with a no-calculator estimation routine that suits the MK subtest, which counts toward the AFQT.

Plan on about 42 minutes. When you finish, take the lesson test to see which formulas and angle rules have stuck before moving on to circles.

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