6.G.A.1: Area of Triangles, Quadrilaterals and Polygons
In plain English: 6.G.A.1 is the Common Core grade 6 math standard that asks students to find the area of right triangles, other triangles, special quadrilaterals and other polygons. Students cut shapes into rectangles and triangles, or put shapes together into rectangles, instead of only memorizing formulas. They use these methods to solve real problems, such as painting a wall or covering a floor.
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Solve real-world and mathematical problems involving area, surface area, and volume. Also written as 6.G.1 · Official standard
Students find the area of triangles, special quadrilaterals and other polygons by changing them into shapes they already know. Area is the number of square units that cover a flat shape, and students already find it for rectangles by multiplying length by width. In this lesson they decompose a shape (cut it into smaller pieces, such as triangles and rectangles) or compose shapes (put pieces or copies together to make a rectangle). Two copies of a right triangle make a rectangle, so the triangle is half of it. A parallelogram can be cut and rearranged into a rectangle. A trapezoid, kite or rhombus can be cut into triangles and rectangles.
The lesson covers all four kinds of figures the standard names: right triangles, other triangles (including ones where the height falls outside the triangle), special quadrilaterals (parallelograms, rhombuses, kites and trapezoids) and other polygons, such as the front wall of a shed. Students then use these methods to solve real problems, such as how much paint or sod (rolls of grass) a surface needs. Side lengths are whole numbers, decimals or simple fractions, as expected in grade 6. Circles are not part of this standard; they come in grade 7.
Learning Objectives
By the end of this lesson, students will be able to:
Explain why the area of a right triangle is half the area of a rectangle with the same base and height
Find the area of any triangle by splitting it into right triangles or by composing it into a rectangle, including triangles whose height falls outside
Find the area of parallelograms, rhombuses, kites and trapezoids by rearranging or decomposing them into rectangles and triangles
Find the area of other polygons by adding or subtracting the areas of rectangles and triangles
Use these methods to solve real-world problems about covering, painting and planting surfaces
Prior Knowledge Required
Students should already be comfortable with:
Finding areas of rectangles and of shapes made of rectangles 3.MD.C.7
Using the area formula for rectangles in real problems 4.MD.A.3
Naming shapes by their sides and angles, such as right triangles and parallelograms 4.G.A.2
Finding the area of a rectangle with fraction side lengths 5.NF.B.4
Give each student a rectangle of centimeter grid paper, 8 cm long and 6 cm wide. Ask:
Warm-Up Prompt
"Cut your rectangle from one corner to the opposite corner. What is the area of the rectangle? What is the area of each triangle? How do you know without counting squares?"
The rectangle has an area of 8 × 6 = 48 square centimeters. The two triangles are the same size and shape (students can place one on top of the other to check), so each one is half: 48 ÷ 2 = 24 square centimeters. Introduce the words: a right triangle has one right angle (a square corner of 90°), and its two legs are the sides that meet at that right angle. Some students will count squares and get stuck on the partial squares along the cut. Point out that the two triangles match exactly, so each one gets half of the 48 squares, and counting and halving agree.
Direct Instruction20-25 minutes
Part 1: Right triangles and other triangles. The base of a triangle can be any side. The height is the distance from the base to the opposite corner, measured along a line that is perpendicular to the base (it meets the base at a right angle). A right triangle is half of a rectangle, so its area is (1/2) × base × height. Any other triangle can be split along its height into two right triangles, as in the second picture of Diagram 1. Each right triangle is half of its own rectangle, so the whole triangle is still (1/2) × base × height. The slanted sides are not the height. In an acute triangle (all three angles less than 90°), the height falls inside. In an obtuse triangle (one angle greater than 90°), the height from the top corner can fall outside, beyond the end of the base; then students compose a bigger right triangle and subtract the extra piece, as in Guided Practice Problem 1.
Right triangle as half a rectangle
A corner garden bed is a right triangle with legs of 6 ft and 4 ft. What is its area?
Equation: Two copies make a 6 ft by 4 ft rectangle: 6 × 4 = 24 sq ft, so the bed is 24 ÷ 2 = 12 sq ft
Other triangle, split by the height
A triangle has a base of 10 cm and a height of 5 cm. The height meets the base 4 cm from the left corner. What is its area?
Equation: (1/2) × 4 × 5 = 10 and (1/2) × 6 × 5 = 15, so 10 + 15 = 25 sq cm, the same as (1/2) × 10 × 5
Parallelogram, cut and slide
A parallelogram has a base of 9 cm, a height of 4 cm and slanted sides of 5 cm. What is its area?
Equation: Cut off the right triangle at one end and slide it to the other end: a 9 cm by 4 cm rectangle, so 9 × 4 = 36 sq cm (the 5 cm side is not used)
Trapezoid, decomposed
The floor of a small stage is a trapezoid. The back edge is 11 m, the front edge is 5 m, and the stage is 4 m deep. The two slanted ends match. What is the floor area?
Equation: Rectangle 5 × 4 = 20 plus two triangles (1/2) × 3 × 4 = 6 each: 20 + 6 + 6 = 32 sq m
Polygon in a real problem
The front wall of a shed is a rectangle 12 ft wide and 8 ft tall with a triangle on top that is 5 ft tall. One quart of paint covers about 100 sq ft. How much wall is there to paint, and how many quarts are needed?
Equation: 12 × 8 = 96 and (1/2) × 12 × 5 = 30, so 96 + 30 = 126 sq ft; 126 is more than 100, so buy 2 quarts
Part 2: Special quadrilaterals. A quadrilateral has four sides. A parallelogram has two pairs of parallel sides (sides that never meet, like train tracks). A rhombus is a parallelogram with four equal sides. A kite has two pairs of equal sides that are next to each other. A trapezoid has at least one pair of parallel sides, called its bases. Work through the third and fourth examples with the class, using Diagram 1 and Diagram 2:
Parallelogram: cut along the height and slide the triangle across. Nothing is added or lost, so the area is base × height.
Trapezoid: draw the heights from the ends of the short base. You get a rectangle in the middle and a triangle at each end. Add the three areas.
Rhombus and kite: the diagonals (segments that join opposite corners) cross at a right angle. Cut along one diagonal into two triangles, or along both into four right triangles, and add the areas.
Part 3: Other polygons. A polygon is a closed flat shape with straight sides, such as a pentagon (5 sides) or a hexagon (6 sides). To find its area, decompose it into rectangles and triangles that do not overlap, find each area, and add. Sometimes it is easier to compose a larger rectangle around the shape and subtract the pieces that are not part of it. The fifth example and the right side of Diagram 2 show the shed wall: a rectangle plus a triangle. Remind students to write square units (sq ft, sq cm) with every area.
Guided Practice15 minutes
Pairs solve four problems on grid paper, one at a time. They draw the shape, show their cuts in color, and write each piece's area inside it. After each problem, one pair shows a different way to cut the same shape.
Guided practice problems (answers are for the teacher)
Problem
Shape
Answer
1
A triangle with a base of 4 cm and a height of 6 cm. The height falls outside the triangle and meets the line of the base 3 cm past its end.
Big right triangle (1/2) × 7 × 6 = 21, minus small right triangle (1/2) × 3 × 6 = 9, gives 12 sq cm
2
A kite for a class project with crossing sticks (the diagonals) of 30 in and 20 in.
Two triangles, each with base 30 and height 10: 150 + 150 = 300 sq in
3
An L-shaped room: 14 ft by 11 ft with a 6 ft by 4 ft corner missing.
14 × 7 = 98 plus 8 × 4 = 32, so 130 sq ft (or 154 - 24 = 130)
4
A trapezoid with parallel sides of 8 m and 13 m, a height of 5 m and one square corner on each base (a right trapezoid).
Rectangle 8 × 5 = 40 plus triangle (1/2) × 5 × 5 = 12.5, so 52.5 sq m
Problem 1 is the hardest: the height is drawn outside the triangle, so students compose a bigger right triangle and subtract. Listen for students who multiply two slanted sides, or who forget the 1/2 for triangles.
Independent Practice10-15 minutes
Students solve five problems on their own and sketch each shape with its cuts. (1) The corner of a school lot is a right triangle with legs of 9 m and 7 m. Find its area. (31.5 sq m.) (2) A parallelogram has a base of 15 in, a height of 8 in and slanted sides of 10 in. Find its area. (15 × 8 = 120 sq in.) (3) A rhombus has diagonals of 12 cm and 9 cm. Cut it into four right triangles and find its area. (Each triangle has legs of 6 cm and 4.5 cm, so 13.5 × 4 = 54 sq cm.) (4) A triangle has a base of 5 1/2 ft and a height of 4 ft. Find its area. ((1/2) × 5 1/2 × 4 = 11 sq ft.) (5) A hexagon-shaped coffee table top is a rectangle 30 in by 24 in with a triangle on each 24 in side. Each triangle has a height of 12 in. Find its area. (720 + 144 + 144 = 1,008 sq in.)
Closure5 minutes
Exit ticket: (1) A triangle has a base of 12 cm and a height of 5 cm. Find its area, and explain in one sentence why you take half. (30 sq cm, because two copies of the triangle make a 12 by 5 rectangle, or its two right-triangle pieces are halves of two rectangles.) (2) A trapezoid has parallel sides of 6 in and 12 in and a height of 3 in. Find its area by cutting it into two triangles along a diagonal. ((1/2) × 6 × 3 = 9 and (1/2) × 12 × 3 = 18, so 27 sq in.)
Differentiation Strategies
For Struggling Students
Give shapes already drawn on centimeter grid paper, with the height drawn as a dashed line, so students can check each area by counting whole and half squares
Let students cut out two copies of a triangle and tape them into a rectangle before they use (1/2) × base × height
Provide a recording chart with columns for each piece (shape, base, height, area) and a final row for the total
For Advanced Students
Ask for two different ways to cut the same trapezoid, then explain why both give the same area
Ask students to draw three different triangles on grid paper that all have an area of 18 square units, including one whose height falls outside the triangle
Give a regular octagon drawn on grid paper and ask students to find its area by subtracting four corner triangles from a square
Assessment Guidance
What to Look For
Check that students use a height that is perpendicular to the base, not a slanted side, and that they can point to the right angle. Look for the 1/2 in every triangle area and for a clear sketch that shows where each shape was cut. When students decompose a polygon, the pieces should cover the shape with no gaps or overlaps, and the areas should be added (or subtracted for a missing corner) correctly. Every answer should have square units. In real-world problems, students should answer the question asked, such as how many quarts or bags to buy, not stop at the area.
02
Classroom Activities
3 Activities
1
Cut, Slide and Double on Grid Paper
20 minPairs
Each pair gets 6 shape cards on centimeter grid paper. For every shape, they cut, slide or double it to make a rectangle, then find the area in square centimeters. This activity shows where each area method comes from.
Shape Cards (6 cards)
Card A, right triangle: legs 8 cm and 3 cm (12 sq cm)
Card B, acute triangle: base 8 cm, height 5 cm (20 sq cm)
Card C, obtuse triangle: base 5 cm, height 4 cm, with the height outside (10 sq cm)
Card D, parallelogram: base 7 cm, height 3 cm (21 sq cm)
Card E, trapezoid: parallel sides 4 cm and 8 cm, height 3 cm (18 sq cm)
Card F, rhombus: diagonals 6 cm and 4 cm (12 sq cm)
Procedure
Cut out each shape. For triangles, cut out a second copy and tape the two copies into a rectangle or a parallelogram
For the parallelogram, cut along a height and slide the triangle to the other side
For the trapezoid and the rhombus, cut along lines you draw (heights or diagonals) and rearrange or add the pieces
Glue each result into a notebook and write the area of every piece and the total
Discussion Questions
Which two cards have the same area, even though they look different? (Cards A and F, 12 sq cm each)
Why did you need two copies of the triangles but only one copy of the parallelogram?
On Card C, where did you draw the height? How did you still make a rectangle?
Modification for Distance Learning
Share the cards as images in a drawing app. Students copy and move the pieces on screen, then post a screenshot of each rectangle with its area written on it.
2
School Garden Plan
20 minGroups of 3-4
Groups receive a printed plan of a school garden with four beds, drawn on grid paper where 1 square stands for 1 square foot. They decompose each bed, find its area, and decide how much seed to buy. Then they measure a real triangle taped on the floor.
Garden Beds (4 beds)
Bed 1, L-shaped: 10 ft by 8 ft with a 4 ft by 3 ft corner missing (68 sq ft)
Bed 2, triangle: base 9 ft, height 6 ft (27 sq ft)
Bed 3, trapezoid: parallel sides 6 ft and 10 ft, height 5 ft (40 sq ft)
Bed 4, pentagon: a 6 ft by 6 ft square with a triangle on one side, base 6 ft and height 3 ft (45 sq ft)
Procedure
One student draws the cuts, one computes each piece, and one checks with a different cut; roles rotate for each bed
Add the four areas to find the total planting area (180 sq ft)
One bag of wildflower seed covers 50 sq ft. Decide how many bags to buy and explain why you round the way you do (180 ÷ 50 = 3.6, so 4 bags)
Measure it: the teacher tapes a large triangle on the floor with masking tape. Measure one side as the base and the perpendicular height with a measuring tape, then find the area in square feet
Discussion Questions
Which bed has the largest area? (Bed 1, the L-shaped bed)
Ana split Bed 1 into two rectangles, 10 × 5 and 6 × 3. Ben subtracted the missing corner from 10 × 8. Do they get the same area? (Yes: 50 + 18 = 68 and 80 - 12 = 68)
For the floor triangle, did every group choose the same base? Did everyone get about the same area?
Challenge Variation
Groups redesign the garden so the total planting area is exactly 150 sq ft, using at least one triangle and one trapezoid, and trade plans with another group to check.
3
Design a Logo with a Target Area
15 minPairs
Each student designs a polygon logo on grid paper with an area of exactly 40 square units. The logo must use at least one triangle and at least one parallelogram or trapezoid. Partners then trade and check the area with a different decomposition.
Sample Design
An arrow: a rectangle 6 units long and 4 units tall, with a triangle on its right end. The triangle has a base of 8 units (the arrowhead edge) and a height of 4 units. Area: 6 × 4 = 24 and (1/2) × 8 × 4 = 16, so 24 + 16 = 40 square units.
Procedure
Sketch the logo with every corner on a grid point
On the back, show your cuts and the area of each piece
Trade with your partner. Your partner finds the area without looking at the back, using a different cut or a surrounding rectangle
If your totals differ, find the piece that caused the difference together
Discussion Questions
Did your partner cut your logo the same way you did? Did you still agree on the area?
When was it easier to subtract from a surrounding rectangle than to add pieces?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Right Triangle, Other Triangle and Parallelogram
Left: a right triangle with legs of 6 ft and 4 ft is half of a 6 by 4 rectangle, so its area is 12 sq ft. Middle: a triangle with a base of 10 cm and a height of 5 cm splits along its height into two right triangles of 10 and 15 sq cm. Right: cutting the end off a parallelogram and sliding it across makes a 9 by 4 rectangle. All shapes are drawn to scale.
Diagram 2: A Trapezoid and a Polygon, Decomposed
Left: the trapezoid stage floor splits into a 5 by 4 rectangle and two triangles with a base of 3 m and a height of 4 m, for 32 sq m in all. Right: the shed wall is a 12 by 8 rectangle plus a triangle with a base of 12 ft and a height of 5 ft, for 126 sq ft. Both are drawn to scale.
04
Homework Assignment
~30 min
6.G.A.1 Homework: Area by Composing and Decomposing
Directions: Sketch every shape and show where you cut it or what you added to it. Write the area of each piece and the total, with square units.
Part 1: Triangles (Problems 1-2)
A corner shelf is a right triangle with legs of 16 inches and 14 inches. Find its area. Draw the rectangle that the shelf is half of, and label its length and width.
(a) A triangular banner has a base of 3 1/2 feet and a height of 2 feet. Find its area. (b) A triangle has a base of 6 cm and a height of 11 cm, and the height falls outside the triangle. Find its area, and sketch how you would compose it into a bigger right triangle.
Part 2: Special Quadrilaterals (Problems 3-4)
(a) An angled parking space is a parallelogram. Its base along the aisle is 11 ft, its height is 18 ft, and its slanted sides are 19 ft. Find its area, and explain why 11 × 19 is not the area. (b) A rhombus-shaped tile has diagonals of 14 cm and 9 cm. Find its area by cutting it into triangles.
A deck is a trapezoid with parallel sides of 16 ft and 10 ft and a height of 12 ft. Find its area two ways: first by cutting it into two triangles along a diagonal, then by cutting it into a rectangle and triangles. (The slanted ends match.)
Part 3: Polygons in Real Problems (Problems 5-6)
The front of a doghouse is a rectangle 3 ft wide and 2 ft tall with a triangle on top that is 1 1/2 ft tall. The door is a rectangle 1 ft wide and 1 1/2 ft tall. How many square feet of the front need paint?
A floor tile is a regular hexagon (6 equal sides of 4 in). It can be cut into 6 matching triangles, each with a base of 4 in and a height of about 3.5 in. (a) Find the area of the tile. (b) Cut it another way, into a rectangle 4 in by 7 in and two triangles with a base of 7 in and a height of 2 in. Do you get the same area?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Triangles
Correct areas with a perpendicular height and the 1/2, and a sketch of the rectangle or composed triangle
One error, or correct areas without a sketch
Slanted side used as the height, or areas missing
Special Quadrilaterals
Parallelogram, rhombus and trapezoid areas correct, with the cuts shown and both trapezoid methods agreeing
One area incorrect or one method missing
Sides multiplied without a height, or work missing
Polygons
Pieces cover the shape with no gaps or overlaps, the door is subtracted, and both hexagon methods are compared
Correct pieces with one arithmetic error
Pieces overlap or leave gaps, or work missing
Units and Explanations
Every area has square units and every explanation answers the question asked
Some units or one explanation missing
No units and no explanations
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Sketch each shape on scrap paper if it helps. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A right triangle has legs of 10 cm and 7 cm. (The legs are the two sides that meet at the right angle.) What is its area?
Answer: C
Two copies of the triangle make a 10 cm by 7 cm rectangle with an area of 70 sq cm, so the triangle is half: (1/2) × 10 × 7 = 35 sq cm. Choice A is the whole rectangle: it forgets the 1/2. Choice B adds the legs, 10 + 7, which is a length, not an area. Choice D is 2 × (10 + 7) = 34, the perimeter of the rectangle (the distance around it).
Question 2 of 20 · Multiple Choice
Why is the area of a right triangle half the area of a rectangle with the same base and height?
Answer: A
A diagonal cuts a rectangle into two identical right triangles, so two copies of the triangle compose the rectangle and each triangle is half of it. Choice B is a true fact about lengths, but it says nothing about area. Choice C is false: the two pieces are the same size, which is exactly why each is half. Choice D is true, but sharing a corner does not explain the half.
Question 3 of 20 · Multiple Choice
A triangle has a base of 14 in and a height of 12 in. Its slanted sides are 13 in and 15 in, and none of its angles is a right angle. What is its area?
Answer: B
Use the base and the height, which are perpendicular: (1/2) × 14 × 12 = 84 sq in. You can check by splitting along the height: the base splits into 5 in and 9 in, and (1/2) × 5 × 12 + (1/2) × 9 × 12 = 30 + 54 = 84. Choice A forgets the 1/2. Choice C uses the 13 in slanted side as the height: (1/2) × 14 × 13 = 91. Choice D uses the 15 in slanted side: (1/2) × 14 × 15 = 105.
Question 4 of 20 · Multiple Choice
A triangle has a base of 6 m and a height of 5 m. One of its angles is greater than 90°, so the height falls outside the triangle: it meets the line of the base 3 m past the end of the base. What is its area?
Answer: D
The area is still (1/2) × base × height = (1/2) × 6 × 5 = 15 sq m. To see why, compose a big right triangle with a base of 6 + 3 = 9 m and a height of 5 m, (1/2) × 9 × 5 = 22.5, then subtract the small right triangle outside, (1/2) × 3 × 5 = 7.5: 22.5 - 7.5 = 15. Choice A forgets the 1/2. Choice B is the big right triangle, with nothing subtracted. Choice C is only the small triangle that was subtracted.
Question 5 of 20 · Multiple Choice
A parallelogram has a base of 12 cm, a height of 6 cm and slanted sides of 7.5 cm. What is its area?
Answer: A
Cut off the right triangle at one end and slide it to the other end: the parallelogram becomes a 12 cm by 6 cm rectangle, so the area is 12 × 6 = 72 sq cm. Choice B multiplies the base by the slanted side, 12 × 7.5 = 90. Choice C takes half, as if the shape were a triangle. Choice D is 2 × (12 + 7.5) = 39, the perimeter in cm, not an area.
Question 6 of 20 · Multiple Choice
Which rearrangement shows that the area of a parallelogram is base × height?
Answer: C
Cutting along the height and sliding the triangle to the other end makes a rectangle with the same base and height, and no area is added or lost. Choice A makes a new shape whose area is still unknown. Choice B is not a rearrangement, and the slanted side is not the height, so it gives too much area. Choice D leaves you with a triangle whose area you still need, so it does not show base × height.
Question 7 of 20 · Multiple Choice
A trapezoid has parallel sides of 8 m and 14 m and a height of 6 m. What is its area?
Answer: D
Cut along a diagonal into two triangles with the same height: (1/2) × 8 × 6 = 24 and (1/2) × 14 × 6 = 42, so 24 + 42 = 66 sq m. Choice A adds the two triangles without the 1/2: 48 + 84 = 132. Choice B uses only the longer side, 14 × 6, as if the shape were a rectangle. Choice C uses only the shorter side, 8 × 6.
Question 8 of 20 · Multiple Choice
A kite for a class project has two crossing sticks (the diagonals) that meet at a right angle. One is 22 in long and the other is 16 in long. What is the area of the kite?
Answer: B
The 22 in stick cuts the kite into two triangles. Each has a base of 22 in and a height of half of 16, which is 8 in: (1/2) × 22 × 8 = 88, and 88 + 88 = 176 sq in. Choice A multiplies the two diagonals, 22 × 16 = 352, which is the rectangle around the kite, twice too big. Choice C adds the diagonals. Choice D is only one of the two triangles.
Question 9 of 20 · Multiple Choice
A rhombus-shaped floor tile has sides of 15 cm and diagonals of 18 cm and 24 cm. What is its area?
Answer: C
The diagonals cut the rhombus into 4 right triangles with legs of 9 cm and 12 cm (half of each diagonal). Each triangle is (1/2) × 9 × 12 = 54 sq cm, and 4 × 54 = 216 sq cm. Choice A treats the rhombus as a square, 15 × 15, but its sides slant, so its height is less than 15 cm. Choice B multiplies the diagonals and forgets the 1/2. Choice D is 4 × 15 = 60, the perimeter in cm.
Question 10 of 20 · Multiple Choice
A patio is L-shaped. It is 20 ft by 16 ft with an 8 ft by 6 ft rectangle missing from one corner. What is the area of the patio?
Answer: A
Compose the full 20 ft by 16 ft rectangle, 320 sq ft, and subtract the missing corner, 8 × 6 = 48 sq ft: 320 - 48 = 272 sq ft. Choice B adds the corner instead of subtracting it. Choice C is 2 × (20 + 16) = 72, the perimeter of the big rectangle in feet. Choice D forgets to remove the missing corner.
Question 11 of 20 · Multiple Choice
A baseball home plate is a pentagon. It is a rectangle 17 in wide and 8.5 in deep, with a triangle on the back. The triangle has a base of 17 in, a height of 8.5 in (rounded) and slanted sides of 12 in. What is the area of home plate?
Answer: D
Rectangle: 17 × 8.5 = 144.5 sq in. Triangle: (1/2) × 17 × 8.5 = 72.25 sq in. Total: 144.5 + 72.25 = 216.75 sq in. Choice A is only the rectangle. Choice B forgets the 1/2 for the triangle: 144.5 + 144.5 = 289. Choice C uses the slanted 12 in side as the height: 144.5 + (1/2) × 17 × 12 = 246.5.
Question 12 of 20 · Multiple Choice
A triangular section of a lawn has a base of 40 ft and a height of 25 ft. Sod (rolls of grass for planting) costs $0.50 per square foot. How much does sod for this section cost?
Answer: B
Area: (1/2) × 40 × 25 = 500 sq ft. Cost: 500 × $0.50 = $250. Choice A forgets the 1/2 and prices 1,000 sq ft. Choice C adds the base and height, 40 + 25 = 65, and takes $0.50 for each. Choice D takes half twice: (1/2) × 500 = 250 sq ft, and 250 × $0.50 = $125.
Question 13 of 20 · Multiple Choice
A school entrance sign is a rectangle 10 ft wide and 6 ft tall with a triangle on top. The triangle has the same 10 ft base and a height of 4 ft. Which expression gives the area of the sign in square feet?
Answer: C
Decompose the sign into the rectangle, 10 × 6 = 60, and the triangle, (1/2) × 10 × 4 = 20, and add: 80 sq ft. Choice A forgets the 1/2 for the triangle (100). Choice B treats the whole 10 ft tall sign as one triangle (50). Choice D subtracts the triangle instead of adding it (40).
Question 14 of 20 · Multiple Choice
A triangular piece of fabric has a base of 3/4 yard and a height of 2/3 yard. What is its area?
Answer: A
(1/2) × 3/4 × 2/3 = 6/24 = 1/4 sq yd. Choice B is 3/4 × 2/3 = 1/2, which forgets the 1/2. Choice C adds the base and height, 3/4 + 2/3 = 17/12, and then halves. Choice D takes half twice.
Question 15 of 20 · Short Answer
The side of a wheelchair ramp is a right triangle. It is 6 ft long along the ground and rises 1/2 ft. Find the area of the side, and name the rectangle that it is half of.
Two copies of the triangle make a 6 ft by 1/2 ft rectangle, which has an area of 6 × 1/2 = 3 sq ft. The ramp side is half of that: (1/2) × 6 × 1/2 = 1 1/2 sq ft.
Question 16 of 20 · Short Answer
A triangle has a base of 8 cm and a height of 7 cm. The height meets the base 3 cm from the left corner. Split the triangle along its height into two right triangles and find the area of each piece and the total.
The base splits into 3 cm and 5 cm. Left piece: (1/2) × 3 × 7 = 10.5 sq cm. Right piece: (1/2) × 5 × 7 = 17.5 sq cm. Total: 10.5 + 17.5 = 28 sq cm, which matches (1/2) × 8 × 7 = 28.
Question 17 of 20 · Short Answer
A parallelogram has a base of 13 in and a height of 6 in. Describe how to cut and move one piece to make a rectangle, then find the area.
Cut along a height at one end to remove a right triangle, and slide it to the other end so its slanted side matches the other slanted side. The result is a 13 in by 6 in rectangle, so the area is 13 × 6 = 78 sq in.
Question 18 of 20 · Short Answer
A table top is a trapezoid with parallel sides of 3 ft and 5 ft and a height of 2 ft. The slanted ends match. Decompose it into a rectangle and two triangles, and find its area.
The rectangle is 3 ft by 2 ft: 6 sq ft. The extra 5 - 3 = 2 ft splits into 1 ft at each end, so each triangle is (1/2) × 1 × 2 = 1 sq ft. Total: 6 + 1 + 1 = 8 sq ft.
Question 19 of 20 · Short Answer
A pentagon-shaped field is a rectangle 40 m by 25 m with a triangle attached along one 25 m side, like the point of an arrow. The triangle has a base of 25 m (the shared side) and a height of 30 m. One bag of grass seed covers 100 sq m. How many bags are needed to seed the whole field?
Rectangle: 40 × 25 = 1,000 sq m. Triangle: (1/2) × 25 × 30 = 375 sq m. Total: 1,375 sq m. 1,375 ÷ 100 = 13.75, so 13 bags are not enough: buy 14 bags.
Question 20 of 20 · Short Answer
Two copies of a trapezoid with parallel sides of 3 cm and 10 cm and a height of 4 cm fit together, one turned upside down, to make a parallelogram. What are the base and the area of the parallelogram? What is the area of one trapezoid?
The two parallel sides line up end to end, so the parallelogram's base is 3 + 10 = 13 cm and its height is 4 cm. Its area is 13 × 4 = 52 sq cm. One trapezoid is half of that: 26 sq cm.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 6.G.A.1 mean?
6.G.A.1 means students find the area of triangles, special quadrilaterals and other polygons by changing them into rectangles and triangles. They can cut a shape into pieces (decompose) or put copies or pieces together into a rectangle (compose). Then they use these methods to solve real problems, such as how much carpet or paint a job needs.
Do students need to memorize area formulas for 6.G.A.1?
No, the standard focuses on understanding where the area comes from. Students should see why a triangle is half of a rectangle and why a parallelogram has the same area as a rectangle with the same base and height. Many classes write (1/2) × base × height and base × height as a summary once students can explain them. For trapezoids, kites and other polygons, cutting into known shapes always works, even without a formula.
What is the difference between the height and a slanted side?
The height is always perpendicular to the base, and a slanted side usually is not. In a parallelogram with a base of 10 cm, a height of 4 cm and slanted sides of 6 cm, the area is 10 × 4 = 40 sq cm, not 10 × 6. Only in a shape with a square corner, such as a right triangle, a rectangle or a right trapezoid, is a side also the height. Ask students to mark the right angle every time they choose a height.
How do you find the area of a triangle when the height is outside it?
Extend the base, draw the height to the extended line, and subtract. The triangle fills a big right triangle except for a small right triangle outside it. The rule (1/2) × base × height still works when the base is the original side, not the extended line.
How do you find the area of a trapezoid without a formula?
Cut it into shapes you know. One way is to cut along a diagonal into two triangles with the same height. Another is to draw heights from the short base, which gives a rectangle and two triangles. A third is to put two copies together into a parallelogram and take half. All three give the same answer.
Which shapes count as special quadrilaterals in 6.G.A.1?
They are the four-sided shapes students learned to name in grades 3-5: parallelograms, rectangles, squares, rhombuses, kites and trapezoids. The standard does not list them, but these are the ones usually taught. Each one can be turned into rectangles and triangles by cutting along a height or a diagonal.
What are common mistakes when finding area?
A common mistake is multiplying a slanted side instead of the height. Others are forgetting the 1/2 for a triangle, adding lengths instead of multiplying, confusing area with perimeter, and leaving out the square units. When students decompose a polygon, watch for pieces that overlap or leave a gap, which count some area twice or miss it.
Do students use fractions and decimals in 6.G.A.1?
Yes, side lengths can be fractions or decimals, which connects to grade 6 work with fractions and decimals. For example, a triangle with a base of 2 1/2 ft and a height of 2 ft has an area of 2 1/2 sq ft. Keep the numbers friendly so the focus stays on choosing the right pieces.
How does 6.G.A.1 connect to grade 7 and high school?
It is the base for the grade 7 area work. In grade 7, students find the area of circles (7.G.B.4) and solve harder problems with composite shapes and surface area (7.G.B.6). In high school geometry, the same idea of cutting and rearranging shapes is used to explain area formulas and to find areas in the coordinate plane.
How can parents help with area at home?
Look for shapes around the house and ask how to find their area. A rug, a garden bed or a room with a closet often splits into rectangles and triangles. Ask your child to sketch the shape, show where to cut it, and explain which measurement is the height. Estimating how much paint or flooring a room needs is good practice too.
07
Related Standards
6 standards
These standards connect to 6.G.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
3.MD.C.7Prerequisite
Relate area to multiplication and addition, including shapes made of rectangles
Lesson coming soon
5.NF.B.4Prerequisite
Multiply fractions, including finding areas of rectangles with fraction sides
Lesson coming soon
Alongside
6.G.A.3Parallel
Draw polygons in the coordinate plane and find side lengths from coordinates
