Triangles and rectangles: area and perimeter

1. The garden is a rectangle, so use A = lw (on the formula sheet).
2. The length is l = 12 ft and the width is w = 8 ft.
3. Multiply: A = 12 \times 8 = 96.
4. Area is measured in square units, so the answer is in square feet.

The area is 96 ft².

Check: think of 8 rows with 12 one-foot squares in each row: 8 \times 12 = 96 squares ✓. Another check: 10 \times 8 = 80 and 2 \times 8 = 16, and 80 + 16 = 96 ✓.

Tip: "Area" means cover the inside — multiply length by width. Adding the sides gives the perimeter, which answers a different question (how much fence you need).

1. Edging goes around the bed, so we need the perimeter.
2. The perimeter of a triangle is the sum of its three sides: P = s_1 + s_2 + s_3.
3. P = 7 + 9 + 12.
4. 7 + 9 = 16, and 16 + 12 = 28.

Dana needs 28 ft of edging.

Check: add in a different order: 12 + 9 = 21, 21 + 7 = 28 ✓. The answer must be more than the longest side (12) and less than three times it (36) — 28 fits.

Tip: Perimeter is always "add every side once." Multiplying sides gives area-type numbers (square units), and a triangle has only three sides to add — no doubling like a rectangle.

1. The panel is a rectangle, so use A = lw (on the formula sheet).
2. Length l = 5.5 ft and width w = 3 ft.
3. Multiply: A = 5.5 \times 3 = 16.5.
4. Area is measured in square units.

The area is 16.5 ft².

Check: 5 \times 3 = 15 and 0.5 \times 3 = 1.5, so 15 + 1.5 = 16.5 ✓. The answer is a little more than 5 \times 3 = 15, which makes sense because the length is a little more than 5 ✓.

Tip: Decimals do not change the rule: area of a rectangle is still length times width. Split the decimal (5 and 0.5) if you want to check it in your head.

1. For a rectangle, A = lw.
2. Substitute what we know: 84 = l \times 7.
3. Undo the multiplication by dividing both sides by 7: l = 84 \div 7.
4. 84 \div 7 = 12.

The storage unit is 12 ft long.

Check: 12 \times 7 = 84 ft² ✓ — that matches the given area.

Tip: To find a missing side from the area of a rectangle, divide the area by the side you know. Multiplying or subtracting the given numbers is the most common "work-backwards" error; plug your answer back into A = lw to be sure.

1. Carpet covers the floor, so we need the area: A = lw.
2. A = 14 \times 11 = 154 ft².
3. Each square foot costs $3.25, so multiply: 154 \times 3.25.
4. 154 \times 3 = 462 and 154 \times 0.25 = 38.50; 462 + 38.50 = 500.50.

The carpet costs $500.50.

Check: estimate 150 \times 3 = 450 plus a bit more for the extra 4 ft² and 25¢ — about $500 ✓.

Tip: For "price per square foot" problems, find the area first, then multiply by the unit price. If a choice equals the perimeter times the price, it is a trap — perimeter is for trim or baseboard, not carpet.

1. Area of a triangle: A = \tfrac{1}{2}bh.
2. Substitute: 54 = \tfrac{1}{2}(12)h.
3. Simplify the right side: \tfrac{1}{2}(12) = 6, so 54 = 6h.
4. Divide by 6: h = 54 \div 6 = 9.

The sail is 9 ft tall.

Check: \tfrac{1}{2}(12)(9) = \tfrac{1}{2}(108) = 54 ft² ✓.

Tip: For a triangle, h = \dfrac{2A}{b}. Doubling the area first "cancels" the \tfrac{1}{2}. If you just divide 54 \div 12 you get 4.5 — exactly half the right answer, a sign the \tfrac{1}{2} was ignored.

1. Use A = \tfrac{1}{2}bh (on the formula sheet).
2. The base is the bottom side, b = 8 m.
3. The height is the dashed segment that meets the extended base at a right angle: h = 6 m. In an obtuse triangle the height can fall outside the triangle.
4. The slanted sides (12.5 m and 6.7 m) are not used for area.
5. Compute: A = \tfrac{1}{2}(8)(6) = \tfrac{1}{2}(48) = 24.

The area is 24 m².

Check: a parallelogram with base 8 and height 6 has area 48; two copies of this triangle make that parallelogram, so one triangle is 48 \div 2 = 24 ✓.

Tip: The height is always measured straight up from the line of the base (look for the right-angle mark). It may sit outside the triangle — that is still the height to use.

1. Area in square feet: A = lw = 12 \times 9 = 108\text{ ft}^2.
2. One square yard is 3\text{ ft} \times 3\text{ ft} = 9\text{ ft}^2.
3. Convert: 108 \div 9 = 12.

The floor needs 12 square yards.

Check (convert first): 12\text{ ft} = 4\text{ yd} and 9\text{ ft} = 3\text{ yd}, so A = 4 \times 3 = 12\text{ yd}^2 ✓ — same answer both ways.

Tip: For square units, divide by 3^2 = 9 (not 3) to go from ft² to yd². The safest method: convert each length to yards before you multiply.

1. Fencing goes around the edge, so find the perimeter: P = 2(120) + 2(85) = 240 + 170 = 410 ft.
2. Each foot costs $4.50, so multiply: 410 \times 4.50 = 1{,}845.

The fencing costs $1,845.

Check: estimate 400 \times \$4.50 = \$1{,}800, and the extra 10 ft add \$45: \$1{,}800 + \$45 = \$1{,}845 ✓.

Tip: Two-step cost problems: (1) find the right measurement (perimeter for fence, area for flooring), (2) multiply by the price per unit. Ask yourself which unit the price uses: "per foot" means perimeter; "per square foot" means area.

1. Let the width be w. Then the length is w + 6.
2. Perimeter: P = 2l + 2w = 2(w + 6) + 2w.
3. Set it equal to 52: 2w + 12 + 2w = 52, so 4w + 12 = 52.
4. Subtract 12: 4w = 40. Divide by 4: w = 10.
5. Length: l = w + 6 = 10 + 6 = 16.

The pool is 16 meters long.

Check: 2(16) + 2(10) = 32 + 20 = 52 ✓, and 16 is 6 more than 10 ✓.

Tip: After solving for the variable, re-read the question. Here w = 10 is the width, but the question asks for the length.

1. Each equal side is 14 + 2 = 16 ft.
2. An isosceles triangle has two equal sides and one base.
3. Perimeter: P = 16 + 16 + 14 = 46 ft.

46 ft of edging are needed.

Check: 2 \times 16 = 32 and 32 + 14 = 46 ✓. The perimeter is more than three times the base (42), which fits because two sides are longer than the base ✓.

Tip: Draw and label all three sides before adding. In an isosceles triangle, the two equal sides are the "legs" — make sure the repeated length is the one that the problem says is equal.

1. Let the width be w. "3 more than twice the width" means the length is 2w + 3.
2. Perimeter: 2l + 2w = 66, so 2(2w + 3) + 2w = 66.
3. Distribute: 4w + 6 + 2w = 66, so 6w + 6 = 66.
4. Subtract 6: 6w = 60. Divide by 6: w = 10.
5. Length: l = 2(10) + 3 = 23.
6. Area: A = lw = 23 \times 10 = 230.

The area is 230 ft².

Check: perimeter 2(23) + 2(10) = 46 + 20 = 66 ✓; and 23 is 3 more than twice 10 ✓.

Tip: In multi-step problems, write down what the question finally asks for (here, area) before you start. Solving for w is only the middle of the problem.

1. Old area: 9 \times 12 = 108 ft².
2. New dimensions: 2 \times 9 = 18 ft and 2 \times 12 = 24 ft.
3. New area: 18 \times 24 = 432 ft².
4. Increase: 432 - 108 = 324 ft².

The new patio has 324 ft² more.

Check: doubling both dimensions multiplies the area by 2 \times 2 = 4: 108 \times 4 = 432 ✓, and 432 - 108 = 324 ✓.

Tip: When both dimensions are multiplied by k, the area is multiplied by k^2 — not by k. Doubling the sides makes the area 4 times as big.

1. Floor area: A = lw = 13 \times 10.5 = 136.5\text{ ft}^2.
2. Boxes needed: 136.5 \div 15 = 9.1.
3. 9 boxes cover only 9 \times 15 = 135\text{ ft}^2, which is 1.5\text{ ft}^2 short.
4. Boxes come whole, so round up: 10 boxes.

10 boxes are needed.

Check: 10 \times 15 = 150\text{ ft}^2 \ge 136.5\text{ ft}^2 ✓, and 9 boxes (135\text{ ft}^2) are not enough ✓.

Tip: When you must cover all of something (paint, tile, fabric, seed), always round up, even if the decimal is small like 9.1.

1. Base AB: both points have y = -1, so the length is 4 - (-2) = 6 units.
2. Height: the vertical distance from C(1, 4) down to the line y = -1 is 4 - (-1) = 5 units.
3. Area: A = \tfrac{1}{2}bh = \tfrac{1}{2}(6)(5) = 15.

The area is 15 square units.

Check: a 6 \times 5 rectangle around the triangle has area 30, and a triangle with the same base and height is half: 30 \div 2 = 15 ✓.

Tip: On a grid, subtract coordinates to get lengths. Counting grid points instead of grid spaces gives 7 and 6 — one too many each time.

1. Use A = lw.
2. Change to improper fractions: 4\tfrac{1}{2} = \tfrac{9}{2} and 2\tfrac{2}{3} = \tfrac{8}{3}.
3. Multiply: \tfrac{9}{2} \times \tfrac{8}{3} = \tfrac{72}{6} = 12.

The area is 12 ft².

Check with decimals: 4.5 \times 2.\overline{6} \approx 4.5 \times 2.667 \approx 12.0 ✓. Also, 4 \times 3 = 12 is a reasonable estimate ✓.

Tip: Never multiply the whole-number parts and the fraction parts separately — (4 \times 2) + (\tfrac{1}{2} \times \tfrac{2}{3}) misses the cross terms. Convert to improper fractions first.

1. Perimeter = base + two equal sides: 50 = 18 + 2s.
2. Subtract 18: 2s = 32.
3. Divide by 2: s = 16.

Each equal side is 16 ft.

Check: 16 + 16 + 18 = 50 ✓. Also, each equal side must be more than half the base (16 > 9), or the two sides could not meet at a peak ✓.

Tip: Work-backwards perimeter = subtract what you know, then split the rest among the equal unknown sides. Stopping at 50 - 18 = 32 gives the total of both sides, not one side.

1. Garden A: area = 15 \times 8 = 120\text{ ft}^2; perimeter = 2(15) + 2(8) = 46 ft.
2. Garden B: area = 11 \times 11 = 121\text{ ft}^2; perimeter = 4 \times 11 = 44 ft.
3. Compare: B has the larger area (121 > 120) but the smaller perimeter (44 < 46).

Garden B has more area but needs less fencing.

Check: a square is the "most efficient" rectangle — for the same perimeter it holds the most area — so it makes sense that B beats A on area while using less fence ✓.

Tip: A larger perimeter does not mean a larger area. On "which statement is true" items, compute every quantity the choices mention, then test each choice.

1. Floor area: 84 \times 52 = 4{,}368\text{ ft}^2.
2. Two coats: 2 \times 4{,}368 = 8{,}736\text{ ft}^2 to cover.
3. Cans: 8{,}736 \div 400 = 21.84, so round up to 22 cans.
4. Cost: 22 \times \$65 = \$1{,}430.

The finish costs $1,430.

Check: 21 cans cover only 21 \times 400 = 8{,}400\text{ ft}^2 < 8{,}736, so 22 cans are really needed ✓. Estimate: about 9{,}000 \div 400 \approx 22 cans ✓.

Tip: Handle multi-step coverage problems in order: area → number of coats → whole containers (round up) → cost. Multiplying 21.84 \times 65 gives a price for part of a can, which a store will not sell.

1. Original area: 5 \times 7 = 35\text{ in}^2. New area: 4 \times 35 = 140\text{ in}^2.
2. Same shape means both sides are multiplied by the same number k, and area is multiplied by k^2.
3. k^2 = 4, so k = 2.
4. New sides: 5 \times 2 = 10 in and 7 \times 2 = 14 in.

The enlarged photo is 10 in by 14 in.

Check: 10 \times 14 = 140 = 4 \times 35 ✓, and 10 : 14 = 5 : 7, so the shape is the same ✓.

Tip: To make area n times larger with the same shape, multiply each side by \sqrt{n}. Multiplying each side by 4 makes the area 16 times larger.

1. Convert to feet: 6\text{ in} = \tfrac{6}{12} = 0.5 ft, so 15\text{ ft }6\text{ in} = 15.5 ft. 9\text{ in} = \tfrac{9}{12} = 0.75 ft, so 11\text{ ft }9\text{ in} = 11.75 ft.
2. Perimeter: P = 2(15.5) + 2(11.75) = 31 + 23.5 = 54.5 ft.
3. Pieces: 54.5 \div 8 = 6.8125, so round up to 7 pieces.

7 pieces must be bought.

Check: 6 pieces give 48 ft (too short), 7 pieces give 56 ft \ge 54.5 ft ✓.

Tip: Inches are twelfths of a foot, not tenths: 9 in is 0.75 ft, not 0.9 ft. Convert mixed units before you add or multiply.