Picture the full 10 ft by 8 ft rectangle. What part of it is missing?
Step-by-step solution
Method 1 — whole minus the missing corner
Full rectangle: 10 \times 8 = 80 ft².
Missing corner: 4 \times 3 = 12 ft².
Floor: 80 - 12 = 68 ft².
Method 2 — split into two rectangles (vertical cut where the top edge ends)
4. Left piece: 6 \times 8 = 48 ft².
5. Right piece: 4 \times 5 = 20 ft² (only 5 ft tall).
6. Total: 48 + 20 = 68 ft².
The area is 68 ft².
Check: both methods give 68 ✓. The answer is less than the full 80 ft² rectangle, as it must be ✓.
Tip: For L-shapes, "whole rectangle minus the missing corner" is often the fastest route. Splitting into two rectangles is a great way to check.
Question 2 of 36
No calculatorFill in
The front wall of a shed is a square with a triangle on top, as shown.
What is the total area of the front wall?
ft²
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find the square's area and the triangle's area separately. The triangle's base is the top of the square.
Step-by-step solution
Square: A = s^2 = 10 \times 10 = 100 ft².
Triangle: its base is the top of the square, b = 10 ft, and its height is the dashed line, h = 4 ft.
A = \tfrac{1}{2}bh = \tfrac{1}{2}(10)(4) = 20 ft².
Add the pieces: 100 + 20 = 120.
The wall's area is 120 ft².
Check: the triangle (20) is half of a 10 by 4 rectangle (40) ✓, and the total is more than the square alone ✓.
Tip: When shapes are joined, find each piece with its own formula and add. Only the triangle gets the \tfrac{1}{2}.
Question 3 of 36
No calculator
A metal plate is a rectangle with a square hole cut out, as shown. The shaded part is metal.
What is the area of the shaded part?
Hint
Find the area of the whole rectangle and the area of the hole. Should you add or subtract them?
Step-by-step solution
Whole rectangle: 9 \times 7 = 63 in².
Square hole: 3 \times 3 = 9 in².
The hole is not metal, so subtract it: 63 - 9 = 54.
The shaded area is 54 in².
Check: the shaded part must be smaller than the whole plate (63) ✓, and adding back the hole gives 54 + 9 = 63 ✓.
Tip: "Shaded region" problems are usually outer shape − unshaded shape. Subtract the hole's area (3 \times 3), not just one of its sides.
Question 4 of 36
No calculator
A sign maker cuts a letter T out of plywood. Its measurements are shown.
What is the area of the letter T?
Hint
Split the T into two rectangles: the bar across the top and the stem below it. How tall is the stem by itself?
Step-by-step solution
Split the T into two rectangles with a line along the bottom of the top bar.
Top bar:12 \times 3 = 36\text{ in}^2.
Stem: it is 4 in wide and 9 in tall (the 9 in is measured below the bar): 4 \times 9 = 36\text{ in}^2.
Total:36 + 36 = 72\text{ in}^2.
The letter T has an area of 72 in².
Check: the T fits inside a 12 \times 12 square (144 in²). The two empty spaces beside the stem are each 4 \times 9 = 36, so 144 - 36 - 36 = 72 ✓.
Tip: Draw one line to break a composite shape into rectangles, and give each rectangle its own length and width. Never multiply the overall width by the overall height unless the shape really is a full rectangle.
Question 5 of 36
A patio is a rectangle with a semicircle on each end, as shown. Use \pi \approx 3.14.
What is the area of the patio?
Hint
The two semicircles together make one whole circle. What is that circle's radius?
Step-by-step solution
Rectangle: 20 \times 10 = 200 ft².
Each semicircle has diameter 10 ft, so r = 5 ft.
Two semicircles = one full circle: \pi r^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 ft².
Total: 200 + 78.5 = 278.5.
The patio's area is 278.5 ft².
Check: the circle part (78.5) is less than a 10 by 10 square (100) that would hold it ✓. Total is between 200 and 300 ✓.
Tip: Two matching semicircles = one circle. Halve the diameter to get the radius before squaring.
Question 6 of 36
Fill in
A window is shaped like a rectangle with a semicircle on top, as shown. Use \pi \approx 3.14.
How many square feet of glass are needed for the window?
ft²
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The semicircle's diameter is the width of the window. Find half of a full circle's area and add the rectangle.
Tip: When you split an L-shape, each rectangle gets its own height. Rectangle B only reaches up to the step, so its height is 5 ft, not 9 ft.
Question 9 of 36
No calculatorSelect all
Select all of the expressions that give the area, in square feet, of the L-shaped figure.
Select all that apply.
Hint
Try different ways to cut the figure: a vertical cut, a horizontal cut, or the full rectangle minus the missing corner. Evaluate each expression to compare.
Step-by-step solution
The correct area is 12 \times 9 - 5 \times 4 = 108 - 20 = 88 ft². Test each expression:
12 \cdot 9 - 5 \cdot 4 = 108 - 20 = 88 ✓ (whole rectangle minus the 5 by 4 corner)
7 \cdot 9 + 5 \cdot 5 = 63 + 25 = 88 ✓ (vertical cut: left 7 by 9, right 5 by 5)
Tip: Every correct way of splitting a figure gives the same area. If two methods disagree, one of them used a wrong side length.
Question 10 of 36
A rectangular yard is covered with grass except for a circular flower bed, as shown. Use \pi \approx 3.14.
What is the area of the grass?
Hint
Grass = whole yard minus the flower bed. Which formula gives the space inside the circle?
Step-by-step solution
Whole yard: 40 \times 25 = 1{,}000 ft².
Flower bed (circle, r = 5): 3.14 \times 5^2 = 3.14 \times 25 = 78.5 ft².
Grass: 1{,}000 - 78.5 = 921.5.
The grass covers 921.5 ft².
Check: the bed is small compared to the yard, so the grass should be a little less than 1,000 ✓.
Tip: "Everything except" means subtract. Use the area of the circle (\pi r^2), not its circumference, and use the radius as given.
Question 11 of 36
No calculatorFill in
A backyard deck is built in a stair-step shape, as shown. One short side is not labeled.
What is the area of the deck?
ft²
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Cut the deck with horizontal lines at each step to make three rectangles stacked on top of each other. How tall is each one?
Step-by-step solution
Cut the deck into three horizontal strips (one for each step).
Bottom strip: 15 ft wide and 4 ft tall (the right side): 15 \times 4 = 60\text{ ft}^2.
Middle strip: the first step goes in 5 ft, so it is 15 - 5 = 10 ft wide and 3 ft tall: 10 \times 3 = 30\text{ ft}^2.
Top strip: 5 ft wide (the top edge). Its height is the unlabeled side: 10 - 4 - 3 = 3 ft. Area 5 \times 3 = 15\text{ ft}^2.
Total:60 + 30 + 15 = 105\text{ ft}^2.
The deck's area is 105 ft².
Check: the full 15 \times 10 rectangle is 150 ft². The empty space at the upper right is a block 5 ft wide and 10 - 4 = 6 ft tall (30) plus a block 5 ft wide and 3 ft tall (15): 150 - 30 - 15 = 105 ✓.
Tip: In a stair-step figure, the vertical pieces on the right must add up to the full height on the left (4 + 3 + 3 = 10). Use that to find any side that is not labeled.
Question 12 of 36
No calculatorDrag and drop
A U-shaped kitchen counter is drawn below. Dashed lines split it into rectangles A, B and C.
Drag the correct value into each box. Not every value is used.
Area of A (ft²)
Area of B (ft²)
Total area (ft²)
Tap a tile, then tap a box to place it.
Hint
Rectangle B sits under the notch. Its height is the full height minus the depth of the notch.
Step-by-step solution
Rectangle A (left arm): 4 ft wide and the full 8 ft tall: 4 \times 8 = 32\text{ ft}^2.
Rectangle B (bottom middle): its width is the notch width, 6 ft. Its height is 8 - 5 = 3 ft (the notch is 5 ft deep). Area 6 \times 3 = 18\text{ ft}^2.
Tip: After you split a figure, label each piece with its own width and height before multiplying. A piece under a notch is shorter than the whole figure.
Question 13 of 36
No calculator
A community garden has an L-shaped flower bed. A gardener splits it into two rectangles, A and B, with the dashed line shown.
What are the dimensions of rectangle B?
Hint
Rectangle B is the upper part only. Which labeled edge is its top, and how far is it from the dashed line up to the top?
Step-by-step solution
Rectangle B is above the dashed line. Its top edge is the labeled 7 m side, so B is 7 m wide.
The dashed line is at the height of the step. The step's vertical edge is labeled 6 m, so B is 6 m tall.
You can also check the height: the left side is 10 m and rectangle A below the line is 4 m tall (the right side), so 10 - 4 = 6 m.
Rectangle B is 7 m by 6 m (area 42 m²).
Check: A is 16 \times 4 = 64 m², B is 7 \times 6 = 42 m², total 106 m². Whole rectangle minus missing corner: 16 \times 10 - 9 \times 6 = 160 - 54 = 106 ✓.
Tip: Each piece of a split figure has its own sides. Trace the outline of just that piece with your finger and read only the labels that touch it.
Question 14 of 36
No calculatorSelect all
Priya is ordering a stone top for a U-shaped reception desk. The shape of the top is shown below.
Priya wants to check the area in more than one way. Which expressions are correct for the area of the desk top, in square feet? Choose every correct one.
Select all that apply.
Hint
Try three strategies: the big rectangle minus the notch, three upright pieces, or a bottom strip plus two arms. Check each expression against the figure.
Step-by-step solution
The area is 11 \times 7 - 5 \times 4 = 77 - 20 = 57\text{ ft}^2. Test each choice.
11 \cdot 7 - 5 \cdot 4 = 77 - 20 = 57 ✓ (whole rectangle minus the 5-by-4 notch).
3 \cdot 7 + 3 \cdot 7 + 5 \cdot 3 = 21 + 21 + 15 = 57 ✓ (two 3-by-7 arms plus the 5-by-3 piece under the notch; 7 - 4 = 3).
11 \cdot 3 + 3 \cdot 4 + 3 \cdot 4 = 33 + 12 + 12 = 57 ✓ (an 11-by-3 bottom strip plus two 3-by-4 arm tops).
11 \cdot 7 + 5 \cdot 4 = 97 ✗ — adds the notch instead of removing it.
3 \cdot 7 + 3 \cdot 7 + 5 \cdot 7 = 77 ✗ — gives the middle piece the full 7 ft height, so the notch is filled in.
11 \cdot 7 - 5 \cdot 3 = 62 ✗ — subtracts a 5-by-3 piece, but the notch is 4 ft deep.
Correct: the first three expressions.
Check: all three correct expressions give the same number, 57 ✓. Different correct splits must always agree.
Tip: On select-all questions, evaluate every expression. The wrong ones usually add the cut-out, use the full height for a short piece, or use the wrong depth.
Question 15 of 36
No calculatorDrop-down
Lena is cutting the front panel of a birdhouse from a board. The panel is a rectangle with a triangle on top.
The area of the panel, in square inches, is + .
Choose an answer from each menu.
Hint
One blank is the rectangle and one is the triangle. Which formula has a one-half in it?
Step-by-step solution
The panel splits into a rectangle (bottom) and a triangle (top) along the dashed line.
Rectangle: 9 in wide and 6 in tall, so its area is 9 \cdot 6 = 54\text{ in}^2.
Triangle: its base is the top of the rectangle, 9 in, and its height is the dashed 4 in. Area \tfrac{1}{2} \cdot 9 \cdot 4 = 18\text{ in}^2.
Check: the whole panel fits in a 9 \times 10 box (90 in²). The two empty triangles beside the roof are \tfrac{1}{2} \cdot 4.5 \cdot 4 = 9 each: 90 - 9 - 9 = 72 ✓.
Tip: The triangle's height is measured from its own base (the top of the rectangle) to the peak, not from the ground. Always keep the \tfrac{1}{2} for a triangle piece.
Question 16 of 36
A school is painting arrow signs to point visitors to the gym. Each sign has the shape and size shown.
What is the area of one arrow sign?
Hint
The arrow is a rectangle plus a triangle. Is the triangle's base the same as the shaft's height?
Step-by-step solution
Shaft (rectangle):18 \times 8 = 144\text{ in}^2.
Head (triangle): its base is the 16 in vertical edge and its height is the 10 in distance from that edge to the tip.
A = \tfrac{1}{2} \times 16 \times 10 = 80\text{ in}^2.
Total:144 + 80 = 224\text{ in}^2.
Each arrow sign has an area of 224 in².
Check: the whole arrow fits in a box 28 in long and 16 in tall (448\text{ in}^2). The sign is roughly half of that box, and 224 = 448 \div 2 — reasonable, since the corners around the shaft and the head are empty ✓.
Tip: In an arrow, the head is wider than the shaft. Use the head's own base (16 in), not the shaft's height (8 in), and remember the \tfrac{1}{2}.
Question 17 of 36
No calculatorFill in
The side of a wooden loading ramp is shown below. The side panel will be covered with sheet metal.
How many square feet of sheet metal are needed for one side panel?
ft²
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The dashed line splits the panel into a triangle and a rectangle. Which lengths go with each piece?
Step-by-step solution
The dashed line splits the side panel into a right triangle (the slope) and a rectangle (the platform).
Triangle: base 8 ft along the ground, height 3 ft. A = \tfrac{1}{2} \times 8 \times 3 = 12\text{ ft}^2.
Rectangle: 6 ft long and 3 ft high. A = 6 \times 3 = 18\text{ ft}^2.
Total:12 + 18 = 30\text{ ft}^2.
One side panel needs 30 ft² of sheet metal.
Check: the panel is a trapezoid with parallel sides 14 ft (bottom) and 6 ft (top) and height 3 ft: \tfrac{1}{2}(14 + 6)(3) = 30 ✓.
Tip: A ramp profile is a rectangle plus a triangle. If you get 14 \times 3 = 42, you have treated the whole panel as a rectangle; 8 \times 3 + 18 = 42 also forgets the \tfrac{1}{2}.
Question 18 of 36
Marcus is painting one wall of his living room. The wall is 14 ft wide and 9 ft tall. It has a door that is 3 ft by 7 ft and a window that is 4 ft by 3 ft. He will not paint the door or the window.
How many square feet of wall will Marcus paint?
Hint
Find the area of the whole wall first. Then think about what must be taken away.
Step-by-step solution
Whole wall:14 \times 9 = 126\text{ ft}^2.
Door:3 \times 7 = 21\text{ ft}^2.
Window:4 \times 3 = 12\text{ ft}^2.
Painted area:126 - 21 - 12 = 93\text{ ft}^2.
Marcus will paint 93 ft².
Check: add back what was removed: 93 + 21 + 12 = 126, the full wall ✓. The answer is less than the whole wall, as it must be ✓.
Tip: For "everything except" problems, find the whole area and subtract every hole. Make a short list of the holes so you do not forget one.
Question 19 of 36
A community theater stage is a rectangle with a semicircle across the front, as shown. Use \pi \approx 3.14.
What is the total floor area of the stage?
Hint
The semicircle's diameter is the front edge of the rectangle. What is its radius?
Step-by-step solution
Rectangle:16 \times 10 = 160\text{ ft}^2.
Semicircle: its diameter is the 16 ft front edge, so r = 16 \div 2 = 8 ft.
Check: the semicircle (100.48) should be less than half of a 16 \times 16 square (128), and it is ✓.
Tip: For a semicircle: halve the diameter to get r, square r, multiply by \pi, then take half. Skipping any one of those steps gives a classic wrong answer.
Question 20 of 36
Four round coasters are cut from a square piece of felt, as shown. Use \pi \approx 3.14.
How many square inches of felt are left over (the shaded part)?
Hint
Find the area of one coaster, then of all four. What is each coaster's radius?
Check: the four circles fill most of the square, so the leftover should be small — about one-fifth of 64 ✓ (13.76 \div 64 \approx 0.21).
Tip: In a "what is left" problem, the last step is always a subtraction. Read the question again before choosing: 50.24 is the coasters, not the leftover.
Question 21 of 36
A running track goes around a field. It has two straight sides and a semicircle at each end, as shown. Use \pi \approx 3.14.
How long is one lap around the outside of the track?
Hint
Walk around the outside edge. Which parts do you walk on? The dashed lines are inside the field.
Step-by-step solution
Straight parts: 100 + 100 = 200 m.
The two semicircles (diameter 60 m each) together form one full circle. Its circumference is \pi d = 3.14 \times 60 = 188.4 m.
The dashed 60 m lines are inside the field, so they are not part of the lap.
Lap: 200 + 188.4 = 388.4.
One lap is 388.4 m.
Check: each curved end (94.2 m) is longer than the straight 60 m diameter it replaces, as a curve should be ✓.
Tip: Perimeter means only the outside edge. Two semicircles = one circumference (\pi d), and inner dividing lines never count.
Question 22 of 36
Fill in
A concrete walkway 3 ft wide is built around all four sides of a rectangular pool that measures 20 ft by 10 ft, as shown.
What is the area of the walkway?
ft²
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find the outside length and width of the pool plus the walkway. The walkway adds 3 ft on each side.
Step-by-step solution
The walkway adds 3 ft on the left and 3 ft on the right: outer length = 20 + 3 + 3 = 26 ft.
It adds 3 ft on the top and bottom: outer width = 10 + 3 + 3 = 16 ft.
Outer rectangle: 26 \times 16 = 416 ft².
Pool: 20 \times 10 = 200 ft².
Walkway = outer − pool: 416 - 200 = 216.
The walkway's area is 216 ft².
Check: split the walkway into strips: two long strips 26 \times 3 = 78 each, two short strips 10 \times 3 = 30 each: 78 + 78 + 30 + 30 = 216 ✓.
Tip: For a border of width x, add 2x to each dimension. Adding only x (getting 23 \times 13) is the most common error in frame and walkway problems.
Question 23 of 36
Fill in
A machine shop starts with a rectangular metal sheet. It cuts a right triangle from one corner and a square hole from the middle, as shown. The shaded part is the metal that is left.
What is the area of the metal that is left?
cm²
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Start with the full sheet. Two pieces are removed, and one of them is a triangle.
Step-by-step solution
Full sheet:20 \times 15 = 300\text{ cm}^2.
Triangle cut from the corner: legs 8 cm and 6 cm. The legs are the base and height, so A = \tfrac{1}{2} \times 8 \times 6 = 24\text{ cm}^2.
Square hole:4 \times 4 = 16\text{ cm}^2.
Metal left:300 - 24 - 16 = 260\text{ cm}^2.
260 cm² of metal is left.
Check:260 + 24 + 16 = 300 ✓. Common wrong results: 300 - 48 - 16 = 236 (forgot the \tfrac{1}{2} for the triangle) and 300 - 24 = 276 (forgot the square hole).
Tip: A right triangle's two legs are its base and height — no extra height is needed. Subtract each removed piece from the original shape.
Question 24 of 36
No calculator
Dana wants to put a fence around her L-shaped yard. Two sides of the yard are not labeled.
How many feet of fence does Dana need to go all the way around the yard?
Hint
Find the two unlabeled sides first. Compare the top with the bottom and the right side with the left side.
Step-by-step solution
Unlabeled horizontal side (the step): the bottom is 40 ft and the top is 25 ft, so the step is 40 - 25 = 15 ft.
Unlabeled vertical side: the left side is 30 ft and the right side is 20 ft, so it is 30 - 20 = 10 ft.
Add all six sides:40 + 20 + 15 + 10 + 25 + 30 = 140 ft.
Dana needs 140 ft of fence.
Check: for an L-shape, the perimeter equals the perimeter of the full rectangle around it: 2(40) + 2(30) = 140 ✓. Pushing the corner in does not change the total distance across and up.
Tip: In a perimeter problem, count the sides first (an L has 6). Every unlabeled side can be found by subtracting: opposite sides of the whole shape must match.
Question 25 of 36
Fill in
Plastic edging will go around the outside of the garden bed shown below. The dashed line is not part of the edge. Use \pi \approx 3.14.
How many feet of edging are needed?
ft
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Walk around the outside of the bed: which straight sides do you pass, and how long is the curved part?
Step-by-step solution
The outside edge is made of three straight sides and one curved side. The dashed line is inside the bed, so it is not counted.
Straight sides:12 + 8 + 12 = 32 ft (top, left end, bottom).
Curved side: half of a circle with diameter 8 ft. Half the circumference: \tfrac{1}{2}\pi d = \tfrac{1}{2} \times 3.14 \times 8 = 12.56 ft.
Total edging:32 + 12.56 = 44.56 ft.
44.56 ft of edging is needed.
Check: the curve (12.56 ft) is longer than the straight 8 ft diameter it replaces, which makes sense because a curve bulges out ✓.
Tip: For perimeter, trace the outside with your finger. Leave out inside (dashed) lines, and use half the circumference for a semicircle. Including the dashed line would give 52.56; a full circle would give 57.12.
Question 26 of 36
No calculator
A photo that is 16 inches by 20 inches is placed in a frame. The frame is 2 inches wide on every side.
What is the area of the frame alone (not including the photo)?
Hint
Find the outer length and width of the frame. Remember the frame is on both sides of the photo.
Step-by-step solution
The frame adds 2 in on each side, so each dimension grows by 2 + 2 = 4 in.
Outer size:16 + 4 = 20 in by 20 + 4 = 24 in.
Outer area:20 \times 24 = 480\text{ in}^2.
Photo area:16 \times 20 = 320\text{ in}^2.
Frame area:480 - 320 = 160\text{ in}^2.
The frame's area is 160 in².
Check: split the frame into strips: two long strips 24 \times 2 = 48 each and two short strips 16 \times 2 = 32 each: 48 + 48 + 32 + 32 = 160 ✓.
Tip: For a border of width w, add 2w to each dimension, then subtract outer minus inner. Multiplying the photo's perimeter by the width (72 \times 2 = 144) misses the four corner squares.
Question 27 of 36
Fill in
A circular fish pond has a radius of 6 ft. A gravel path 2 ft wide is built around the entire pond. Use \pi \approx 3.14.
What is the area of the gravel path?
ft²
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The pond and path together make a bigger circle. What is the radius of that big circle?
Step-by-step solution
The path adds 2 ft all around, so the outer circle has radius 6 + 2 = 8 ft.
Tip: A ring (circular border) is the big circle minus the small circle. Do not square the path width alone: 3.14 \times 2^2 = 12.56 is the area of a tiny 2-ft circle, not the path. Also, the radius grows by 2 ft (not 4 ft) because the radius goes from the center out to just one side.
Question 28 of 36
The Garcias are putting new carpet in their L-shaped family room, shown below. Carpet costs $2.75 per square foot, installed.
What will the carpet cost?
Hint
Find the floor area first. The missing corner does not get carpet.
Check: split instead: bottom 18 \times 7 = 126 plus top 12 \times 5 = 60 gives 186 ✓. Estimate: 186 \times 3 \approx 560, a bit more than $511.50 because $2.75 is less than $3 ✓.
Tip: In a two-step cost problem, get the area exactly right first, then multiply by the price per square unit. Pricing the full rectangle (216 \times 2.75 = \$594) pays for floor that is not there.
Question 29 of 36
Fill in
A park lawn is a rectangle 50 ft long and 30 ft wide with a semicircle attached to one 30-ft side. One bag of grass seed covers 400 square feet. Bags are sold whole. Use \pi \approx 3.14.
What is the least number of bags needed to seed the whole lawn?
bags
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find the total area first. Then think about whether a part of a bag can be bought.
Step-by-step solution
Rectangle:50 \times 30 = 1{,}500\text{ ft}^2.
Semicircle: diameter 30 ft, so r = 15 ft. \tfrac{1}{2} \times 3.14 \times 15^2 = \tfrac{1}{2} \times 3.14 \times 225 = 353.25\text{ ft}^2.
Total lawn:1{,}500 + 353.25 = 1{,}853.25\text{ ft}^2.
Bags:1{,}853.25 \div 400 \approx 4.63.
Four bags cover only 1{,}600\text{ ft}^2, which is not enough, so round up.
Tip: When you buy whole items (bags, cans, boxes) to cover an area, always round up, even if the decimal is small. Rounding 4.63 down to 4 would leave part of the lawn bare.
Question 30 of 36
No calculator
An L-shaped patio has an area of 136 square feet. Its shape is shown below.
What is the value of x?
Hint
Write the area as the big rectangle minus the missing corner, set it equal to 136, and solve.
Step-by-step solution
Big rectangle:14 \times 12 = 168\text{ ft}^2.
Missing corner:4 \times x = 4x.
Equation: 168 - 4x = 136.
Subtract: 168 - 136 = 32, so 4x = 32.
Divide: x = 32 \div 4 = 8.
x = 8 ft.
Check: missing corner 4 \times 8 = 32, and 168 - 32 = 136 ✓. Also, x = 8 is less than the 12-ft height, so the corner fits ✓.
Tip: Work-backwards area problems become one-step equations: (known area) − (unknown piece) = total. After you find the missing area (32), divide by the known side to get the missing length.
Question 31 of 36
No calculator
An L-shaped figure is drawn on the coordinate grid below. Its vertices are (-4, -3), (4, -3), (4, 1), (0, 1), (0, 4) and (-4, 4).
What is the area of the figure, in square units?
Hint
Find side lengths by subtracting coordinates. Then split the L with a horizontal line at y = 1.
Step-by-step solution
Split the figure with a horizontal line at y = 1.
Bottom rectangle: from x = -4 to x = 4 is 4 - (-4) = 8 units wide; from y = -3 to y = 1 is 1 - (-3) = 4 units tall. Area 8 \times 4 = 32.
Top rectangle: from x = -4 to x = 0 is 4 units wide; from y = 1 to y = 4 is 3 units tall. Area 4 \times 3 = 12.
Total:32 + 12 = 44 square units.
The area is 44 square units.
Check: bounding rectangle 8 \times 7 = 56 minus the missing corner from x = 0 to 4 and y = 1 to 4: 4 \times 3 = 12. 56 - 12 = 44 ✓.
Tip: On a grid, length = difference of coordinates. Subtracting a negative adds: 4 - (-4) = 8. Counting grid points instead of spaces gives one too many in each direction.
Question 32 of 36
A circular patio has a radius of 10 ft. It contains two circular planters, each 6 ft in diameter, which will not be paved. Paving costs $4.50 per square foot. Use \pi \approx 3.14.
What is the cost to pave the shaded part of the patio?
Hint
Paved area = big circle − both planters. Watch out: the patio gives a radius, but the planters give a diameter.
Tip: In multi-piece problems, list each piece with its radius before computing. Mixed information (one radius, one diameter) is a deliberate GED trap.
Question 33 of 36
Fill in
A church doorway is a rectangle 6 ft wide topped by a semicircle whose diameter is the top of the rectangle. The total area of the doorway is 56.13 square feet. Use \pi \approx 3.14.
How tall is the rectangular part of the doorway?
ft
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find the area of the semicircle first. What area is left for the rectangle?
Step-by-step solution
Semicircle: diameter 6 ft, so r = 3 ft.
\tfrac{1}{2} \times 3.14 \times 3^2 = \tfrac{1}{2} \times 3.14 \times 9 = 14.13\text{ ft}^2.
Rectangle's area:56.13 - 14.13 = 42\text{ ft}^2.
Rectangle's height: area = 6 \times h, so h = 42 \div 6 = 7 ft.
The rectangular part is 7 ft tall.
Check:6 \times 7 + 14.13 = 42 + 14.13 = 56.13 ✓.
Tip: Peel off the parts you can compute (the semicircle), and what remains belongs to the unknown part. Common slips: using a full circle (56.13 - 28.26 = 27.87, then \div 6 \approx 4.6) or dividing the whole area by 6 (\approx 9.4).
Question 34 of 36
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A building lot is made of a rectangle and a right triangle, as shown. The total area of the lot is 210 square feet.
The area of the triangle part is square feet, so the missing length x is feet.
Choose an answer from each menu.
Hint
Subtract the rectangle's area from the total. Then use the triangle formula with the 10-ft height to find the base.
Step-by-step solution
Rectangle:15 \times 10 = 150\text{ ft}^2.
Triangle part:210 - 150 = 60\text{ ft}^2.
Triangle formula: \tfrac{1}{2} \times x \times 10 = 60, so 5x = 60.
Tip: When working backwards through a triangle, the \tfrac{1}{2} is still there: base × height must equal twice the area. Forgetting it gives 60 \div 10 = 6, half of the real base.
Question 35 of 36
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A five-sided figure is drawn on the coordinate grid. Its vertices are (-3, -2), (3, -2), (3, 2), (0, 5) and (-3, 2).
What is the area of the figure, in square units?
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Draw a horizontal line at y = 2. You get a rectangle and a triangle. Find each base and height from the coordinates.
Step-by-step solution
A horizontal line at y = 2 splits the figure into a rectangle (bottom) and a triangle (top).
Triangle: base along y = 2 from x = -3 to x = 3, so base = 6. Height from y = 2 up to the peak at y = 5: 5 - 2 = 3. Area \tfrac{1}{2} \times 6 \times 3 = 9.
Total:24 + 9 = 33 square units.
The area is 33 square units.
Check: the bounding box is 6 \times 7 = 42. The two empty corner triangles beside the roof are each \tfrac{1}{2} \times 3 \times 3 = 4.5: 42 - 4.5 - 4.5 = 33 ✓.
Tip: The triangle's height is measured from its base line (y = 2), not from the x-axis. Using 5 as the height would give 15 + 24 = 39.
Question 36 of 36
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A city planner draws a park on a coordinate grid, where each unit is 10 meters. The park's vertices are (-5, -4), (5, -4), (5, 0), (1, 4) and (-5, 4).
What is the area of the park, in square units of the grid?
Hint
Picture the full rectangle from x = -5 to 5 and y = -4 to 4. Which corner piece is missing, and what shape is it?
Step-by-step solution
Full rectangle around the park: width 5 - (-5) = 10, height 4 - (-4) = 8. Area 10 \times 8 = 80.
The top-right corner is cut off along the slanted side from (5, 0) to (1, 4). The missing piece is a right triangle with corners (1, 4), (5, 4) and (5, 0).
Its legs: 5 - 1 = 4 (across) and 4 - 0 = 4 (down). Area \tfrac{1}{2} \times 4 \times 4 = 8.
Park:80 - 8 = 72 square units.
The park's area is 72 square units.
Check: split another way: bottom rectangle 10 \times 4 = 40, top rectangle from x = -5 to 1: 6 \times 4 = 24, plus the triangle between x = 1 and 5: \tfrac{1}{2}(4)(4) = 8. 40 + 24 + 8 = 72 ✓.
Tip: A slanted side on a grid often means "rectangle minus a right triangle". The triangle's legs run along grid lines, so you can read them by subtracting coordinates. (In real units this park would be 72 \times 100 = 7{,}200 m², because each square unit is 10 \times 10 m.)
