Practice area, polygons on the grid, volume with fractional edges, and surface area from nets — solve each problem right here, then check your answer and reveal the steps.
Score: 0 / 18
🟢 Easy
E1 · Triangle A triangle has a base of 8 cm and a height of 5 cm. What is its area? (A = ½ · b · h)
💡 Hint
Think: A triangle is exactly half of which simple shape that shares its base and height?
A = ½ · b · h → triangle area formula
A = ½ · 8 · 5→ put in base 8 and height 5
A = ½ · 40 → 8 × 5 = 40
A = 20 cm²→ half of 40
Why: The ½ appears because the triangle is exactly half of the 8-by-5 rectangle that boxes it in.
E2 · Parallelogram A parallelogram has a base of 7 m and a height of 4 m. Find its area. (A = b · h)
💡 Hint
Think: Does a parallelogram need the ½ that a triangle uses, or does it fill the whole rectangle?
A = b · h → parallelogram fills the whole rectangle (no ½)
A = 7 · 4→ base 7, height 4
A = 28 m²→ 7 × 4 = 28
Why: No ½ here — the parallelogram leans over to fill the entire 7-by-4 rectangle, so base × height is the whole area.
E3 · Triangle A triangular flag has a base of 10 in and a height of 6 in. How much cloth (area) does it use?
💡 Hint
Think: Find base × height first — then what one step turns a rectangle's area into a triangle's?
A = ½ · b · h → triangle
A = ½ · 10 · 6 → base 10, height 6
A = ½ · 60 → 10 × 6 = 60
A = 30 in²→ half of 60
Why: The cloth covers a triangle, which is half the 10-by-6 rectangle around it, so you halve 60.
E4 · Volume A box (right rectangular prism) is 4 cm long, 3 cm wide, and 2 cm tall. What is its volume? (V = l · w · h)
💡 Hint
Think: To fill a 3D space, do you add the three edges together or multiply all three?
V = l · w · h → volume of a box
V = 4 · 3 · 2 → length 4, width 3, height 2
V = 12 · 2 → 4 × 3 = 12
V = 24 cm³→ 12 × 2 = 24
Why: Multiplying all three edges counts every 1-cm cube that packs the box, which is what volume measures.
E5 · Parallelogram A garden bed is a parallelogram with a base of 9 ft and a height of 5 ft. Find its area.
💡 Hint
Think: A parallelogram's area is just base × height — no halving. Why does it skip the ½?
A = b · h → parallelogram
A = 9 · 5 → base 9, height 5
A = 45 ft²→ 9 × 5 = 45
Why: A parallelogram fills its whole surrounding rectangle, so it uses base × height with no ½.
E6 · Surface Area A net is made of 6 identical squares, each one 2 cm by 2 cm (the net of a cube). What is the total surface area?
💡 Hint
Think: Find the area of one square face first — then how many faces does a cube have?
Area of one square = 2 · 2 = 4 cm² → side × side
Surface area = 6 · 4 → a cube net has 6 faces
SA = 24 cm²→ 6 × 4 = 24
Why: All six faces of a cube are identical, so one face's area times 6 gives the whole outer skin.
🟡 Medium
M1 · Trapezoid A trapezoid has parallel bases of 6 cm and 10 cm, and a height of 4 cm. Find its area. (A = ½ · (b₁ + b₂) · h)
💡 Hint
Think: A trapezoid has two different bases — what should you do with them before multiplying by the height and halving?
A = ½ · (b₁ + b₂) · h → trapezoid formula
A = ½ · (6 + 10) · 4 → add the two parallel bases
A = ½ · 16 · 4 → 6 + 10 = 16
A = ½ · 64 → 16 × 4 = 64
A = 32 cm²→ half of 64
Why: Half the sum of the two bases is their average — it turns the slanted trapezoid into one equivalent rectangle.
M2 · Volume A jewelry box is 5 in long, 2 in wide, and 1½ in tall. Find its volume. (Enter your answer as a decimal, e.g. 7.5)
💡 Hint
Think: Multiply the two whole edges first; then how do you handle the 1½?
V = l · w · h → box volume, fractional edge is fine
V = 5 · 2 · 1½ → rewrite 1½ as 3/2
V = 10 · 3/2 → 5 × 2 = 10
V = 30 / 2 → 10 × 3 = 30, over 2
V = 15 in³→ 30 ÷ 2 = 15
Why: Rewriting 1½ as 3/2 lets you multiply cleanly — 10 × 3/2 is just 30 split in half.
M3 · Coordinate grid A rectangle has corners at A(2, 1), B(2, 5), C(7, 5), and D(7, 1). What is the length of side AB? (count units, A and B share the same x)
💡 Hint
Think: A and B share the same x, so the side is vertical — which coordinates do you subtract?
A(2, 1) and B(2, 5) have the same x, so AB is vertical. → subtract the y-values
Length = |5 − 1| → distance between the y-coordinates
AB = 4 units→ 5 − 1 = 4
Why: Because the x's are equal, only the up-down (y) gap can change the length, so you subtract the y's.
M4 · Coordinate area Using the same rectangle from M3 — A(2, 1), B(2, 5), C(7, 5), D(7, 1) — what is its area? (width × height)
💡 Hint
Think: Find the vertical side and the horizontal side first — then what do you do with them for a rectangle?
Height (AB) = |5 − 1| = 4 → vertical side
Width (AD) = |7 − 2| = 5 → horizontal side
A = width · height = 5 · 4 → rectangle area
A = 20 units²→ 5 × 4 = 20
Why: The corners line up into a clean rectangle on the grid, so width × height gives its area directly.
M5 · Composite A "house" shape is a rectangle 8 cm wide and 4 cm tall with a triangle on top (base 8 cm, height 3 cm). Find the total area.
💡 Hint
Think: Split the house into a rectangle and a triangle — then how do you combine their two areas?
Rectangle: A = 8 · 4 = 32 cm² → base × height
Triangle: A = ½ · 8 · 3 = 12 cm² → ½ · base · height
Total = 32 + 12 → add the two parts
A = 44 cm²→ 32 + 12 = 44
Why: Breaking a compound shape into a familiar rectangle and triangle lets you add areas you already know how to find.
M6 · Surface Area The net of a cube is made of 6 squares, each 3 cm × 3 cm. Find the surface area.
💡 Hint
Think: Find the area of one 3×3 face first — then multiply by how many faces a cube has.
One face = 3 · 3 = 9 cm² → side × side
SA = 6 · 9 → 6 faces on a cube
SA = 54 cm²→ 6 × 9 = 54
Why: A cube has six identical square faces, so one face's 9 cm² times 6 covers the whole surface.
🔴 Hard
H1 · Fractional volume A small box has edges 1½ in × 2½ in × 2 in. Find its volume. (Answer as a decimal, e.g. 7.5)
💡 Hint
Think: Turn each mixed number into a fraction, then multiply all the tops together and all the bottoms together. What do you get?
V = l · w · h → rewrite mixed numbers as fractions
V = 3/2 · 5/2 · 2 → 1½ = 3/2, 2½ = 5/2
V = (3 · 5 · 2) / (2 · 2) → multiply tops and bottoms
V = 30 / 4 → 30 over 4
V = 7.5 in³→ 30 ÷ 4 = 7.5
Why: Multiplying the fractions 3/2 × 5/2 × 2 gives 30/4; dividing top by bottom converts it to the decimal 7.5.
H2 · Trapezoid A trapezoid has parallel bases of 5 cm and 11 cm and a height of 6 cm. Find its area.
💡 Hint
Think: Add the two parallel bases first, then multiply by the height and take half — why the half?
A = ½ · (b₁ + b₂) · h → trapezoid formula
A = ½ · (5 + 11) · 6 → add the bases
A = ½ · 16 · 6 → 5 + 11 = 16
A = ½ · 96 → 16 × 6 = 96
A = 48 cm²→ half of 96
Why: Halving the sum of the bases gives their average width, so the trapezoid behaves like a rectangle of that average width.
H3 · Surface Area A rectangular prism is 5 cm long, 3 cm wide, and 2 cm high. Find its surface area from the net. (SA = 2(lw + lh + wh))
💡 Hint
Think: Find the area of one of each face (lw, lh, wh) — then why does the formula double each one?
The net has 3 pairs of matching rectangles. → top/bottom, front/back, two sides
lw = 5 · 3 = 15, lh = 5 · 2 = 10, wh = 3 · 2 = 6 → one of each face
SA = 2 · (15 + 10 + 6) → each face appears twice
SA = 2 · 31 → 15 + 10 + 6 = 31
SA = 62 cm²→ 2 × 31 = 62
Why: Every face on a box has an identical twin on the opposite side, so each of the three areas is counted twice.
H4 · Composite (L-shape) An L-shaped room fits inside a 10 ft × 8 ft rectangle, but a 4 ft × 3 ft rectangular corner is cut out. Find the area of the L-shape.
💡 Hint
Think: Find the full rectangle's area, then deal with the missing corner — do you add it or subtract it?
Why: The L-shape is the full rectangle with one piece removed, so you subtract the cut-out instead of adding.
H5 · Coordinate area A polygon has corners at (−3, −2), (4, −2), (4, 3), and (−3, 3). Find its area. (it's a rectangle — find width and height by counting units)
💡 Hint
Think: Subtract the x's for the width and the y's for the height — but watch the negative signs. Then what?
Width: from x = −3 to x = 4 → |4 − (−3)| = 7 → horizontal side
Height: from y = −2 to y = 3 → |3 − (−2)| = 5 → vertical side
A = width · height = 7 · 5 → rectangle area
A = 35 units²→ 7 × 5 = 35
Why: Taking the absolute difference handles the negative coordinates, turning the four corners into a clean 7-by-5 rectangle.
H6 · Fractional volume A planter box is ¾ ft long, 4 ft wide, and 2 ft tall. Find its volume. (Answer as a decimal)
💡 Hint
Think: Multiply the two whole edges first, then multiply by ¾ — what does multiplying by ¾ do to the result?
V = l · w · h → box volume
V = 3/4 · 4 · 2 → ¾ = 3/4
V = 3/4 · 8 → 4 × 2 = 8
V = 24 / 4 → 3 × 8 = 24, over 4
V = 6 ft³→ 24 ÷ 4 = 6
Why: Multiplying 8 by 3/4 takes three-quarters of it, which is why the answer (6) is smaller than 8.
Grounded in CA CCSS-M, Grade 6 · 6.G, California Department of Education. Image generated with Gemini Nano Banana Pro.
