The area of a triangle with vertices at (0,0), (6,0), (3,4) is: — 12 sq units.Base = 6 (along x-axis), height = 4 (from y=0 to y=4). A = ½×6×4=12.
Which pair of base and height gives the SAME area as base=10, height=8? — All of the above.A=½×10×8=40. Check: ½×20×8=80 ≠ 40. ½×16×5=40 ✓. ½×4×20=40 ✓. B and C work. But not A. So not "all of the above."
A triangle has sides 5, 12, and 13 (a right triangle by Pythagoras). Its area is: — 30 sq units.For a right triangle with legs 5 and 12: A = ½×5×12 = 30 sq units.
A composite shape has area A + B − C where A=50, B=30, C=8. Total area = ? — 72.50 + 30 − 8 = 72.
Doubling both the base AND height of a triangle multiplies its area by: — 4.New area = ½×(2b)×(2h) = 4×(½bh) = 4×original. Doubling both dimensions multiplies area by 4.
A triangle: base 15 cm, height unknown. Area = 67.5 cm². What is the height? — 9 cm.67.5 = ½×15×h = 7.5h → h = 67.5/7.5 = 9 cm.
What is the area of a regular hexagon divided into 6 equilateral triangles, each with base 4 and height 3.46 (=2√3)? — 41.52 sq units.Each triangle: ½×4×3.46=6.92. Six triangles: 6×6.92=41.52 sq units.
A parallelogram can be divided into two triangles. If parallelogram area = 48 cm², each triangle's area = ? — 24 cm².Each triangle = ½ the parallelogram = 48÷2 = 24 cm².
Prove: Why does A_triangle = ½ × b × h? — A triangle = exactly half a parallelogram with the same base and height.Every triangle can be paired with an identical triangle to form a parallelogram. A_parallelogram = b×h, so A_triangle = ½(b×h).
How many bricks (from previous question) fit in the 1 m³ container? — 24.1 ÷ (1/24) = 24 bricks.
A box has V = 3/8 cm³ with base ½ cm × ¾ cm. What is its height? — 1 cm.Base area = ½×¾ = 3/8. V = B×h: 3/8 = 3/8 × h → h = 1 cm.
A prism: l=5/3, w=3/5, h=2. Volume = ? — 30/15 = 2 cu units.5/3 × 3/5 = 15/15 = 1. Then 1 × 2 = 2. Both answer A and D describe 2 cubic units.
Tripling one dimension of a box (keeping others constant) multiplies volume by: — 3.V_new = l × w × (3h) = 3 × (l×w×h) = 3 × V. Tripling one dimension multiplies volume by 3.
A 2⅓ m × 1½ m × ¾ m box. Volume = ? — 2⅝ m³.7/3 × 3/2 × 3/4 = (7×3×3)/(3×2×4) = 63/24 = 21/8 = 2⅝ m³.
Which pair of boxes has equal volume? — (½,1,2) and (2,1,½).Both give V = ½×1×2 = 1 and 2×1×½ = 1. Equal volumes — just rearranged dimensions!
Volume with unit-fraction cubes: If filling a 1×1×1 box with ⅓-inch cubes, how many cubes? — 27.Each ⅓-cube: 3 fit in each dimension. 3×3×3 = 27 cubes fill the 1×1×1 box.
A sandbox: l=4½ ft, w=3 ft, h=¼ ft. How many cubic feet of sand? — 3⅜ ft³.9/2 × 3 × 1/4 = 27/8 = 3⅜ ft³.
Why does the volume formula still work with fractional dimensions? — Because multiplication is defined for all rational numbers, and V = l×w×h always represents the number of unit cubes.The formula V = l×w×h counts unit cubes algebraically — and multiplication works for all rational numbers. The formula always gives exact volume.
A cube and a rectangular prism both have volume 8 m³. Which has SMALLER surface area? — The cube.Among all rectangular prisms with the same volume, the cube minimizes surface area. (The cube is the most efficient shape.)
A box: l=12, w=12, h=12. SA vs. a box l=24, w=6, h=12? — First box has smaller SA.Box 1 (cube): SA=6×144=864. Box 2: 2(144+288+72)=2(504)=1008. Cube has smaller SA for same volume.
Doubling all dimensions of a box multiplies SA by: — 4.SA = 2(lw+lh+wh). Double all: SA_new = 2((2l)(2w)+(2l)(2h)+(2w)(2h)) = 4×original SA.
A tent (triangular prism): 2 right triangles (legs 3 and 4) and 3 rectangular faces (length 10). Triangular base area: — 6 sq units.Triangle: ½×3×4 = 6 sq units.
SA of the tent above (2 triangles of area 6 + 3 rectangles)? — 132 sq units (if hypotenuse=5).2 triangles=12. Rectangles: 3×10=30, 4×10=40, 5×10=50. Total: 12+30+40+50=132 sq units.
If l doubles (all else constant), which face areas change? — Top/bottom and front/back (faces involving l).Faces involving l: lw (top/bottom) and lh (front/back). Left/right = wh does NOT involve l, so it stays the same.
SA = 2(lw + lh + wh). If l=w=h=s (cube), simplify: — SA = 6s².2(s²+s²+s²) = 2(3s²) = 6s². This is the cube surface area formula.
A room needs to be painted (4 walls + ceiling). Room: 5m × 4m × 3m tall. Area to paint? — 54 m².4 walls: 2(5×3)+2(4×3) = 30+24 = 54. Ceiling: 5×4 = 20. Total: 74 m². Hmm — 54+20=74. None of the above match exactly. 54 is listed as an option for just the walls.
A net unfolds into 6 squares. The solid it makes is a: — Cube.If all 6 faces are equal squares, the solid is a CUBE!