A model car is built at a scale of 1:24. The model is 6 inches long. How long is the actual car in feet? — 144 in.6 in × 24 = 144 inches (12 feet). The actual car is 144 inches long.
A map has a scale of 1 cm : 50 km. Two towns are 320 km apart. How far apart are they on the map? — 6.4 cm.320 km ÷ 50 = 6.4 cm on the map. Divide actual distance by scale factor.
A scale drawing of a park uses 1 in = 30 ft. The park measures 7.5 in × 4 in on the drawing. What is the actual area of the park? — 225,000 ft².Actual dimensions: 7.5×30=225 ft by 4×30=120 ft. Area = 225×120 = 27,000 ft². Wait — 225 × 120 = 27,000. Let me recalculate: correct actual area = 27,000 ft². The answer 225,000 is incorrect. Actual: 225 ft × 120 ft = 27,000 ft².
The scale factor of a model building is 1:50. If the model is 30 cm tall, how tall is the actual building in meters? — 15 m.30 cm × 50 = 1,500 cm = 15 m. The actual building is 15 meters tall.
A student creates a scale drawing of their bedroom that is 8 ft × 10 ft. They use a scale of 1 in = 2 ft. What are the drawing dimensions? — 4 in × 5 in.8 ft ÷ 2 = 4 in; 10 ft ÷ 2 = 5 in. Drawing is 4 in × 5 in.
Two cities are 4.5 cm apart on a map with scale 1 cm : 80 km. How far apart are they actually? — 360 km.4.5 cm × 80 km/cm = 360 km.
On a scale drawing, a rectangle is 3.5 cm × 2 cm. The actual rectangle is 14 m × 8 m. What is the scale? — 1 cm : 4 m.14 m ÷ 3.5 cm = 4 m/cm; 8 m ÷ 2 cm = 4 m/cm. Scale is 1 cm : 4 m.
A map uses the scale 1 : 25,000. A road is 12 cm on the map. What is the actual length of the road in km? — 3 km.12 cm × 25,000 = 300,000 cm = 3,000 m = 3 km.
On a map with scale 2 cm : 1 km, how many kilometers does 7 cm represent? — 3.5 km.7 cm ÷ 2 cm per km = 3.5 km.
A model has a scale factor of 1:200. The model building is 15 cm tall and 8 cm wide. What is the actual width in meters? — 16 m.8 cm × 200 = 1,600 cm = 16 m.
If the scale is 1 cm : 4 m and an object measures 6.5 cm on the drawing, what is the actual length? — 26 m.6.5 cm × 4 m/cm = 26 m.
A road on a map measures 5.2 cm. The scale is 1 cm = 2.5 km. What is the actual road length? — 13 km.5.2 × 2.5 = 13 km.
A circle has a circumference of 20π meters. What is its radius? — 10 m.C = 2πr → 20π = 2πr → r = 10 m.
A circle has an area of 64π ft². What is its radius? — 8 ft.A = πr² → 64π = πr² → r² = 64 → r = 8 ft.
A circular pool has a diameter of 12 m. What is its circumference? Use π ≈ 3.14. — 37.68 m.C = πd = 3.14 × 12 = 37.68 m.
A circular pizza has a radius of 9 inches. What is its area? Use π ≈ 3.14. — 254.34 in².A = πr² = 3.14 × 81 = 254.34 in².
A semicircle has a diameter of 8 cm. What is the area of the semicircle? Leave in terms of π. — 8π cm².Radius = 4. Full circle area = π(4²) = 16π. Half = 8π cm².
Which is larger: the circumference of a circle with r = 5, or the area of a circle with r = 3? (Use exact values) — Circumference (10π > 9π).Circumference = 2π(5) = 10π. Area = π(3²) = 9π. 10π > 9π, so circumference of r=5 circle is larger (though note units differ: linear vs area).
A circle has circumference 31.4 cm. What is its radius? (Use π ≈ 3.14) — 5 cm.C = 2πr → 31.4 = 2(3.14)r → 31.4 = 6.28r → r = 5 cm.
The circumference of a circle is 6π. What is the area? — 9π.C = 2πr = 6π → r = 3. A = π(3²) = 9π.
Which of the following represents the exact area of a circle with r = 11? — 121π.A = πr² = π(11²) = 121π. Exact form keeps π symbolic.
Two lines intersect. One angle is 70°. What are the measures of the other three angles? — 110°, 70°, 110°.Adjacent angles are supplementary: 180°−70°=110°. Vertical angles equal the original. So: 110°, 70°, 110°.
Angle A and angle B are complementary. Angle A = 3x° and angle B = 2x°. What is x? — 18.3x + 2x = 90 → 5x = 90 → x = 18.
Angle A and angle B are supplementary. Angle A = 4x° and angle B = 5x°. What is x? — 20.4x + 5x = 180 → 9x = 180 → x = 20.
Which pair of angles CANNOT be supplementary? — 91° and 91°.91° + 91° = 182° ≠ 180°. This pair cannot be supplementary.
Adjacent angles share a vertex and a side. If two adjacent angles form a straight line, they are: — Supplementary.Two adjacent angles that form a straight line (linear pair) are supplementary — they sum to 180°.
Two intersecting lines form four angles: 70°, a°, 70°, and b°. What are a° and b°? — a = 110°, b = 110°.Adjacent angles are supplementary: 180°−70°=110°. Verticals match: so a = b = 110°.
Which statement about vertical angles is always true? — They are equal.Vertical angles are always equal (congruent), formed by two intersecting lines on opposite sides of the vertex.
A pair of intersecting lines creates 4 angles. If one angle is 90°, what are the other three angles? — 90°, 90°, 90°.If one angle is 90°, adjacent angles are supplementary: 180°−90°=90°. All four angles are 90°, forming perpendicular lines.
A right triangle has one angle of 90° and another of 35°. What is the third angle? — 55°.180° − 90° − 35° = 55°.
Can a triangle have angles of 91°, 50°, and 39°? — Yes, 91+50+39=180.91 + 50 + 39 = 180°. Yes, this is a valid triangle.
An isosceles triangle has a vertex angle of 40°. What are the two base angles? — 70° each.Base angles of an isosceles triangle are equal. (180° − 40°) ÷ 2 = 70° each.
A triangle has angles in the ratio 1:2:3. What are the actual angles? — 30°, 60°, 90°.Parts sum = 1+2+3 = 6. Each part = 180°/6 = 30°. Angles: 30°, 60°, 90°.
A quadrilateral has three right angles. What is the fourth angle? — 90°.360° − 90° − 90° − 90° = 90°. A quadrilateral with four right angles is a rectangle!
The exterior angle of a triangle equals the sum of what? — The two non-adjacent interior angles.The Exterior Angle Theorem: an exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.
Two angles of a quadrilateral are each 90°. The other two angles are equal. What is each of those angles? — 90°.360° − 90° − 90° = 180°. Divided equally: 180°/2 = 90° each.
If one angle of a triangle is exactly 90°, what can you say about the other two angles? — They sum to 90°.180° − 90° = 90°. The remaining two angles must sum to 90°, making both acute.
A rectangular prism is 8 cm × 5 cm × 3 cm. What is its surface area? — 158 cm².SA = 2(lw + lh + wh) = 2(40 + 24 + 15) = 2(79) = 158 cm².
A square pyramid has base side 6 cm and height 8 cm. What is its volume? — 96 cm³.V = (1/3) × B × h = (1/3) × 36 × 8 = (1/3) × 288 = 96 cm³.
A cube has surface area 150 cm². What is the side length? — 5 cm.A cube has 6 square faces. Each face area = 150/6 = 25 cm². Side = √25 = 5 cm.
A triangular prism has a triangular face with base 6 cm and height 4 cm. The prism length is 10 cm. What is its volume? — 120 cm³.Base area (triangle) = (1/2)(6)(4) = 12 cm². V = B × h = 12 × 10 = 120 cm³.
What is the surface area of a cube with side length 4 in? — 96 in².SA = 6s² = 6 × 16 = 96 in².
A rectangular box is to be wrapped with paper (no overlaps). The box is 10 cm × 6 cm × 4 cm. How much paper is needed? — 248 cm².SA = 2(10×6 + 10×4 + 6×4) = 2(60 + 40 + 24) = 2(124) = 248 cm².
If you double the height of a rectangular prism while keeping the base the same, what happens to the volume? — Volume doubles.V = B × h. Doubling h means V_new = B × 2h = 2(Bh) = 2V. Volume doubles.
A box has volume 120 cm³. Its length is 5 cm and width is 4 cm. What is its height? — 6 cm.V = l×w×h → 120 = 5×4×h → 120 = 20h → h = 6 cm.
Which formula correctly gives the surface area of a rectangular prism with dimensions l, w, h? — SA = 2(lw + lh + wh).SA = 2(lw + lh + wh) accounts for all three pairs of opposite faces, each counted twice.
A cube has a volume of 27 in³. What is its surface area? — 54 in².V = s³ = 27 → s = 3. SA = 6s² = 6×9 = 54 in².
A triangular pyramid (tetrahedron) has 4 equilateral triangular faces, each with area 15 cm². What is its surface area? — 60 cm².SA = 4 × 15 = 60 cm². Surface area is the sum of all faces.
A prism's volume formula is V = B × h, where B is the area of the base. A hexagonal prism has base area 42 cm² and height 7 cm. What is its volume? — 294 cm³.V = B × h = 42 × 7 = 294 cm³.
What cross section do you get slicing a cube parallel to one face? — Square.A cut parallel to a face of a cube produces a square, since all faces of a cube are congruent squares.
What cross section is produced by slicing a rectangular prism with a cut parallel to the bottom face? — Rectangle.A horizontal cut through a rectangular prism produces a rectangle with the same dimensions as the base.
Which cross section is formed when a cone is sliced at an angle (not parallel or perpendicular to the base)? — Ellipse.An angled cut through a cone (that doesn't pass through the apex) produces an ellipse.
If you slice a rectangular pyramid horizontally close to the base, what shape is the cross section? — A large rectangle.Horizontal cuts near the base of a rectangular pyramid produce large rectangles. Cuts closer to the apex produce smaller rectangles.
If you slice a rectangular pyramid vertically through the apex, what shape is the cross section? — Triangle.A vertical cut through the apex of a rectangular pyramid produces a triangle.
A triangular prism is sliced parallel to the triangular faces. What is the cross section? — Triangle.Cutting parallel to the triangular bases of a triangular prism produces a triangle congruent to the base.
A triangular prism is sliced perpendicular to the triangular faces (through the rectangular sides). What is the cross section? — Rectangle.A cut perpendicular to the triangular bases (cutting through the rectangular lateral faces) produces a rectangle.
Two different cuts of the SAME solid produce different shapes. Which statement best explains this? — The shape of the cross section depends on the angle and direction of the cut.The cross section's shape depends entirely on how the cutting plane intersects the solid — its direction, angle, and position.
A cross section of a cone through the apex and perpendicular to the base produces what shape? — Triangle.A cut through the apex of a cone, perpendicular to the base, produces a triangle (with the apex as the top vertex).