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Perimeter area circles surface area and volume - Mathematics Year 10

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Introduction

Perimeter, area and volume answer different questions and use different dimensions. A visible face is not always an exposed face once solids touch, and a drilled hole changes more than just volume.

Complete eight mensuration questions and two composite-solid tasks. Keep exact π answers, convert squared/cubed units correctly and use independent face-count or cross-section checks.

Complete eight mensuration questions with exact units

Circumference is a length around a circle; area measures its enclosed two-dimensional region. With radiusr, use $C = 2 π r, A = π r^{2} .$ The source radius 6 cm gives 12πcm circumference and 36πcm² area. A diameter 6 cm would instead mean radius 3 cm, so the stated radius/diameter role matters.

Eight complete mensuration questions. Each problem states its own shape, dimensions and whether a solid is closed. Keep exact results involvingπ; the following chapters explain the dimensional and composite checks rather than treating the answer table as evidence of independent work.

Task	Given information and request	Calculation	Exact result with units
1	Circle radius 6 cm; circumference and area	C=2π×6; A=π×6²	12πcm; 36πcm²
2	Trapezium parallel sides 7 and 11 cm, height 5 cm	A=(7+11)×5/2	45 cm²
3	90°sector, radius 8 cm; arc and area	s=(90/360)×2π×8; A=(90/360)×π×8²	4πcm; 16πcm²
4	8 by 5 cm rectangle, upper-right 3 by 2 cm corner removed	A=8×5−3×2; P=8+3+3+2+5+5	34 cm²; 26 cm
5	Closed cuboid 4 by 3 by 2 cm; volume and surface	V=4×3×2; S=2×(12+8+6)	24 cm³; 52 cm²
6	Right-triangular prism, legs 3 and 4 cm, hypotenuse 5 cm, length 10 cm	V=(3×4/2)×10; S=2×6+10×(3+4+5)	60 cm³; 132 cm²
7	Closed cylinder, radius 3 cm, height 7 cm	V=π×3²×7; S=2π×3²+2π×3×7	63πcm³; 60πcm²
8	Convert 0.75 m² and 0.002 m³ to centimetre units	0.75×10000; 0.002×1000000	7500 cm²; 2000 cm³

For the trapezium, the perpendicular height 5 cm separates parallel sides 7 and 11 cm. Their mean 9 cm multiplied by 5 cm gives 45 cm². The formula uses the perpendicular separation, not a sloping side which merely looks like a height. Doubling the trapezium into a parallelogram also gives(7+11)×5=90 cm², then halving gives 45.

A sector is a fraction of a full circle. For central angleθ measured in degrees, use $s = \frac{θ}{36 0^{\circ}} 2 π r, A_{s} = \frac{θ}{36 0^{\circ}} π r^{2} .$ A 90°sector of radius 8 cm is one quarter of the circle, so the arc is 4πcm and the area 16πcm². The full sector perimeter adds both radii: 4π+16 cm. Arc length alone is not the entire boundary length.

For the cut-corner rectangle, area subtracts the removed 3×2=6 cm² from 40 cm², giving 34 cm². The following original unit-cell model has 1 cm sides per cell. Filled cells remain; dots mark the removed upper-right corner.

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Text alternative: two rows of 5 remaining cells and three rows of 8, giving 34 cells. Trace the six boundary lengths 8, 3, 3, 2, 5, 5 cm to get 26 cm. The new inner horizontal/vertical edges replace equal removed outer lengths, so this particular corner cut leaves the perimeter unchanged. That is a property of this specified cut, not a rule that removing area never changes perimeter.

Explain solid faces, volumes and unit conversions

For a prism, volume is cross-sectional area multiplied by perpendicular extrusion length: $V = B ℓ .$ Here B is an area, not one side of the cross-section. Surface area counts every exposed face once. Its units are squared lengths; volume units are cubed lengths. Both can involve the same dimensions while producing different numerical expressions.

For the closed 4×3×2 cm cuboid, volume is 24 cm³. Its six faces occur in equal pairs with areas 12, 8 and 6 cm², so surface area 2(12+8+6)=52 cm². Multiplying all three edges finds volume, not a surface. Forgetting to double the three different face areas would count only half the closed box.

For the right-triangular prism, cross-sectional area 3×4/2=6 cm² and length 10 cm give volume 60 cm³. Its two triangular ends contribute 12 cm². The three rectangles have areas 30, 40 and 50 cm², giving total 132 cm². A second surface method uses two end areas plus cross-section perimeter times length: 2×6+(3+4+5)×10=132.

The actual original Corbettmaths Surface Area - Other Prisms example has a 5-12-13 right-triangle cross-section and length 20 cm. Follow how both triangles and all three rectangles are counted. Its numerical values differ from our 3-4-5 cross-section and length 10 cm, so our 132 cm² result requires our own data. Original: corbettmaths, Surface Area of Prisms - Corbettmaths. The written explanations and native tasks work without sound or the external player.

For the closed cylinder, radius 3 cm gives circular base area 9πcm². Height 7 cm makes volume 63πcm³. The curved wall unfolds into a rectangle of width 2πr=6πcm and height 7 cm, so its area is 42πcm². Add two end circles 18πcm² for the closed total 60πcm². An open-topped vessel would have a different stated surface problem; do not silently remove a lid.

Unit conversions also depend on dimension. One metre is 100 cm, so one square metre is 10000 cm² and one cubic metre is 1000000 cm³. Thus.75 m²=7500 cm² and.002 m³=2000 cm³=2 litres. Multiplying an area or volume only by 100 would apply a length conversion to the wrong dimension.

Keep πexact until any requested final approximation. Do not round every intermediate face or use a rounded radius before combining surfaces. These are mathematical calculations from supplied dimensions, not a claim that actual objects were measured accurately or that workshop measurement competence has been observed.

Solve two composite solids by independent methods

Two complete composite-solid tasks. Count volume only where material exists. For surface area, state which faces are exposed and whether an interior hole wall is included. The two tasks below explicitly require all exposed surfaces, including a through-hole’s inner wall.

Task	Exact construction and dimensions	Volume calculation	Exposed-surface calculation	Final exact results
A	Base 6×4×2 cm; top 3×4×1 cm, full 4 cm width, contact 3×4 cm	48+12	88+38−2×12	V 60 cm³; S 102 cm²
B	8×6×4 cm cuboid; radius 1 cm hole through height 4 cm, centre of 8×6 face	192−4π	208−2π+8π	V 192−4πcm³; S 208+6πcm²

In taskA, a 6×4×2 cm base supports a 3×4×1 cm top on one end, covering half the base length and the full 4 cm width. The volumes 48 and 12 cm³ do not overlap, so the total is 60 cm³. The base’s separate surface area is 88 cm² and the top’s 38 cm². Their 12 cm² contact rectangle is counted in both separate surfaces but is not exposed after joining.

Thus $S = S_{1} + S_{2} - 2 A_{c o n t a c t}$ gives 88+38−24=102 cm². Subtracting the contact only once would leave one hidden face counted as exposed. This method assumes the stated clean full-face contact, with no gaps, overlap volume or additional coatings.

An independent cross-section method checks both answers. The unit-cell cross-section below has base length 6, height 2, and a 3-unit-wide top one unit high. Extrude it through the stated width 4 cm. Filled cells are material and dots are empty.

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Text alternative: the cross-section has 3+6+6=15 square units and boundary lengths 6, 2, 3, 1, 3, 3, totalling 18 cm. Volume 15×4=60 cm³, and surface 2×15+18×4=102 cm² agree with the separate-solid method. The two cross-sectional ends and all exposed extrusion sides are counted exactly once.

In taskB, a cylinder of radius 1 cm is removed vertically through the entire 4 cm height of an 8×6×4 cm cuboid. The hole is centred on the 8×6 face and does not touch an outer side. Volume 192−π×1²×4=192−4πcm³. A different route takes cross-sectional material area 48−π and multiplies by height 4, agreeing exactly.

For the exposed surface, start with the closed cuboid area 208 cm². The two circular mouths remove 2πcm². The new inner cylindrical wall adds 2π×1×4=8πcm². Total 208+6πcm² includes that inner wall. Removing material can increase exposed area; volume decrease does not imply surface decrease.

A second surface check counts the remaining two horizontal faces 2(48−π), the four outer vertical walls 112, and the hole wall 8π. This sums to 208+6πagain. The original box net helps identify the old face groups, but the hole wall is new and cannot be found by merely subtracting circles from a six-face total.

Sphere, cone and pyramid formulae remain later GCSE progression in this allocation; their full surface/volume work is not claimed as covered by these tasks. Retain eight basic solutions, two complete composite tasks, unit/dimension checks and independent alternate routes. Numerical model work does not certify actual machining, measurement or safety competence.

Interactive Tasks

Quiz: Test Your Knowledge

Circle radius 6 cm. Circumference? (12πcm) (!6πcm) (!36πcm²) (!24πcm)

Same circle. Area? (36πcm²) (!12πcm²) (!6πcm²) (!72πcm²)

Trapezium parallel sides 7 and 11 cm, height 5 cm. Area? (45 cm²) (!90 cm²) (!35 cm²) (!55 cm²)

90°sector, radius 8 cm. Arc length? (4πcm) (!16πcm) (!4πcm²) (!8πcm)

Same sector. Area? (16πcm²) (!4πcm²) (!64πcm²) (!8πcm²)

Corner-cut 8 by 5 rectangle, corner 3 by 2 removed. Area? (34 cm²) (!40 cm²) (!6 cm²) (!26 cm²)

Closed 4 by 3 by 2 cuboid. Surface area? (52 cm²) (!24 cm²) (!26 cm²) (!48 cm²)

Closed cylinder radius 3, height 7. Volume? (63πcm³) (!60πcm³) (!21πcm³) (!18πcm³)

0.75 m² equals? (7500 cm²) (!75 cm²) (!750 cm²) (!75000 cm²)

Joined cuboids with contact area 12: what surface adjustment? (Subtract 24 cm² from their separate surface sum) (!Subtract 12 cm² only) (!Add 24 cm²) (!Subtract 12 cm³ from volume)

Memory Game

Perimeter	Length of the whole boundary
Area	Two-dimensional region measured in square units
Volume	Three-dimensional space measured in cubic units
Sector	Circle region cut by two radii and their arc
Prism	Solid with a constant cross-section along an extrusion
Contact face	Shared face hidden from the exposed outer boundary

Drag and Drop

12πcm	Circumference of the source radius 6 circle
36πcm²	Area of that same source circle
45 cm²	Source trapezium area
102 cm²	Stepped composite exposed surface
192−4πcm³	Drilled cuboid remaining volume

Crossword Puzzle

PERIMETER	Length around the full boundary
CIRCUMFERENCE	Perimeter of a circle
SECTOR	Region bounded by two radii and an arc
PRISM	Constant-cross-section solid
CYLINDER	Solid with circular cross-section and curved wall
CONTACT	Shared hidden face in a joined solid

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Distinguish dimensions: Name the length, area or volume unit for each request.
Read radius versus diameter: Use the centre-to-edge segment in the circle reference.
Read a circle fraction: Explain why 90°is one quarter.
Count unit cells: Check 34 remaining cells in the corner-cut shape.

Standard

Complete all eight basic questions: Retain exact πanswers and units.
Check prism and cylinder surfaces: Count ends and side walls with closed-solid assumptions.
Solve compositeA: Use separate volumes and remove both contact faces.
Solve compositeB: Remove cylinder volume and count all exposed surfaces.

Advanced

Verify the stepped solid independently: Use cross-section area 15 and perimeter 18.
Verify the drilled solid independently: Use remaining horizontal faces, outer walls and inner wall.
Challenge a surface intuition: Explain why removing material may increase surface area.
Evaluate scope and measurement: Separate exact supplied dimensions from observed practical competence.

Learning Check

Eight basic results: Keep exact values and dimensional units.
Circle and cut-corner checks: Explain whole boundary versus partial boundary.
CompositeA: Count material and exposed faces.
CompositeB: Count the hole wall as an exposed surface.
Units and first errors: Explain how incorrect formula choice is exposed.
Evidence and scope: Retain full working and independent routes.

Worked checks and assessment guidance

Circle 12πcm and 36πcm²; trapezium 45 cm²; sector arc 4πcm and area 16πcm²; cut-corner area 34 cm²/perimeter 26 cm; cuboid 24 cm³/52 cm²; triangular prism 60 cm³/132 cm²; cylinder 63πcm³/60πcm²; 7500 cm² and 2000 cm³.
Sector perimeter is 4π+16 cm because both radii join the arc. Corner cut removes 6 cm² but replaces outer 3+2 lengths with equal inner lengths, leaving this perimeter 26 cm. This does not mean every area removal leaves perimeter unchanged.
Volumes 48+12=60 cm³. Separate surfaces 88+38 minus twice contact 12 give 102 cm². Independent cross-section has area 15 and perimeter 18, so 15×4=60 and 2×15+18×4=102.
Remaining volume 192−4πcm³. Surface 208−2π+8π=208+6πcm². Independent grouping 2×(48−π)+112+8π agrees. The hole is centred and stays within the box.
1 m²=10000 cm² and 1 m³=1000000 cm³; 2000 cm³=2 L. Radius is not diameter, arc is not full sector perimeter, and volume is not surface. Closed-cylinder surface includes both end circles.
Keep eight basic questions, two composite tasks, unit checks and alternate calculations. Preserve exact πuntil requested final approximation. Sphere/cone/pyramid formulae remain later progression here; model calculations do not certify practical measurement, machining or safety competence.

Learning evidence

Keep eight complete mensuration solutions, two full composite-solid tasks, exact πresults, dimensional conversions, face/contact reasoning and independent cross-section checks.

OER and Wikipedia

Department for Education:GCSE Mathematics subject content and assessment objectives. Source area:GCSE Mathematics subject content > Scope of study > Geometry and measures - Mensuration and calculation items 14 to 19. Year10 allocation is editorial within the GCSE pathway;Higher-only extensions are explicitly labelled. Original examples,teaching and feedback;Commons images retain their actual attribution,and videos identify original publishers. Reviewed12November2026.

Circle-withsegments.svg, Jokes Free4Me, Public domain. Unchanged original; caption and learning activities are original.

Triangular prism.svg, Triangular_prism.png: mathematics derivative work: Lovasoa (talk), CC BY-SA 3.0. Unchanged original; caption and learning activities are original.

Net of cuboid.png, A2569875, CC BY-SA 3.0. Unchanged original; caption and learning activities are original.