Perimeter and area

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Identify sides, vertices and parallel edges
Use properties rather than appearance to classify a shape

Vocabulary: polygon · vertex · parallel · perpendicular · regular

Retrieve previous learning

How many vertices does a quadrilateral have?

Check: Four.

Learn the concept

A polygon is a closed 2D shape formed from straight line segments. A square is both a rectangle, because it has four right angles, and a rhombus, because its four sides are equal. Classifications can overlap; a drawing's orientation does not change its properties.

When preparing to calculate perimeter or area, first identify the shape and its useful properties. A regular pentagon has five equal sides, so if one side is 4 cm, its perimeter is 5 × 4 = 20 cm. Its area cannot be found from the side length by the rectangle formula.

Worked model

Identify five straight sides and five vertices in the stated regular pentagon.
Regular means the sides are equal, so each is 4 cm.
Add all five lengths: perimeter = 20 cm.

Guided practice

A four-sided figure has four right angles and sides 3 cm and 7 cm alternately. Classify it and find its perimeter.

Hint: The opposite sides are equal.

Check the guided answer

It is a rectangle, with perimeter 3 + 7 + 3 + 7 = 20 cm.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“A tilted square becomes a diamond and stops being a square.”

Correction: Rotation preserves side lengths and angles; 'diamond' is an informal drawing description.

Extend and reflect

Is every square a parallelogram?

Guidance: Yes. Its two pairs of opposite sides are parallel.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Identify sides, vertices and parallel edges. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Geometry and measures · original lesson pack

DASKEREL beta · internal AI review pending

Separate boundary length from enclosed area

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Use linear units for perimeter
Use square units for area

Vocabulary: perimeter · area · linear unit · square unit

Retrieve previous learning

What is 3 cm × 4 cm as an area?

Check: 12 cm².

Learn the concept

Perimeter measures a boundary, so a 3 cm by 4 cm rectangle has perimeter 3 + 4 + 3 + 4 = 14 cm. Area measures how many square units cover the interior, so its area is 3 × 4 = 12 cm². The two numbers and units answer different questions.

Rectangles with equal area can have different perimeters. A 2 cm by 6 cm rectangle also has area 12 cm² but perimeter 16 cm. Do not infer a boundary length from area alone without additional shape information.

Worked model

Sketch a rectangle labelled 3 cm and 4 cm.
Add its four boundary lengths to get 14 cm.
Multiply adjacent lengths to get 12 cm² and keep the squared unit.

Guided practice

Find perimeter and area of a 5 m by 8 m rectangle.

Hint: Boundary addition and interior multiplication use different units.

Check the guided answer

Perimeter = 2(5 + 8) = 26 m; area = 5 × 8 = 40 m².

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Perimeter and area use the same units because both describe a shape.”

Correction: Perimeter is one-dimensional length; area counts two-dimensional square units.

Extend and reflect

Can a square of area 25 cm² have perimeter 25 cm?

Guidance: Its side is 5 cm and perimeter is 20 cm; the equal numerical area and perimeter would require different side length.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Use linear units for perimeter. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Geometry and measures · original lesson pack

DASKEREL beta · internal AI review pending

Derive perimeter rules from equal side properties

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Add all exterior side lengths
Generalise a perimeter formula

Vocabulary: perimeter · regular · side length · formula

Retrieve previous learning

How many sides has a triangle?

Check: Three.

Learn the concept

Perimeter is the length of a complete trip around a shape. A regular n-sided polygon with side length s has perimeter P = ns, because there are n equal contributions of s. An irregular polygon needs its individual exterior side lengths, including any sides not drawn horizontally.

A rectangle with adjacent sides a and b has P = a + b + a + b = 2(a + b). This is a derived rule: opposite sides match. For a 7 cm by 4 cm rectangle, P = 22 cm. Always state the length unit.

Worked model

List an irregular quadrilateral's exterior side lengths as 3, 5, 4 and 6 cm.
Add 3 + 5 + 4 + 6 = 18.
Report perimeter 18 cm; no area has been calculated.

Guided practice

A regular octagon has 3.2 cm sides. Find its perimeter.

Hint: Use the number of equal sides.

Check the guided answer

8 × 3.2 = 25.6 cm.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Perimeter is found by multiplying any two side lengths.”

Correction: That product may be an area for a rectangle; perimeter adds boundary lengths.

Extend and reflect

A regular n-gon has perimeter 36 cm and side 4 cm. Find n.

Guidance: n = 36 ÷ 4 = 9 sides.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Add all exterior side lengths. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Geometry and measures · original lesson pack

DASKEREL beta · internal AI review pending

Use a known perimeter to find an unknown side

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Represent a missing side with a letter
Solve a perimeter equation and check it

Vocabulary: unknown length · perimeter equation · inverse · constraint

Retrieve previous learning

What is 30 − 18?

Check: 12.

Learn the concept

If a quadrilateral's sides are 5 cm, 7 cm, 8 cm and x cm, and the perimeter is 26 cm, then 5 + 7 + 8 + x = 26. The known sides total 20, so x = 6 cm. Check 5 + 7 + 8 + 6 = 26.

Shape properties can reduce unknowns. A rectangle has opposite sides equal. If its perimeter is 34 cm and one side is 6 cm, then 2(6 + w) = 34, so 6 + w = 17 and w = 11 cm. A negative length would show that data or interpretation is invalid.

Worked model

Write 2(6 + w) = 34.
Divide both sides by 2: 6 + w = 17.
Subtract 6: w = 11 cm; check 2(6 + 11) = 34.

Guided practice

A rectangle has perimeter 40 m and length 12 m. Find its width.

Hint: The semi-perimeter is 20 m.

Check the guided answer

2(12 + w) = 40, so 12 + w = 20 and w = 8 m.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“A rectangle has four copies of its longer side in the perimeter.”

Correction: It has two copies of each adjacent side, unless it is a square.

Extend and reflect

A rectangle's width is half its length and its perimeter is 48 cm. Find both dimensions.

Guidance: Let length be L and width L/2; 2(L + L/2) = 48 gives L = 16 cm and width 8 cm.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Represent a missing side with a letter. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Geometry and measures · original lesson pack

DASKEREL beta · internal AI review pending

Trace the outside of a composite right-angled shape

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Count every exterior edge once
Deduce a missing inset length

Vocabulary: composite · rectilinear · inset · boundary

Retrieve previous learning

What is the perimeter of an 8 cm by 6 cm rectangle?

Check: 28 cm.

Learn the concept

Consider an 8 cm by 6 cm rectangle with a 3 cm by 2 cm corner removed from the top right. The L-shape's exposed boundary has lengths 5, 2, 3, 4, 8 and 6 cm in order. Their sum is 28 cm. The removed top and right edges are replaced by equal inset edges, so in this special corner-cut case the perimeter stays equal to the outer rectangle's.

Do not apply that result to every cut-out. A rectangular notch cut inward from the middle of an edge adds two extra depth edges and increases perimeter. Trace the actual exterior path, label unknown spans by subtracting aligned lengths, and exclude internal shared edges.

Worked model

Top remaining span = 8 − 3 = 5 cm.
Right remaining span = 6 − 2 = 4 cm.
Add boundary 5 + 2 + 3 + 4 + 8 + 6 = 28 cm.

Guided practice

A 10 cm by 7 cm rectangle loses a 2 cm by 3 cm top-right corner. Find the L-shape perimeter.

Hint: List 8, 3, 2, 4, 10 and 7 around the exterior.

Check the guided answer

8 + 3 + 2 + 4 + 10 + 7 = 34 cm.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Any indentation decreases the perimeter because material was removed.”

Correction: A notch can create new exposed edges; trace the complete boundary.

Extend and reflect

Explain algebraically why a corner cut from a rectangle preserves perimeter.

Guidance: A removed width w and height h are replaced by inner edges w and h, so the total boundary change is −w − h + w + h = 0.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Count every exterior edge once. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Geometry and measures · original lesson pack

DASKEREL beta · internal AI review pending

Derive triangle area from a matching parallelogram

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Identify a base and its perpendicular height
Derive A = bh/2

Vocabulary: base · perpendicular height · parallelogram · area

Retrieve previous learning

What is the area of an 8 cm by 5 cm rectangle?

Check: 40 cm².

Learn the concept

Two congruent copies of a triangle can form a parallelogram with the same base b and perpendicular height h. The parallelogram area is bh, so one triangle occupies half: A = bh/2. The height is measured at right angles to the chosen base, even when it falls outside an obtuse triangle.

For base 8 cm and perpendicular height 5 cm, A = 8 × 5 ÷ 2 = 20 cm². A sloping side labelled 6 cm is not automatically the height; using it without a right-angle relationship may produce the wrong area.

Worked model

Choose base 8 cm and its perpendicular height 5 cm.
Find a matching rectangle or parallelogram area 8 × 5 = 40 cm².
Halve it to get triangle area 20 cm².

Guided practice

Find a triangle's area with base 10 m and perpendicular height 7 m.

Hint: Multiply then halve.

Check the guided answer

A = 10 × 7 ÷ 2 = 35 m².

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Any two labelled sides can be multiplied and halved for triangle area.”

Correction: The two measurements must be a base and its perpendicular height.

Extend and reflect

Can different base-height pairs give the same triangle area?

Guidance: Yes. A base of 8 and height 5 gives the same 20 square units as a base of 10 and height 4.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Identify a base and its perpendicular height. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Geometry and measures · original lesson pack

DASKEREL beta · internal AI review pending

Rearrange triangle area to find a missing dimension

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Solve A = bh/2 for b or h
Check dimensional units

Vocabulary: rearrange · base · height · inverse operation

Retrieve previous learning

What is twice 24?

Check: 48.

Learn the concept

If a triangle has area 24 cm² and base 8 cm, then 24 = 8h/2 = 4h, so h = 6 cm. Equivalently, multiply the area by 2 and divide by the base: h = 2A/b. The result is a length, not square centimetres.

When the height is given instead, b = 2A/h. For area 30 m² and height 5 m, b = 60/5 = 12 m. A zero height produces zero area, and the inverse formula is used only when the given nonzero dimension permits division.

Worked model

Write 24 = 8h/2.
Multiply by 2: 48 = 8h.
Divide by 8: h = 6 cm; check 8 × 6 ÷ 2 = 24 cm².

Guided practice

A triangle's area is 45 cm² and its perpendicular height is 9 cm. Find its base.

Hint: Use b = 2A/h.

Check the guided answer

b = 2 × 45 ÷ 9 = 10 cm.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“The missing height is area divided by base.”

Correction: Triangle area includes a half, so multiply area by 2 before dividing by base.

Extend and reflect

If area is fixed and base doubles, what happens to height?

Guidance: From h = 2A/b, doubling b halves h.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Solve A = bh/2 for b or h. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Geometry and measures · original lesson pack

DASKEREL beta · internal AI review pending

Find a composite area by splitting or subtracting

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Partition an L-shape into rectangles
Cross-check by subtracting a cut-out

Vocabulary: composite area · partition · subtract · square unit

Retrieve previous learning

What is 8 × 6 − 3 × 2?

Check: 48 − 6 = 42.

Learn the concept

Take an 8 cm by 6 cm outer rectangle with a 3 cm by 2 cm top-right corner removed. The L-shape area is 8 × 6 − 3 × 2 = 42 cm². A second method splits it into a 5 cm by 6 cm left rectangle and a 3 cm by 4 cm lower-right rectangle: 30 + 12 = 42 cm².

Both methods work because the pieces do not overlap and together cover the shape exactly. Mark all dimensions before calculating; a length missing from the drawing can be inferred from aligned outer and cut-out lengths.

Worked model

Outer area is 8 × 6 = 48 cm².
Removed corner area is 3 × 2 = 6 cm².
Subtract to get 42 cm², then verify by 5 × 6 + 3 × 4 = 42 cm².

Guided practice

A 10 m by 7 m rectangle loses a 2 m by 3 m top-right corner. Find the area.

Hint: Subtract the missing small rectangle.

Check the guided answer

10 × 7 − 2 × 3 = 70 − 6 = 64 m².

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Composite area always requires adding every visible rectangle.”

Correction: For a cut-out, subtract the removed area from the enclosing shape or split the remaining part without overlap.

Extend and reflect

If the cut-out depth doubles but width stays 4 cm, how much does area decrease?

Guidance: The removed area rises by 4 × the extra depth, subject to the depth fitting inside the outer shape.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Partition an L-shape into rectangles. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Geometry and measures · original lesson pack

DASKEREL beta · internal AI review pending

Find an unknown cut-out size from composite area

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Set up an outer-area-minus-cut-out equation
Check a found length fits the shape

Vocabulary: unknown depth · cut-out · area equation · constraint

Retrieve previous learning

What is 9 × 6?

Check: 54.

Learn the concept

An outer 9 cm by 6 cm rectangle has a top-right rectangular cut-out of width 3 cm and unknown depth h. If the remaining area is 45 cm², then 54 − 3h = 45, so 3h = 9 and h = 3 cm. The depth fits within the 6 cm outer height.

A numerical answer still needs a geometric check. If the calculation yielded h = 8 cm, it would not fit a 6 cm-high outer rectangle, indicating inconsistent data or a wrong model. Use square units for area and linear units for h.

Worked model

Compute outer area 9 × 6 = 54 cm².
Subtract remaining area: missing area = 54 − 45 = 9 cm².
Divide by cut-out width 3 cm: h = 9 ÷ 3 = 3 cm.

Guided practice

An 8 m by 5 m rectangle has a 2 m wide corner cut-out. Remaining area is 34 m². Find cut-out depth.

Hint: The missing area is 40 − 34.

Check the guided answer

Missing area is 6 m², so depth = 6 ÷ 2 = 3 m.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Every solution of the area equation is a physically possible cut-out.”

Correction: Check the found dimensions against the outer shape and the model's constraints.

Extend and reflect

Write a formula for the remaining area of an L-shape with outer width W, height H, cut-out width w and depth h.

Guidance: A = WH − wh, with 0 ≤ w ≤ W and 0 ≤ h ≤ H.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Set up an outer-area-minus-cut-out equation. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Geometry and measures · original lesson pack

DASKEREL beta · internal AI review pending

Derive and use the trapezium area formula

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Identify parallel sides and perpendicular height
Derive A = (a + b)h/2

Vocabulary: trapezium · parallel sides · perpendicular height · average

Retrieve previous learning

What is the average of 6 and 10?

Check: 8.

Learn the concept

Join two congruent copies of a trapezium to form a parallelogram. Its base is the sum of the two parallel side lengths, a + b, and its perpendicular height is h. One trapezium is half the parallelogram, so A = (a + b)h/2. This also means average parallel width times height.

For parallel sides 6 cm and 10 cm and perpendicular height 4 cm, A = (6 + 10) × 4 ÷ 2 = 32 cm². The sloping nonparallel sides are not substituted for h. A rectangle is a special case when a = b.

Worked model

Identify parallel lengths 6 cm and 10 cm.
Add and halve: (6 + 10)/2 = 8 cm average width.
Multiply by perpendicular height 4 cm: A = 32 cm².

Guided practice

Find area for parallel sides 5 m and 9 m, perpendicular height 6 m.

Hint: Average the parallel sides, then multiply by height.

Check the guided answer

(5 + 9) × 6 ÷ 2 = 42 m².

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Any side joining the parallel sides can be used as the height.”

Correction: Height is the shortest, perpendicular distance between the parallel lines.

Extend and reflect

Why does the formula become bh for a rectangle?

Guidance: Set a = b; then (b + b)h/2 = bh.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Identify parallel sides and perpendicular height. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Geometry and measures · original lesson pack

DASKEREL beta · internal AI review pending

Rearrange trapezium area for an unknown parallel side

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Solve A = (a + b)h/2 for a side
Check that the found side gives the stated area

Vocabulary: rearrange · parallel side · height · constraint

Retrieve previous learning

What is 2 × 36 ÷ 4?

Check: 18.

Learn the concept

If a trapezium has area 36 cm², height 4 cm and one parallel side 7 cm, then 36 = (7 + b) × 4/2. Twice the area divided by height gives the sum of parallel sides: 72/4 = 18. Therefore b = 18 − 7 = 11 cm.

The general rearrangement is b = 2A/h − a, provided h is nonzero. Check the result by substituting: (7 + 11) × 4/2 = 36 cm². A negative result would require examining the measurements and diagram.

Worked model

Write 36 = (7 + b)4/2.
Multiply both sides by 2 and divide by 4 to get 7 + b = 18.
Subtract 7: b = 11 cm; check the original formula.

Guided practice

Area is 45 m², height 5 m and one parallel side 8 m. Find the other.

Hint: The two parallel sides sum to 2A/h.

Check the guided answer

2 × 45/5 = 18 m total, so the other side is 18 − 8 = 10 m.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Divide the area by height and subtract the known side.”

Correction: The area formula halves the product, so first use 2A/h to obtain the sum of parallel sides.

Extend and reflect

For fixed A, h and one parallel side a, how does increasing a change b?

Guidance: Since a + b = 2A/h is fixed, b falls by the same amount, until a valid positive length constraint is reached.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Solve A = (a + b)h/2 for a side. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 7 · Geometry and measures · original lesson pack

DASKEREL beta · internal AI review pending

Choose the right measure in a composite design

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Select perimeter or area for a stated purpose
Combine formulae with unit conversions and a reasoned check

Vocabulary: boundary · coverage · composite shape · conversion

Retrieve previous learning

How many centimetres are in 2 m?

Check: 200 cm.

Learn the concept

A fictional 8 m by 6 m rectangular garden has a 3 m by 2 m top-right corner excluded. Its planting area is 8 × 6 − 3 × 2 = 42 m². Its outer L-shaped boundary is 5 + 2 + 3 + 4 + 8 + 6 = 28 m. Mulch is bought by area; edging is bought by length.

If mulch covers 6 m² per bag, exactly 42 ÷ 6 = 7 bags cover the modelled area. If edging is sold in 5 m lengths, 28 ÷ 5 = 5.6, so six whole lengths are needed before accounting for cuts or waste. Real purchases need a stated allowance and product specifications.

Worked model

State which quantity the task asks: mulch needs m², edging needs m.
Calculate 42 m² and 28 m from the described L-shape.
Divide by package coverage and round whole items up only where required.

Guided practice

A 10 m by 4 m rectangle needs grass seed at one bag per 8 m² and one continuous border. How many bags and border metres?

Hint: Area and perimeter are separate.

Check the guided answer

Area = 40 m², so 5 bags. Perimeter = 2(10 + 4) = 28 m of border.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“The same numerical answer can be used for both material coverage and edging length.”

Correction: Coverage is area in square units; edging follows a boundary in linear units.

Extend and reflect

If a 4 m by 3 m notch is cut from the middle of a 10 m edge, how does it change area and perimeter?

Guidance: Area decreases by 12 m² and perimeter increases by two 3 m notch depths, or 6 m.