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Triangle base, height and area

Find a perpendicular height and connect triangle area with half a parallelogram.

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Enlarged diagram

Which line meets your chosen base at a right angle?

Match the base and height

Choose AB as the base of triangle ABC. Its height reaches from the opposite vertex C to the line containing AB, at a right angle. The sloping sides do not make that right angle with AB.

Base AB and its perpendicular height

The dashed segment joins C to AB at the square right-angle mark.

Each grid step is 1 cm. The square mark identifies a right angle; h names the dashed segment.
Figure data

This table is included in print while it is open.

Triangle ABC. Each grid step represents 1 cm
Point or marked featurex (grid steps)y (grid steps)
A05
B85
C20
Line h start20
Line h end25
Right-angle mark 1 vertex25
Right-angle mark 1 first arm85
Right-angle mark 1 second arm20
Perpendicular height for base AB

A base and its matching perpendicular height belong together. A base length alone cannot determine the area.

Why a triangle uses half the area

Join two congruent copies of triangle ABC along BC. A half-turn maps one copy onto the other. Together, they form a parallelogram. Its horizontal base is 8 cm and its perpendicular height is 5 cm.

Follow the cut piece. What changes, and what stays the same?

Two equal triangle copies

Triangles ABC and BCD share BC. The dashed cut from C to H is perpendicular to AB. Each grid step is 1 cm.

The joined pair keeps the original 8 cm base and 5 cm perpendicular height. BC splits it into two triangles with the same shape and size.
Figure data

This table is included in print while it is open.

Two matching triangle copies share BC
Point or marked featurex (grid steps)y (grid steps)
A05
B85
D100
C20
H25
Line 1 start85
Line 1 end20
Line 2 start20
Line 2 end25
Right-angle mark 1 vertex25
Right-angle mark 1 first arm85
Right-angle mark 1 second arm20

Move a piece without changing its area

The moved triangle fills BED. The outside is now rectangle HEDC, 8 cm by 5 cm. The interior line BD marks the moved piece.

Both drawings use the same grid scale. Moving the cut piece changes the outline, while preserving the area of the pair.
Figure data

This table is included in print while it is open.

Move the left triangular end eight grid steps right
Point or marked featurex (grid steps)y (grid steps)
H25
E105
D100
C20
B85
Line 1 start85
Line 1 end100
Right-angle mark 1 vertex25
Right-angle mark 1 first arm105
Right-angle mark 1 second arm20
  1. Two equal triangles make a parallelogram. Its base is 8 centimetres, perpendicular height 5 centimetres.
  2. Cut along C H. Move triangle A, C, H eight grid steps right.
  3. A reaches B. C reaches D. H reaches E.
  4. You filled an 8 centimetre by 5 centimetre rectangle. The area stays the same.

The two original triangles share that area equally. Find the area of one triangle. For these area and perimeter fields, enter the numerical value in the unit shown. Exact fractions can use /.

Show workingHide answer

The pair has area 8 × 5 = 40 cm². One triangle has area 40÷2 = 20 cm².

triangle area = 1/2 × base × perpendicular height.

The right angle matters: base × height measures the enclosing parallelogram’s area. A sloping side paired with that base does not generally give this area.

Heights can lie outside the triangle

Keep AB as the base. The opposite vertex now lies beyond one end of AB. Extend the line through AB to draw the perpendicular height. The triangle’s boundary stays the same.

The height reaches the extended base line

The height h meets an extension of the line through AB at a right angle. The extension e is outside the triangle.

Each grid step is 1 cm. AB is 8 cm long and h is 5 cm long. Segment e extends the base line; it does not extend side AB.
Figure data

This table is included in print while it is open.

Height outside triangle ABC. Each grid step represents 1 cm
Point or marked featurex (grid steps)y (grid steps)
A05
B85
C-20
Line h start-20
Line h end-25
Line e start05
Line e end-25
Right-angle mark 1 vertex-25
Right-angle mark 1 first arm05
Right-angle mark 1 second arm-20
Show workingHide answer

1/2 × 8 × 5 = 20 cm². The extension helps locate the height; it is not an extra side or an extra region to count. A height can lie inside, along a side, or outside a triangle.

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Check your understanding 1

Choose a height for a turned triangle

Use AB as the base. Name a side that is its perpendicular height, then give both lengths. Each grid step is 1 cm.

A vertical base

AB is vertical and AC is horizontal, with a right-angle mark at A.

Turning the triangle does not change which two directions are perpendicular.
Figure data

This table is included in print while it is open.

Triangle with vertical base AB. Each grid step represents 1 cm
Point or marked featurex (grid steps)y (grid steps)
A00
B06
C80
Right-angle mark 1 vertex00
Right-angle mark 1 first arm06
Right-angle mark 1 second arm80
Height paired with AB
Using AB as base, why is AC the matching height?
Show answerHide answer

AC (or CA) is the height: it is perpendicular to AB. Base AB is 6 cm and height AC is 8 cm, so the area is 6 × 8÷2 = 24 cm². The sloping side BC does not meet AB at a right angle.

Which parts cover the shape, and which sides run around it?

Split the shape to find its area

A 12 cm by 7 cm rectangle loses a triangular corner with base 4 cm and perpendicular height 3 cm. The new sloping side is shown as 5 cm. Find the remaining area and the exposed perimeter.

A rectangle with a triangular corner removed

The outside boundary follows A, B, C, D, E and back to A. Dashed inside lines divide the remaining region into two rectangles and a triangle.

Each grid step is 1 cm. BC is shown as 5 cm. The dashed partition lines are inside the figure, not part of its exposed perimeter.
Figure data

This table is included in print while it is open.

Rectangle with a triangular corner removed. Each grid step represents 1 cm
Point or marked featurex (grid steps)y (grid steps)
A00
B80
C123
D127
E07
P03
Q83
Line 1 start03
Line 1 end123
Line 2 start80
Line 2 end83
Show workingHide answer

Area by subtraction: 12 × 7-1/2 × 4 × 3 = 84-6 = 78 cm².

Area by parts: the dashed lines give rectangles of areas 12 × 4 = 48 cm² and 8 × 3 = 24 cm², plus a triangle of area 4 × 3÷2 = 6 cm². Their total is 48 + 24 + 6 = 78 cm². They meet without overlapping.

Perimeter: follow AB, BC, CD, DE and EA: 8 + 5 + 4 + 12 + 7 = 36 cm. Alternatively, start from the old rectangle’s perimeter and replace the cut edges: 38-4-3 + 5 = 36 cm.

Area uses cm²; perimeter uses cm. A triangle’s area does not determine its sloping side length. Use the shown 5 cm; counting horizontal or vertical grid steps along that slope would not measure it.

Check your understanding 2

Keep area and perimeter separate

A 9 cm by 6 cm rectangle loses a triangular corner with base 3 cm and perpendicular height 4 cm. The new sloping boundary is shown as 5 cm. Find the remaining area and perimeter, with units.

A different corner cut

The five-sided remaining region has one sloping boundary shown as 5 cm.

Each grid step is 1 cm. Choose your own area decomposition or subtraction, and trace the exposed boundary for perimeter.
Figure data

This table is included in print while it is open.

A 9 cm by 6 cm rectangle with a triangular corner removed. Each grid step represents 1 cm
Point or marked featurex (grid steps)y (grid steps)
A00
B60
C94
D96
E06
How does the triangular corner cut change the perimeter?
Show answerHide answer

Area: 9 × 6-3 × 4÷2 = 54-6 = 48 cm².

Perimeter: the old rectangle has 2 × (9 + 6) = 30 cm. Replace the removed 3 cm and 4 cm lengths with the shown 5 cm side: 30-3-4 + 5 = 28 cm. Check by tracing the five exposed sides: 6 + 5 + 2 + 9 + 6 = 28 cm.

The cut changes both measures, but they count different features.

Which base and height belong together?

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Check your understanding 3

Which length is the height?

A triangle has base 10 cm, perpendicular height 6 cm and a sloping side 8 cm. A learner uses 10 × 8÷2. Find the area and explain the error.

Which length matches the 10 cm base in the area formula?
Show answerHide answer

10 × 6÷2 = 30 cm². The 8 cm side is not perpendicular to the chosen base. Using it with that base does not measure the enclosing parallelogram’s area.

Explain which length is the perpendicular height for your base. Then fix the calculation.

Work backwards

A triangle has area 42 cm² and base 12 cm. Find its perpendicular height. Check your result by rebuilding the known area.

Which calculation recovers the triangle’s height?
Show answerHide answer

Twice the triangle’s area is 42 × 2 = 84 cm². That is the area of a rectangle with base 12 cm and the required height. Its height is 84÷12 = 7 cm.

Check 1/2 × 12 × 7 = 42 cm². Keep the units distinct: 42 cm² is an area; 7 cm is a length.

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