Practice Maths

Area of Triangles

Key Terms

Triangle
A polygon with three sides and three angles.
Base (b)
Any side of the triangle chosen as the bottom reference.
Perpendicular height (h)
The vertical distance from the base to the opposite vertex, measured at 90° to the base. For right-angled triangles, the two shorter sides (legs) act as the base and height.
Area
The amount of space inside a shape, measured in square units (cm², m², mm²).
base b h Acute triangle base b h Right triangle (legs = base & height)

The height h is always perpendicular (90°) to the base.

Area Formula

The area of a triangle is:

A = ½ × base × height    (A = ½bh)

  • The height must be perpendicular to the base.
  • For right-angled triangles: the two legs are the base and height.
  • A triangle is exactly half a parallelogram with the same base and height.
Hot Tip
Always check which measurement is the perpendicular height — it might not be the longest side or the slant side. For obtuse triangles, the height may fall outside the triangle; you still use the perpendicular height from the base’s line.

Worked Example 1

Base 10 cm, height 6 cm.

A = ½ × 10 × 6 = 30 cm²

Worked Example 2

Base 8 m, height 5 m.

A = ½ × 8 × 5 = 20 m²

Worked Example 3 — Right-Angled Triangle

Legs 9 cm and 4 cm.

A = ½ × 9 × 4 = 18 cm²

Parallelogram: A = b × h = 2 identical triangles → Triangle: A = ½bh = half the parallelogram

Cut a parallelogram along its diagonal — both triangles are identical, each with half the area.

The Triangle Area Formula

The area of a triangle is: A = ½ × base × height (also written as A = ½bh)

The base (b) is any side of the triangle. The height (h) is the perpendicular distance from that base to the opposite vertex (measured at 90° to the base).

  • Triangle with base 8 cm and height 5 cm: A = ½ × 8 × 5 = 20 cm²
  • Triangle with base 12 m and height 7 m: A = ½ × 12 × 7 = 42 m²

Why Is It Half a Parallelogram?

Every triangle is exactly half of a parallelogram (or rectangle). If you take any triangle and make an identical copy, you can always arrange them together to form a parallelogram with the same base and height. Since the area of the parallelogram is b × h, the triangle is half of that: ½ × b × h.

This means the formula works for ALL triangles — right-angled, acute, obtuse, equilateral, scalene — as long as you use the correct perpendicular height.

Identifying the Base and Perpendicular Height

This is the tricky part for different triangle orientations:

  • For a right-angled triangle: the two shorter sides (legs) serve as base and height. A = ½ × leg1 × leg2.
  • For an acute triangle: the height is inside the triangle, drawn as a dotted line from a vertex to the opposite side at 90°.
  • For an obtuse triangle: the height may fall outside the triangle — extend the base line and draw the perpendicular from the vertex to that extended line.

Composite Areas Using Triangles and Rectangles

Complex shapes can be split into triangles and rectangles. Find the area of each part separately, then add them together.

  • A house shape (rectangle + triangle on top): Area = (l × w) + (½ × b × h)
  • An arrow shape can be split into a rectangle and a triangle.
Key tip: The height in the triangle area formula must be perpendicular to the base — it must form a 90° angle. If the slant side of a triangle is given but the perpendicular height is not labelled, do not use the slant side as h. Look for the right angle symbol or the dotted height line in the diagram.

Mastery Practice

  1. Calculate the Area Fluency

    Find the area of each triangle. Use A = ½ × b × h.

    1. b = 12 cm h = 5 cm
    2. b = 8 m h = 9 m
    3. b = 20 cm h = 7 cm
    4. b = 6.4 m h = 5 m
    5. b = 15 mm h = 8 mm

  2. Right-Angled Triangles Fluency

    Find the area of each right-angled triangle. Use A = ½ × leg1 × leg2.

    1. 8 cm 6 cm
    2. 12 m 5 m
    3. 4 cm 3 cm
    4. 9 m 9 m
    5. 10 mm 7 mm

  3. Find the Missing Dimension Understanding

    Use A = ½bh to find the missing value.

    1. Area = 30 cm², base = 10 cm. Find the height.
    2. Area = 45 m², height = 9 m. Find the base.
    3. Area = 28 cm², base = 8 cm. Find the height.
    4. Area = 66 mm², height = 11 mm. Find the base.
    5. Area = 42 m², base = 12 m. Find the height.

  4. Triangles and Parallelograms Understanding

    1. A parallelogram has base 14 cm and height 6 cm. A triangle is drawn with the same base and height. What is the area of the triangle?
    2. A rectangle is 10 cm × 8 cm. A diagonal is drawn, making two triangles. What is the area of each triangle?
    3. A square has side 7 cm. One diagonal creates two triangles. What is the area of each triangle?
    4. Explain why the area of a triangle is always exactly half the area of a parallelogram with the same base and height.

  5. Mixed Units Understanding

    1. Base = 60 mm, height = 40 mm. Find area in cm².
    2. Base = 1.5 m, height = 80 cm. Find area in m².
    3. Base = 3 m, height = 250 cm. Find area in m².
    4. Base = 45 mm, height = 2 cm. Find area in mm².

  6. Problem Solving Problem Solving

    1. A triangular garden plot has base 8 m and perpendicular height 5 m. Find the area.
    2. A triangular sail has base 6 m and height 9 m. Find its area.
    3. A triangular piece of glass costs $15 per m². It has base 1.2 m and height 0.8 m. Find the cost.
    4. A right-angled triangular plot has legs 15 m and 20 m. Find its area in m². If fencing costs $8 per m of perimeter, and the hypotenuse is 25 m, find the total fencing cost.
    5. A triangular wall panel has area 7.5 m². Its height is 3 m. Find its base.

  7. More Calculate the Area Fluency

    Find the area of each triangle. Use A = ½ × b × h.

    1. b = 14 cm h = 9 cm
    2. b = 5 m h = 12 m
    3. b = 18 mm h = 7 mm
    4. b = 4 cm h = 4 cm
    5. b = 22 m h = 5 m

  8. Spot the Error Understanding

    Each calculation contains an error. Identify the mistake and give the correct answer.

    b = 10 cm h = 6 cm
    (a)
    8 cm 5 cm
    (b) right-angled
    b = 9 m h = 4 m
    (c)
    b = 7 mm h = 6 mm
    (d)
    1. Triangle with base 10 cm and height 6 cm. A student writes: A = 10 × 6 = 60 cm².
    2. Right-angled triangle with legs 5 cm and 8 cm. A student writes: A = 5 × 8 = 40 cm².
    3. Triangle with base 9 m and height 4 m. A student writes: A = ½ × 9 + 4 = 8.5 m².
    4. Triangle with base 7 mm and height 6 mm. A student writes: A = ½ × (7 + 6) = 6.5 mm².

  9. Comparing Triangle Areas Understanding

    b = 8 cm h = 6 Triangle 1 b = 12 cm h = 4 Triangle 2
    Part (a): which has the greater area?
    1. Triangle 1 has base 8 cm and height 6 cm. Triangle 2 has base 12 cm and height 4 cm. Which has the greater area?
    2. Two triangles both have area 30 cm². Triangle A has base 10 cm. Triangle B has height 6 cm. Find the height of A and the base of B.
    3. A triangle has base 8 m and height 5 m. If the base is doubled but the height stays the same, how does the area change?
    4. A triangle has base b and height h. Write an expression for the area if both the base and height are doubled. How many times larger is the new area compared to the original?

  10. Extended Problem Solving Problem Solving

    Δ A Δ B 30 m 20 m diagonal fence
    Part (c): rectangular field with diagonal fence
    1. A triangular park has base 120 m and height 80 m. Council wants to turf the entire park at a cost of $4.50 per m². Find the total cost.
    2. A triangular roof truss has a base of 8 m and rises to a height of 3.5 m. Find the cross-sectional area of the truss. If the building is 12 m long, find the volume of the triangular prism-shaped roof space (area × length).
    3. A rectangular field measures 30 m by 20 m. A diagonal fence is built dividing it into two triangles. Find the area of each triangular section. Show that they add up to the rectangle’s area.
    4. A triangle has perimeter 30 cm. Two of its sides are 12 cm and 10 cm. The height corresponding to the 8 cm base is 7.5 cm. Find the area. (Hint: first find the base.)
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