Area of a Triangle

Q: Why is the area of a triangle half the area of a rectangle?

A diagonal line across a rectangle splits it into two identical right-angled triangles. Each triangle covers exactly half the rectangle. Both shapes share the same base and height. So the area of the triangle equals ½ × base × height. This relationship holds for all triangles, not just right-angled ones.

Q: Does the area formula work for all triangles?

Yes. The formula area = ½ × base × height works for every triangle, including right-angled, isosceles, equilateral, and scalene triangles. As long as you use a base and its corresponding perpendicular height, the formula gives the correct area. The triangle does not need to be right-angled for the formula to apply.

Q: How do you choose the correct base and height of a triangle?

You can pick any side of the triangle as the base. The corresponding height is the perpendicular distance from that base to the opposite vertex. The height must form a right angle with the chosen base. If you pick a different side as the base, the corresponding height changes, but the calculated area stays the same.

Key concept

The area of a triangle = ½ × base × height. Multiply the base by the height, then divide by 2, so a base of 6 and height of 5 gives ½ × 6 × 5 = 15. The height must be perpendicular, meeting the base at a right angle.

Area of a Triangle - introduction visual

Video Lesson

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$Area of a Triangle poster$

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Flashcards

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Diagram explaining the area of triangles formula 1/2 x b x h, examples of perpendicular height, and notes emphasising any side as base.

Illustration of finding the area of a triangle with a base of 6 cm and height of 5 cm using the formula 1/2 x base x height, resulting in 15 cm².

The area question of a right-angled triangle with a base of 5 cm and height of 12 cm. Calculations demonstrate the area formula, yielding 30 cm².

How to calculate the height of a triangle using its area, with given dimensions and a worked-out formula solution.

What is the Area of a Triangle

The area of a triangle is given by: $Area = \frac{1}{2} \times$ base × height
The height must be perpendicular to the base.

Practice Questions

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Progress1 / 6

Q1Easy

The base of a triangle is $10 cm$ , and the corresponding height is $5 cm$ . What is the area?

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Interactive Activity

Drag the top point to change the shape, or use sliders below

Base (b)

15 cm

Height (h)

10 cm

Calculation

A=0.5×15cm×10cm

=75cm²

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Students Also Ask

The questions students bump into most on this topic

What is the formula for the area of a triangle?

The formula for the area of a triangle is area = ½ × base × height. The base is any side of the triangle you choose. The height is the perpendicular distance from that base to the opposite vertex. Multiply the base by the height, then halve the result to get the area in squared units.

Why is the area of a triangle half the area of a rectangle?

A diagonal line across a rectangle splits it into two identical right-angled triangles. Each triangle covers exactly half the rectangle. Both shapes share the same base and height. So the area of the triangle equals ½ × base × height. This relationship holds for all triangles, not just right-angled ones.

Does the area formula work for all triangles?

Yes. The formula area = ½ × base × height works for every triangle, including right-angled, isosceles, equilateral, and scalene triangles. As long as you use a base and its corresponding perpendicular height, the formula gives the correct area. The triangle does not need to be right-angled for the formula to apply.

Can the height of a triangle be outside the triangle?

Yes. Sometimes the opposite vertex sits beyond the base line. In that case, the perpendicular height falls outside the triangle. This happens in obtuse triangles where one angle is greater than 90°. The height still forms a right angle with the base, and the formula area = ½ × base × height still applies.

How do you choose the correct base and height of a triangle?

You can pick any side of the triangle as the base. The corresponding height is the perpendicular distance from that base to the opposite vertex. The height must form a right angle with the chosen base. If you pick a different side as the base, the corresponding height changes, but the calculated area stays the same.

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